?? hmmm, seems not at all clear how to transport "homotopy fiber" idea from dimensional doctrine to multi-dimensional ... ???? so then how is "adele" concept supposed to work ?? .... maybe should try just looking it up ... ?? ...
??how does "kernel" get along with "group algebra" ??? ....
Friday, June 24, 2011
from here :
??? any possibility that progression from abelian class field theory ("artin reciprocity") to non-abelian ("langlands reciprocity") might have to do with moving comma construction / homotopy fiber idea from dimensional doctrine to AG doctrine ???? ..... ???? .... ???? hmmm, possibility of generalized "ramification index" here ???? ......
?? possibility of .... ?? understanding stuff about archimedean ramification in terms of ... ?? extending of "differential calculus" / "blow-up" (???? ... ???relationship and/or non-relationship to homotopy fiber idea ... ??? ... ??? bit about ... ????_(cartier) divisor_ as already blown-up ... ???? .....) from AG to "AG without -1" doctrine ??? ..... ????......
??stuff that todd was trying to tell me about .... trying to unify archimedean with non-archimidean factors of zeta function .... ??? .... ??? "gaussian as self-dual under fourier transform" ... ????? ..... (?? relationship to "poisson summation" ??? ..... ????? ......) ..... "tate's thesis" ... ???...
[end quote]
??? "adeles" as (??? limiting case of ... ???) some decategorification of such homotopy fiber of AG theories ??? ..... ????? .....
??relationship to "automorphic representation" ???? .... ???
galois representation ...
??? jugendtraum as giving equivalence between certain maximal abelian extension and certain "field of moduli", roughly ... ??? .... ?? not at all clear any nice way to interpret the two sides of this equivalence as two sides ("galois" and "automorphic") of langlands .... ???? ..... ??? maybe both more on galois side ??? ....
?? artin reciprocity as "better" than jugendtraum ?? ... ??? or something (??) as "better" than langlands reciprocity ??? ....
?? taking seriously AG theory of "j-adeles" for j "level of ramification" ... ????.... and its decategorification of some sort ?? ....
?? "reciprocity" ... ?? between elliptic variable and modular variable ?? ... ???....
??? any possibility that progression from abelian class field theory ("artin reciprocity") to non-abelian ("langlands reciprocity") might have to do with moving comma construction / homotopy fiber idea from dimensional doctrine to AG doctrine ???? ..... ???? .... ???? hmmm, possibility of generalized "ramification index" here ???? ......
?? possibility of .... ?? understanding stuff about archimedean ramification in terms of ... ?? extending of "differential calculus" / "blow-up" (???? ... ???relationship and/or non-relationship to homotopy fiber idea ... ??? ... ??? bit about ... ????_(cartier) divisor_ as already blown-up ... ???? .....) from AG to "AG without -1" doctrine ??? ..... ????......
??stuff that todd was trying to tell me about .... trying to unify archimedean with non-archimidean factors of zeta function .... ??? .... ??? "gaussian as self-dual under fourier transform" ... ????? ..... (?? relationship to "poisson summation" ??? ..... ????? ......) ..... "tate's thesis" ... ???...
[end quote]
??? "adeles" as (??? limiting case of ... ???) some decategorification of such homotopy fiber of AG theories ??? ..... ????? .....
??relationship to "automorphic representation" ???? .... ???
galois representation ...
??? jugendtraum as giving equivalence between certain maximal abelian extension and certain "field of moduli", roughly ... ??? .... ?? not at all clear any nice way to interpret the two sides of this equivalence as two sides ("galois" and "automorphic") of langlands .... ???? ..... ??? maybe both more on galois side ??? ....
?? artin reciprocity as "better" than jugendtraum ?? ... ??? or something (??) as "better" than langlands reciprocity ??? ....
?? taking seriously AG theory of "j-adeles" for j "level of ramification" ... ????.... and its decategorification of some sort ?? ....
?? "reciprocity" ... ?? between elliptic variable and modular variable ?? ... ???....
Thursday, June 23, 2011
?? so how did that stuff go about ... ??? various sorts of "zeta function" ... ??? only partially overlapping in meaning ... ???? .... ???? ....
??relationship to various sorts of "l-function" ?????? ..... ?????? ...
?? people's names ... ???? ....
?? hasse-weil ... dedekind .... riemann ....
artin ... hecke .... ?? "grossencharacter" ... .... ??? ....
??? "automorphic l-function" .... ????? .......
???? ... "motivic" .... ????? .....
??relationship to various sorts of "l-function" ?????? ..... ?????? ...
?? people's names ... ???? ....
?? hasse-weil ... dedekind .... riemann ....
artin ... hecke .... ?? "grossencharacter" ... .... ??? ....
??? "automorphic l-function" .... ????? .......
???? ... "motivic" .... ????? .....
?? maybe i'll take a stab at trying to re-invent langlands reciprocity (??? ...) here ....
?? so... ?? maybe we're supposed to start with a "galois representation" of some sort (?? probably really a functor of some kind ....) ... ??? and then get from this an "automorphic representation", whatever that is .... ??? ...
??? but let's try fleshing it out a bit ... ???....
?? "galois representation" meaning something like representation of "absolute galois group" of certain "base field" k .... ??? really some sort of functor from some sort of commutative k-algebras to some sort of vector spaces .... ????say over some field
(???) k' ... ???? .....
??now what does "automorphic representation" mean here ???? .....
?? well, first let me try picking some plausible guesses as to what k and k' and so forth might be in some simple but maybe not too simple example ....
k = imaginary quadratic field ....
k' = p-adics ?? ....
galois representation = ... ???? torsion points ... ???.... on corresponding (...) elliptic curve ... ?? ....
???? and then .... "automorphic representation" being representation (?????) of _something_ "over k" ???? ..... ????
??any idea how to get "l-function" of galois rep and/or of automorphic rep here ??? ....
??? "adeles" .... ????? of k ????? .... ????
???trying to get "abelian variety with generalized complex multiplication" (??? ...) from ... "number-flavored dimensional theory" .... ????? .....
?? try making table of galois reps ....
p-torsion of gl(1) ... ???? ??? gl(1)'s involved here ??? ...
p-torsion of elliptic curve with complex multilication ... ???....
????? ....
?? so... ?? maybe we're supposed to start with a "galois representation" of some sort (?? probably really a functor of some kind ....) ... ??? and then get from this an "automorphic representation", whatever that is .... ??? ...
??? but let's try fleshing it out a bit ... ???....
?? "galois representation" meaning something like representation of "absolute galois group" of certain "base field" k .... ??? really some sort of functor from some sort of commutative k-algebras to some sort of vector spaces .... ????say over some field
(???) k' ... ???? .....
??now what does "automorphic representation" mean here ???? .....
?? well, first let me try picking some plausible guesses as to what k and k' and so forth might be in some simple but maybe not too simple example ....
k = imaginary quadratic field ....
k' = p-adics ?? ....
galois representation = ... ???? torsion points ... ???.... on corresponding (...) elliptic curve ... ?? ....
???? and then .... "automorphic representation" being representation (?????) of _something_ "over k" ???? ..... ????
??any idea how to get "l-function" of galois rep and/or of automorphic rep here ??? ....
??? "adeles" .... ????? of k ????? .... ????
???trying to get "abelian variety with generalized complex multiplication" (??? ...) from ... "number-flavored dimensional theory" .... ????? .....
?? try making table of galois reps ....
p-torsion of gl(1) ... ???? ??? gl(1)'s involved here ??? ...
p-torsion of elliptic curve with complex multilication ... ???....
????? ....
?? so what about conceptual interpretation of stable 2-group obtained from dimensional category ???? ....
??? but which stable 2-group do i mean, and what about relationships among them, such as are they all the same ???? .....
"picard ..." .... ???
"jacobian ..." ... ??? ...
?? in toric case, "the torus" .... ????? ....
????..... left adjoint vs right adjoint .... smart vs stupid vs sesqui-clever ....
????? .....
dimensional analysis .... ???? .....
??? but which stable 2-group do i mean, and what about relationships among them, such as are they all the same ???? .....
"picard ..." .... ???
"jacobian ..." ... ??? ...
?? in toric case, "the torus" .... ????? ....
????..... left adjoint vs right adjoint .... smart vs stupid vs sesqui-clever ....
????? .....
dimensional analysis .... ???? .....
?? hmm, so it just occurred to me to try to make a galois representation (over (??? ..) Q ...) from p-torsion points on gl(1) .... ???? then occurred to me that this might be one of those ideas that i've seen ("without seeing" ...) discussed a bunch of times before .... in connection with "artin reciprocity as abelian special case of langlands program" .... ??and then it seems like a bunch of stuff is threatening to make sense when i pursue this idea ...
?? for one thing, i just mentioned the other day certain "coincidence" ... 2 apparently somewhat different occurrences of "gl(1)" in cyclotomy phenomenon ... and now i'm suddenly realizing that that seems to fit with something that used to bug me about trying to use artin reciprocity as alleged "abelian special case" springboard towards understanding langlands reciprocity ... namely that the alleged "fourier dual interpretation of artin reciprocity" (at least in the cyclotomy special case, which i may have gotten confused about being presented as the _only_ case ... ????...) sounds confusingly similar-but-different to the "forwards" interpretation ... because of occurrence of gl(1) in both forwards and dual interpretations, in way that seems hard to disentangle to what extent it's a coincidence .... ?? whereas now maybe i'm beginning to see more clearly how it really is just a coincidence in a certain way ...
anyway it now seems like it should be really interesting to try to understand artin-reciprocity-as-abelian-special-case-of-langlands-reciprocity in cyclotomic special case ... but not confusing it for the whole of the abelian case; instead trying to look at it alongside perhaps even more interesting jugendtraum / imaginary quadratic base (?? and more general taniyama-shimura development ...) cases ...
(?? vague memories ... ??? maybe from b f wyman article about some toe-in-non-abelian-water "solvable reciprocity law" due to shimura ?? ... ???? .... but then maybe tying in with ... stuff baez told me about ... binary icosahedral groups showing up in langlands program in certain context .... ??? ....finite group with 2d rep .... ???? ..... ??something about "icosahdral case" in particular .... ?????? ..... not sure this tying-in attempt makes any sense ... ???
?? "solvable reciprocity law" always suggested to me mundane approach involving "successive abelian extensions" (of course! ... considering how galois allegedly invented solvability in first place ....), though here i was trying to vaguely imagine some different interpretation .... ??? .... "elliptic curve not quite having complex multiplication, but maybe coming close to it in some sense ..." .... ???....)
?? whne todd and i were talking this morning ... stumbling onto idea of galois representation associated with p-torsion points ... ??? realizing now that ... how we stumbled onto it involved functor f_x of "extraneous" variable k varying over number fields (?? but then probably interestingly over much larger category ... letting k be a finite field for example ... ??? ... maybe just arbitrary commutative ring ...).... assigning to k vector space (...) of p-torsion points in given abelian variety x over k .... ?? relaizing now how non-extraneous this variable k and its extended variability (not restricting k to be algebraically closed ...) really seems to be ... "galois representation" concept as misconceived version of functor defined on non-algebraicially closed things as well ... ??? ??? relationship to "galois stack" idea ????? ..... ????? ..... ?? ....
???so what in the world _is_ an "automorphic representation" ??? ... and why can't anyone give a straight answer to the question .... ???? .....
??? cyclotomic reciprocity as seeming to not really fit in as part of taniyama-shimura development of kronecker's jugendtraum, despite having in significant part inspired the jugendtraum, but then does very much fit in as part of langlands program ??? .... ??? so idea of langlands program as some sort of development of jugendtraum maybe makes good sense ??? .... (??maybe "better" development than shimura-taniyama development (temporarily assuming that this latter development really is limited in certain way ... ???? ....) ... ???) ??? ... non-abelian generalization of jugendtraum ... ??? vs non-abelian generalization of artin reciprocity ???? ..... ??artin reciprocity as in some ways more "general" than jugendtraum, but jugendtraum as "better" in some ways ??? .... ??? maybe in particular in way of more clearly hinting at non-abelian generalization ??? .... ???simply (?? ...) switching from abelian variety with complex multiplication to one without ??? .... ????? ...... ???? switching from considering _endomorphisms_ of abelian varieties to considering _non_-endo-morphisms ??? ..... ?? hecke operators .... ????.... ??so what _about_ how hecke operators manifest in special case of abelian variety with complex multiplication ???? ..... ??? as something to be generalized .... ???? .....
??? p1-torsion points mod p2 ... ???? .... ????.....
??? so is "shimura variety" going to end up having to do with "homotopy fiber of dimensional functor" ?????? ...... ????? ....
?? for one thing, i just mentioned the other day certain "coincidence" ... 2 apparently somewhat different occurrences of "gl(1)" in cyclotomy phenomenon ... and now i'm suddenly realizing that that seems to fit with something that used to bug me about trying to use artin reciprocity as alleged "abelian special case" springboard towards understanding langlands reciprocity ... namely that the alleged "fourier dual interpretation of artin reciprocity" (at least in the cyclotomy special case, which i may have gotten confused about being presented as the _only_ case ... ????...) sounds confusingly similar-but-different to the "forwards" interpretation ... because of occurrence of gl(1) in both forwards and dual interpretations, in way that seems hard to disentangle to what extent it's a coincidence .... ?? whereas now maybe i'm beginning to see more clearly how it really is just a coincidence in a certain way ...
anyway it now seems like it should be really interesting to try to understand artin-reciprocity-as-abelian-special-case-of-langlands-reciprocity in cyclotomic special case ... but not confusing it for the whole of the abelian case; instead trying to look at it alongside perhaps even more interesting jugendtraum / imaginary quadratic base (?? and more general taniyama-shimura development ...) cases ...
(?? vague memories ... ??? maybe from b f wyman article about some toe-in-non-abelian-water "solvable reciprocity law" due to shimura ?? ... ???? .... but then maybe tying in with ... stuff baez told me about ... binary icosahedral groups showing up in langlands program in certain context .... ??? ....finite group with 2d rep .... ???? ..... ??something about "icosahdral case" in particular .... ?????? ..... not sure this tying-in attempt makes any sense ... ???
?? "solvable reciprocity law" always suggested to me mundane approach involving "successive abelian extensions" (of course! ... considering how galois allegedly invented solvability in first place ....), though here i was trying to vaguely imagine some different interpretation .... ??? .... "elliptic curve not quite having complex multiplication, but maybe coming close to it in some sense ..." .... ???....)
?? whne todd and i were talking this morning ... stumbling onto idea of galois representation associated with p-torsion points ... ??? realizing now that ... how we stumbled onto it involved functor f_x of "extraneous" variable k varying over number fields (?? but then probably interestingly over much larger category ... letting k be a finite field for example ... ??? ... maybe just arbitrary commutative ring ...).... assigning to k vector space (...) of p-torsion points in given abelian variety x over k .... ?? relaizing now how non-extraneous this variable k and its extended variability (not restricting k to be algebraically closed ...) really seems to be ... "galois representation" concept as misconceived version of functor defined on non-algebraicially closed things as well ... ??? ??? relationship to "galois stack" idea ????? ..... ????? ..... ?? ....
???so what in the world _is_ an "automorphic representation" ??? ... and why can't anyone give a straight answer to the question .... ???? .....
??? cyclotomic reciprocity as seeming to not really fit in as part of taniyama-shimura development of kronecker's jugendtraum, despite having in significant part inspired the jugendtraum, but then does very much fit in as part of langlands program ??? .... ??? so idea of langlands program as some sort of development of jugendtraum maybe makes good sense ??? .... (??maybe "better" development than shimura-taniyama development (temporarily assuming that this latter development really is limited in certain way ... ???? ....) ... ???) ??? ... non-abelian generalization of jugendtraum ... ??? vs non-abelian generalization of artin reciprocity ???? ..... ??artin reciprocity as in some ways more "general" than jugendtraum, but jugendtraum as "better" in some ways ??? .... ??? maybe in particular in way of more clearly hinting at non-abelian generalization ??? .... ???simply (?? ...) switching from abelian variety with complex multiplication to one without ??? .... ????? ...... ???? switching from considering _endomorphisms_ of abelian varieties to considering _non_-endo-morphisms ??? ..... ?? hecke operators .... ????.... ??so what _about_ how hecke operators manifest in special case of abelian variety with complex multiplication ???? ..... ??? as something to be generalized .... ???? .....
??? p1-torsion points mod p2 ... ???? .... ????.....
??? so is "shimura variety" going to end up having to do with "homotopy fiber of dimensional functor" ?????? ...... ????? ....
?? since we seem to be dealing with abelian extensions of imaginary quadratic number fields rather than of Q ... well, i guess it's not as though we never had any idea about that before .... but nevertheless ... ??? ideas associated with this .... ??? .... ?? instead of just single zeta function or single l-function giving information only about splitting behavior over Z-prime, getting complex of them giving finer information, about splitting behavior over primes of higher base.... ???? ......
??zeta function vs l-functions .... p-torsion galois representation .... ??extent to which get full information about whole maximal abelian extension .... ????
???isogenous elliptic curves .... p-torsion galois representations for each .... ?? putting together information coming from all of them .... ???? ......
??? level slips here .... base vs total .... ???? ..... ??? ... ??fiber ... ???? ....
???solutions over number fields vs solutions over finite fields ... ?? interplay ... ??? ... for varieties of various dimensions .... ???? ...... ?? geometric dimensions vs arithmetic dimension .... ???? ....
??zeta function vs l-functions .... p-torsion galois representation .... ??extent to which get full information about whole maximal abelian extension .... ????
???isogenous elliptic curves .... p-torsion galois representations for each .... ?? putting together information coming from all of them .... ???? ......
??? level slips here .... base vs total .... ???? ..... ??? ... ??fiber ... ???? ....
???solutions over number fields vs solutions over finite fields ... ?? interplay ... ??? ... for varieties of various dimensions .... ???? ...... ?? geometric dimensions vs arithmetic dimension .... ???? ....
?? todd mentions "splitting the difference" in connection with "fourier analysis"... ?? this idea of "splitting the difference" as sort of resolution of "twin paradox" in special relativity ..... ???? also "arrow of time paradox" in thermodynamics (??? ....) .... ???? forwards vs backwards transition probabilities .... ???? ..... ?? relationships ?? .....
??also talks about gaussian "carried to itself by fourier transform" .... ??? but mustn't this sort of self-duality (?? ...) be riding a higher level of self-duality ??? .....
??also talks about gaussian "carried to itself by fourier transform" .... ??? but mustn't this sort of self-duality (?? ...) be riding a higher level of self-duality ??? .....
?? hmmm... used to joke sometimes about ... ?? when modular curve x turns out to be elliptic, could look at point on that curve corresponding to x itself ..... ???? ..... well, regardless of whether anything like that ever turns up, might be interesting to at least try looking at torsion points on x-as-elliptic-curve and relate their ionterpretation as [ ?? ... torsion points of elliptic curve ... ??? relating to decorated ideals in corresponding imaginary quadratic number field ... ??? which i guess means that i'm suddenly assuming that x-as-elliptic-curve has complex multiplication .... ???? ....] to their interpretation as [decorated elliptic curves and/or lattices ... coming from x-as-modular-curve ... ] ... ??? ...
??? special point of terminal (??? ...) modular curve as ideal in imaginary quadratic number fields .... ?? sort of ... better, invertible module ??? .....
??? special point of non-terminal modular curve as such invertible module, but with extra decoration ... ?????? ...... hmmmmmmm ..... ????did we already know/understand about how this ties in with bit about "artin reciprocity" and "ramification index" and "homotopy fiber of dimensional functor" ????? ..... ??? and double-meaning of "congruence subgroup" ???????? ...... ?????? .......
??? special point of elliptic-curve-with-complex-multiplication as .... ??? embodiment (???) of sort of extra decoration mentioned above .... ??????? ..... ?????? ......
??? almost sounds like we're trying to suggest .... ???? local section of tautological bundle of elliptic curves over walking elliptic curve... analytically continuing to multi-valued section whose natural domain of definition is "hobbling elliptic curve" ( = non-terminal modular curve ...) .... ???? ..... ?? confusion between "analytically continuing local section to twisted global section vs to multi-valued "global" section" ???? ..... ????? ..... ??? relationship to "cohomology" ??? .....
??? special point of terminal (??? ...) modular curve as ideal in imaginary quadratic number fields .... ?? sort of ... better, invertible module ??? .....
??? special point of non-terminal modular curve as such invertible module, but with extra decoration ... ?????? ...... hmmmmmmm ..... ????did we already know/understand about how this ties in with bit about "artin reciprocity" and "ramification index" and "homotopy fiber of dimensional functor" ????? ..... ??? and double-meaning of "congruence subgroup" ???????? ...... ?????? .......
??? special point of elliptic-curve-with-complex-multiplication as .... ??? embodiment (???) of sort of extra decoration mentioned above .... ??????? ..... ?????? ......
??? almost sounds like we're trying to suggest .... ???? local section of tautological bundle of elliptic curves over walking elliptic curve... analytically continuing to multi-valued section whose natural domain of definition is "hobbling elliptic curve" ( = non-terminal modular curve ...) .... ???? ..... ?? confusion between "analytically continuing local section to twisted global section vs to multi-valued "global" section" ???? ..... ????? ..... ??? relationship to "cohomology" ??? .....
?? generalized toric structures on a _discrete_ algebraic variety ???? ....
?? generalized toric embeddings of such into ordinary toric varieties ??? ...
???for example into torus itself ... ??? ...
?? walking idempotent .... ?????....
?? "fourier duality" for semi-lattices .... ??? dual bialgebras ???? ..... ??? and /or for commutative monoids, marrying semi-lattice case with abelian group case ... ??? .....
?? generalized toric embeddings of such into ordinary toric varieties ??? ...
???for example into torus itself ... ??? ...
?? walking idempotent .... ?????....
?? "fourier duality" for semi-lattices .... ??? dual bialgebras ???? ..... ??? and /or for commutative monoids, marrying semi-lattice case with abelian group case ... ??? .....
?? so at the moment (after discussion with todd this morning ...) the idea seems to be something like ... ?? for a nice (?? in sense described by shimura, maybe??) abelian-variety-with-complex-multiplication, the galois representation (??wrt the absolute galois group of the associated number field, that is ... ??? ... ?? rather than of the absolute galois group of Q, for example ...) that you get from p-torsion points of the variety breaks up into 1d representations of that galois gp ... meaning that it's really just a rep of the abelianized galois group ... ???? and kronecker's jugendtraum (???as generalized to some extent by taniyama and shimura, for example ?? ... ?? and intermingled with artin reciprocity ... ?? ...) can be interpreted as giving some sort of nice explicit description of those 1d reps .... ????? ....
(meanwhile todd and i are struggling to get even the most basic calculations along these lines to work out .... lemniscate inflection points ... ?? ....)
??? and then maybe the langlands program will have a lot to do with what happens in the case of an abelian-variety-without-complex-multiplication ... presumably now the 2d rep is typically irreducible .... ????? ....
??? and maybe the modularity theorem as specialized to the complex-multiplication case will have to do with relationship between "elliptic" (?? ... evaluating elliptic functions at torsion points ...) and "modular" (?? ... evaluating modular functions at ideals in imaginary quadratic number fields .... "turning ideal numbers into actual numbers" ....) versions of jugendtraum .... ??? .... ??????? ...... ?? but then will somehow also be very interesting in without-complex-multiplication case .... ???? .....
?? seems promising to try to understand stuff about ... ??? hecke operators acting on modular forms .... and so forth ...
??but also ... ??? i want to try again with my semi-ancient homegrown attempt to "directly use hecke operators associated to geometry of finite galois group to construct higher-dim galois rep" .... ???? ..... ??? 3! as galois group ??? ..... ???hmm, but does langlands program / conjectures make _any_ sort of claim about _this_ kind of galois rep ??? ....
???possibility of relationship to issue of "galois rep" as misconceived version of some sort of functor defined not only on algebraically closed fields but on some more general class of fields and / or rings ... ??? .... (?? see further discussion in later posts ... ???? .....)
(meanwhile todd and i are struggling to get even the most basic calculations along these lines to work out .... lemniscate inflection points ... ?? ....)
??? and then maybe the langlands program will have a lot to do with what happens in the case of an abelian-variety-without-complex-multiplication ... presumably now the 2d rep is typically irreducible .... ????? ....
??? and maybe the modularity theorem as specialized to the complex-multiplication case will have to do with relationship between "elliptic" (?? ... evaluating elliptic functions at torsion points ...) and "modular" (?? ... evaluating modular functions at ideals in imaginary quadratic number fields .... "turning ideal numbers into actual numbers" ....) versions of jugendtraum .... ??? .... ??????? ...... ?? but then will somehow also be very interesting in without-complex-multiplication case .... ???? .....
?? seems promising to try to understand stuff about ... ??? hecke operators acting on modular forms .... and so forth ...
??but also ... ??? i want to try again with my semi-ancient homegrown attempt to "directly use hecke operators associated to geometry of finite galois group to construct higher-dim galois rep" .... ???? ..... ??? 3! as galois group ??? ..... ???hmm, but does langlands program / conjectures make _any_ sort of claim about _this_ kind of galois rep ??? ....
???possibility of relationship to issue of "galois rep" as misconceived version of some sort of functor defined not only on algebraically closed fields but on some more general class of fields and / or rings ... ??? .... (?? see further discussion in later posts ... ???? .....)
Wednesday, June 22, 2011
y = f(x)^(1/2)
y' = (1/2)*f(x)^(-1/2)*f'(x)
y'' = (1/2)*(((-1/2)*f(x)^(-3/2)*f'(x)^2)+(f(x)^(-1/2)*f''(x)))
(1/2)*f(x)^(-3/2)*f'(x)^2 = f(x)^(-1/2)*f''(x)
(1/2)*f(x)^(-1)*f'(x)^2 = f''(x)
f(x) = x^3-x
f'(x) = 3*x^2-1
f''(x) = 6*x
(3*x^2-1)^2/(2*x^3-2*x) = 6*x
9*x^4 - 6*x^2 + 1 = 12*x^4 - 12*x^2
6*x^2 + 1 = 3*x^4
y' = (1/2)*f(x)^(-1/2)*f'(x)
y'' = (1/2)*(((-1/2)*f(x)^(-3/2)*f'(x)^2)+(f(x)^(-1/2)*f''(x)))
(1/2)*f(x)^(-3/2)*f'(x)^2 = f(x)^(-1/2)*f''(x)
(1/2)*f(x)^(-1)*f'(x)^2 = f''(x)
f(x) = x^3-x
f'(x) = 3*x^2-1
f''(x) = 6*x
(3*x^2-1)^2/(2*x^3-2*x) = 6*x
9*x^4 - 6*x^2 + 1 = 12*x^4 - 12*x^2
6*x^2 + 1 = 3*x^4
?? was thinking about "moduli stack of elliptic curves" and various mental pictures of it .... ??? .... ?? got to thinking that it wasn't purely just matter of "a's classified up to b-equivalence vs b's classified up to a-equivalence" .... ????.... ?? rather ... ???? situation where certain actual space of a's gets re-intepreted as certain actual space of b's .... ???? .... ?? "gauge-fixing" and "in the presence of a c, a's and b's are equivalent" .... ????for example in the presence of a frame, just about anything is equivalent to just about anything else .... ?????? ......
??? specific example in mind here .... upper half plane as corresponding to .... ????what ???? .... well, points in the upper half plane, i guess, but how did they get involved here, exactly??? ..... they generate lattices extending the real integers .... ?????.... ??? but meanwhile i think that i have this other sort-of equivalent picture of the unit disk representing nice quadratic forms on something or other .... ????? ...... .... ?????...... ??so what _is_ going on here?? ... ???what is the space of lattices in C extending the real integers like ??? .... vs certain hopefully obvious usual quotient space .... ???? ......
presumably i've known the answers to some of theze questions before ... ???
well, actually there's enough similar questions that it's hard to remember which ones i actually knew the answers to ...
?? for example space of all lattices in the plane ....
?? trefoil complement .....
??? ..... ????? ......
??? various kinds of "normalization" / "gauge-fixing" ... ??? ..... projectivization .... real vs complex .... ????? .....
??? consider .... ??? most (?? ...) general sort of equivalence of stacks between pair of orbit stacks ... ??? whether in that (?? ...) generality it can be given some sort of conceptual interpretation .... ????? ..... ??? ...
??decoration on torsor ... ??? abelian case .... ?? "stripes" .... ????? ..... ??? "normal form for decoration" .... ???? .... ??? .....
??? specific example in mind here .... upper half plane as corresponding to .... ????what ???? .... well, points in the upper half plane, i guess, but how did they get involved here, exactly??? ..... they generate lattices extending the real integers .... ?????.... ??? but meanwhile i think that i have this other sort-of equivalent picture of the unit disk representing nice quadratic forms on something or other .... ????? ...... .... ?????...... ??so what _is_ going on here?? ... ???what is the space of lattices in C extending the real integers like ??? .... vs certain hopefully obvious usual quotient space .... ???? ......
presumably i've known the answers to some of theze questions before ... ???
well, actually there's enough similar questions that it's hard to remember which ones i actually knew the answers to ...
?? for example space of all lattices in the plane ....
?? trefoil complement .....
??? ..... ????? ......
??? various kinds of "normalization" / "gauge-fixing" ... ??? ..... projectivization .... real vs complex .... ????? .....
??? consider .... ??? most (?? ...) general sort of equivalence of stacks between pair of orbit stacks ... ??? whether in that (?? ...) generality it can be given some sort of conceptual interpretation .... ????? ..... ??? ...
??decoration on torsor ... ??? abelian case .... ?? "stripes" .... ????? ..... ??? "normal form for decoration" .... ???? .... ??? .....
?? .... so ... ?? trying to flesh out plan to ... ??? experiment (via mathematica, mainly ... ?? ...) with 3-torsion / inflection points of elliptic curves ... to begin with especially those with complex multiplication, trying to to tie in / together artin reciprocity and jugendtraum .... ???..... ???? continue with plan to work out method to explicitly find inflection points, and then evaluate canonical elliptic functions there, and try to test this against artin reciprocity predictions about nature of these values, in sense of how they transform under absolute galois group .... ?? prediction should work in some pretty simple uniform way for all (??) elliptic curves with complex multiplication ??? .....
( ?? "pretty uniform" i think, but hopefully not so trivial as to be disappointing ... ?? seems like it ought to go somewhat beyond (or at least beside ... ???), for example, just plain "quadratic reciprocity" ... ?? and things similar (?? ...) to that ...?? though of course (...??...) perhaps not beyond full artin reciprocity ... ??? ....)
??? but then also ... try to simultaneously proceed with other plan (again, mainly mathematica-based), involving using _modular_ functions (?? including some sort of hecke (???) modular function/form (???) specially relating to 3-torsion case .... ???? .... ) as "machine for turning ideal number into actual number" .... ???? .... ??? and try to get these two (?? ...) plans to mesh .... ???? .....
???field of moduli (??) of elliptic curve without complex multiplication, beside that of those without ... ????? ..... ??? "field of moduli" (????) of modular curve .... ???? ..... ???"special moduli" .... ???? up to "ramification limit/index" .... ???? any meaningfulness in "modular" context ???? ..... .... evaluating modular function at elliptic curve _without_ complex multiplication .... evaluating [elliptic function living on elliptic curve _without_ complex multiplication] at torsion points .... ???? .... ??? ... relationships .... ???? ....
( ?? "pretty uniform" i think, but hopefully not so trivial as to be disappointing ... ?? seems like it ought to go somewhat beyond (or at least beside ... ???), for example, just plain "quadratic reciprocity" ... ?? and things similar (?? ...) to that ...?? though of course (...??...) perhaps not beyond full artin reciprocity ... ??? ....)
??? but then also ... try to simultaneously proceed with other plan (again, mainly mathematica-based), involving using _modular_ functions (?? including some sort of hecke (???) modular function/form (???) specially relating to 3-torsion case .... ???? .... ) as "machine for turning ideal number into actual number" .... ???? .... ??? and try to get these two (?? ...) plans to mesh .... ???? .....
???field of moduli (??) of elliptic curve without complex multiplication, beside that of those without ... ????? ..... ??? "field of moduli" (????) of modular curve .... ???? ..... ???"special moduli" .... ???? up to "ramification limit/index" .... ???? any meaningfulness in "modular" context ???? ..... .... evaluating modular function at elliptic curve _without_ complex multiplication .... evaluating [elliptic function living on elliptic curve _without_ complex multiplication] at torsion points .... ???? .... ??? ... relationships .... ???? ....
?? relationship between modular curves and children's drawings as mediated by ... gauss, eisenstein, cusp trilogy .... ??? ....
???making it seem at first like ... ?? the relationship's a bit one way ... modular curve as special case of children's drawing rather than vice versa ... ??? but maybe it's not that far from being 2-way ?? .... ???? ..... ???? ....
??? bit about ... ???danger of believing in easy way to prove modularity theorem, for example?? ... ?? that grothendieck quote .... ????? ..... ????? ..... ?? where did i read that attempt at helpful explicit warning about how not to over-interpret .... ??? "congruence subgroup" vs ... ??? ?? some more general class of subgroup ??? .... ????? ...... ?? on the other hand, what about ... ??? nevertheless seeing what happens if you try to modularize some typical elliptic curve by drawing a child's drawing for/on it .... ??? .... ??????? .....
???relationship between alleged action of absolute galois group on (??? ...) children's drawings and "one person's decoration as another's graffiti" ??? .... ???? ......
???sl(2) as maybe quasi-projective in particular maybe sort of interesting way ??????? ..... ?????
?? ... heisenberg .... theta .... ???? .....
?? algebraic group (over Q ???) given by "multiplicative group of particular number field" .... ???question what number field/s it "splits" over ?? .... ??? .... ?? maybe obvious in some sense but need to understand .... ?????.....
???? "number-theory-flavored dimensional categories in somewhat general" .... ???? ...
??? various meanings of "field of moduli" ??? .... "elliptic" ... "abelian" ... "modular" ..... ????? ???_can_ "modular" concept of "field of moduli" (as mentioned in wpa on children's drawings ... ?? working via galois correspondence ??? .....) be interpreted in way that ties in with jugendtraum version involving generating (?? j-, for j some "ramification index" ??? ...)maximal abelian extensions of given number fields by special values of _modular_ functions rather than of _elliptic_ functions .... ???? ..... ( ??? even if these (...) end up being pretty directly more or less the same thing, via ... actually function of both "modular" variable and "elliptic" one .... ????....) ... ?? thus perhaps somewhat unifying two ideas about what "field of moduli" might mean in "modular" context .... ????......
??? analog of "p-torsion" in "modular" context ???? ..... ???relationship to ... stuff in brown's book ... "iwahori-hecke algebra" ....??? ..... ???? ....
??? confusion about ... ???some stuff here (...) getting bigger vs getting co-bigger ... modular curve or maybe discrete-ish subspace inside of it ... ??? .... hecke operator .... ???? .... "hecke modular form" ??? .... "hecke modular curve" ??? .... "correspondence" .... "torsion point" .... "on generic elliptic curve" ... elliptic variable vs modular variable .... ???? .....
???? "zeta/theta" and ... ??? structure type on a set given by value of categorified polynomial at that set .... .... ??? "coefficient-value duality" ....
... ???categorified hypergeometric function ??? ..... ???? "q-hypergeometric" ??? .... ???? .....
?? hopf ring structures on ring of polynomials in 2 variables ... corresponding to multiplying binomials "a+bx" according to rule f(x)=0 for certain quadratic polynomial f ... (??or maybe even ... ???binomials "ax+by" according to rule f(x,y)=0 for certain binary quadratic form f .... ??? does that make any sense ??????? .....) .... ??? seems like maybe we're close here to interpreting moduli space of elliptic curves as moduli space of some other kind of (??? maybe related????) algebraic group ... ??????? ... ???again, questions about "splitting" .... over various "base"s .... ??? ..... ???? ....
???? some stuff here ... or something ... reminding me of .... bit about .... pictures we had .... ???? "mass hyperboloids" in 2+1 special relativity .... ???? ??? discretized structure .... "discriminant of binary quadratic form" as ternary quadratic form which almost (?? "up to annoying factor of 2" ??? ...????? ...) acts like "universal" honorary binary quadratic form ...... ?????? ...... ???weird ideas that we had about this .... "conceptual circularity" ... "modularity theorem" ... "evaluating modular form at modular curve" .... ?????? not sure i said that last bit the best way ... ??? ...... .... ???conway .... ????? .... ??? that (?? ...) stuff about .... well, that stuff in conway's book ... that gunnarsen also talk about, i think .... ??? ....
???? light cone itself as degenerate hyperboloid ???? ..... ????? any relationship to archimedean prime ???? .... ??? .... ???? ....
?? trying to remember whether allegedly obvious way of relating ellipse to elliptic curve is essentially same as historical way ... ??? vaguely think that the answer turned out to be close to yes ... ??? was there an (??that ???) annoying factor of two in there ?? gauss's quadratic forms vs someone else's ??? .... ??interpretation in terms of slightly differing modular curves ????? ..... ??? "polarization" ??? .... ????? ......
??? using "gauss/eisenstein trade-off" to act on forms of _other_ discriminants ???? ..... coxeter presentation ... ??? .... or is it important to include cusp as third coxeter generator, or is it better treated as slightly different sort of presentation, or something ??? ....
??? bunch of ways of viewing double coset space / stack / groupoid as orbit space / stack /groupoid ... ... ??versus viewing as more just space / stack / groupoid .... ???? .... ?? perhaps in several yet other ways ???? .... ... ???? .... ?? as maybe interesting to consider here .... ???? ..... i mean, moduli stack of elliptic curves as a double coset stack .... and so forth ...... ???? .....
???hmmm, what _about_ "genericity classification" here .... ??? seems maybe somewhat straightforward .... ????? well, or is this a slightly different pattern than we see in coxeter geometry situations, for instance ?? .... ??? pun on "generic" ??? .... ??? lots of more generic double cosets ???? .... and few less generic ones ... ???? in addition to the generic ones being individually "bigger" ... ??? .... ???? (??any situations where this pun backfires (??) and there's a sort of "population inversion" ????? .... ???? .....)
?? in general how many ways to express _triple_ coset stack as orbit stack ??? .... ??? ...
??? are we sure that ... the things that we're talking about here ... give equivalent orbit spaces pretty much just when they give equivalent orbit stacks ??? .... ????? .....
???making it seem at first like ... ?? the relationship's a bit one way ... modular curve as special case of children's drawing rather than vice versa ... ??? but maybe it's not that far from being 2-way ?? .... ???? ..... ???? ....
??? bit about ... ???danger of believing in easy way to prove modularity theorem, for example?? ... ?? that grothendieck quote .... ????? ..... ????? ..... ?? where did i read that attempt at helpful explicit warning about how not to over-interpret .... ??? "congruence subgroup" vs ... ??? ?? some more general class of subgroup ??? .... ????? ...... ?? on the other hand, what about ... ??? nevertheless seeing what happens if you try to modularize some typical elliptic curve by drawing a child's drawing for/on it .... ??? .... ??????? .....
???relationship between alleged action of absolute galois group on (??? ...) children's drawings and "one person's decoration as another's graffiti" ??? .... ???? ......
???sl(2) as maybe quasi-projective in particular maybe sort of interesting way ??????? ..... ?????
?? ... heisenberg .... theta .... ???? .....
?? algebraic group (over Q ???) given by "multiplicative group of particular number field" .... ???question what number field/s it "splits" over ?? .... ??? .... ?? maybe obvious in some sense but need to understand .... ?????.....
???? "number-theory-flavored dimensional categories in somewhat general" .... ???? ...
??? various meanings of "field of moduli" ??? .... "elliptic" ... "abelian" ... "modular" ..... ????? ???_can_ "modular" concept of "field of moduli" (as mentioned in wpa on children's drawings ... ?? working via galois correspondence ??? .....) be interpreted in way that ties in with jugendtraum version involving generating (?? j-, for j some "ramification index" ??? ...)maximal abelian extensions of given number fields by special values of _modular_ functions rather than of _elliptic_ functions .... ???? ..... ( ??? even if these (...) end up being pretty directly more or less the same thing, via ... actually function of both "modular" variable and "elliptic" one .... ????....) ... ?? thus perhaps somewhat unifying two ideas about what "field of moduli" might mean in "modular" context .... ????......
??? analog of "p-torsion" in "modular" context ???? ..... ???relationship to ... stuff in brown's book ... "iwahori-hecke algebra" ....??? ..... ???? ....
??? confusion about ... ???some stuff here (...) getting bigger vs getting co-bigger ... modular curve or maybe discrete-ish subspace inside of it ... ??? .... hecke operator .... ???? .... "hecke modular form" ??? .... "hecke modular curve" ??? .... "correspondence" .... "torsion point" .... "on generic elliptic curve" ... elliptic variable vs modular variable .... ???? .....
???? "zeta/theta" and ... ??? structure type on a set given by value of categorified polynomial at that set .... .... ??? "coefficient-value duality" ....
... ???categorified hypergeometric function ??? ..... ???? "q-hypergeometric" ??? .... ???? .....
?? hopf ring structures on ring of polynomials in 2 variables ... corresponding to multiplying binomials "a+bx" according to rule f(x)=0 for certain quadratic polynomial f ... (??or maybe even ... ???binomials "ax+by" according to rule f(x,y)=0 for certain binary quadratic form f .... ??? does that make any sense ??????? .....) .... ??? seems like maybe we're close here to interpreting moduli space of elliptic curves as moduli space of some other kind of (??? maybe related????) algebraic group ... ??????? ... ???again, questions about "splitting" .... over various "base"s .... ??? ..... ???? ....
???? some stuff here ... or something ... reminding me of .... bit about .... pictures we had .... ???? "mass hyperboloids" in 2+1 special relativity .... ???? ??? discretized structure .... "discriminant of binary quadratic form" as ternary quadratic form which almost (?? "up to annoying factor of 2" ??? ...????? ...) acts like "universal" honorary binary quadratic form ...... ?????? ...... ???weird ideas that we had about this .... "conceptual circularity" ... "modularity theorem" ... "evaluating modular form at modular curve" .... ?????? not sure i said that last bit the best way ... ??? ...... .... ???conway .... ????? .... ??? that (?? ...) stuff about .... well, that stuff in conway's book ... that gunnarsen also talk about, i think .... ??? ....
???? light cone itself as degenerate hyperboloid ???? ..... ????? any relationship to archimedean prime ???? .... ??? .... ???? ....
?? trying to remember whether allegedly obvious way of relating ellipse to elliptic curve is essentially same as historical way ... ??? vaguely think that the answer turned out to be close to yes ... ??? was there an (??that ???) annoying factor of two in there ?? gauss's quadratic forms vs someone else's ??? .... ??interpretation in terms of slightly differing modular curves ????? ..... ??? "polarization" ??? .... ????? ......
??? using "gauss/eisenstein trade-off" to act on forms of _other_ discriminants ???? ..... coxeter presentation ... ??? .... or is it important to include cusp as third coxeter generator, or is it better treated as slightly different sort of presentation, or something ??? ....
??? bunch of ways of viewing double coset space / stack / groupoid as orbit space / stack /groupoid ... ... ??versus viewing as more just space / stack / groupoid .... ???? .... ?? perhaps in several yet other ways ???? .... ... ???? .... ?? as maybe interesting to consider here .... ???? ..... i mean, moduli stack of elliptic curves as a double coset stack .... and so forth ...... ???? .....
???hmmm, what _about_ "genericity classification" here .... ??? seems maybe somewhat straightforward .... ????? well, or is this a slightly different pattern than we see in coxeter geometry situations, for instance ?? .... ??? pun on "generic" ??? .... ??? lots of more generic double cosets ???? .... and few less generic ones ... ???? in addition to the generic ones being individually "bigger" ... ??? .... ???? (??any situations where this pun backfires (??) and there's a sort of "population inversion" ????? .... ???? .....)
?? in general how many ways to express _triple_ coset stack as orbit stack ??? .... ??? ...
??? are we sure that ... the things that we're talking about here ... give equivalent orbit spaces pretty much just when they give equivalent orbit stacks ??? .... ????? .....
Tuesday, June 21, 2011
?? x^3 - x = x^2 .... x^2 - 1 = x ?? ...
?? "inflection point" ?? ...
??so ... ??try to understand inflection points in general here ??? ....
??second google hit on "eight inflection points on" is about "elliptic curves with isomorphic 3-torsion over Q" .... ?????? ......
??so ... ?? artin reciprocity tells us galois group of 3-maximal abelian extension of imaginary quadratic number field of discriminant d (?? ...) is ... ???? ....
??multiplicative group of imaginary quadratic number field f, mod n .... ????? ....
?? arbitrary ring r as algebraic ring given by functor taking commutative ring x to ring r tensor x ???? .... (limit-preservation properties of such functor, in general ??? .... ??? or in less general ?? ...???? ... ???? distributivity of cartesian product of affine schemes over .... ??? finite colimits of affine schemes ... = finite limits of commutative rings ... ???? .... ???? .... ???? .... ????? .....)
?? then giving rise to algebraic group by taking multiplicative group ... ???
?? then specializing to case x = Z/n ???? .....
?? commutativity of tensor product of commutative rings as maybe giving some sort of "reciprocity" here ??? ..... ???? .....
??what _is_ going on here ??? ... ??? any lawvere-theory morphism t -> t_[comm ring] (??how crucial is comm here ???) as giving nice algebraic ... ??? maybe level (??) slip ??? .... mult gp of any (comm ?) ring as nice alg group ... ??? ... more general ... ???? ..... ??? ... ... ??? ....
??actually maybe "flatness" issues here ??? ..... ????? .....
?? hmm, yes ... ??? ... and ring r need not be commutative .... ???? .....
??? multiplicative group of flat ring as nice ( = affine ??) algebraic group ... ???
?? in non-flat case ... ?? not so nice algebraic group ??? .... ??? maybe "stacky" and / or .... ???? ..... ??? maybe somewhat different versions depending on .... ????? ..... ????? ...... ??? "coarse vs fine" ??? .... ???? ..... ?? maybe "higher-affine" sometimes ... ??? ....
??well, so what about "multiplicative group of Z/n" as attempted algebraic group here ??? .... ??? with some sort of "correction" ... ??? ....
?? "inflection point" ?? ...
??so ... ??try to understand inflection points in general here ??? ....
??second google hit on "eight inflection points on" is about "elliptic curves with isomorphic 3-torsion over Q" .... ?????? ......
??so ... ?? artin reciprocity tells us galois group of 3-maximal abelian extension of imaginary quadratic number field of discriminant d (?? ...) is ... ???? ....
??multiplicative group of imaginary quadratic number field f, mod n .... ????? ....
?? arbitrary ring r as algebraic ring given by functor taking commutative ring x to ring r tensor x ???? .... (limit-preservation properties of such functor, in general ??? .... ??? or in less general ?? ...???? ... ???? distributivity of cartesian product of affine schemes over .... ??? finite colimits of affine schemes ... = finite limits of commutative rings ... ???? .... ???? .... ???? .... ????? .....)
?? then giving rise to algebraic group by taking multiplicative group ... ???
?? then specializing to case x = Z/n ???? .....
?? commutativity of tensor product of commutative rings as maybe giving some sort of "reciprocity" here ??? ..... ???? .....
??what _is_ going on here ??? ... ??? any lawvere-theory morphism t -> t_[comm ring] (??how crucial is comm here ???) as giving nice algebraic ... ??? maybe level (??) slip ??? .... mult gp of any (comm ?) ring as nice alg group ... ??? ... more general ... ???? ..... ??? ... ... ??? ....
??actually maybe "flatness" issues here ??? ..... ????? .....
?? hmm, yes ... ??? ... and ring r need not be commutative .... ???? .....
??? multiplicative group of flat ring as nice ( = affine ??) algebraic group ... ???
?? in non-flat case ... ?? not so nice algebraic group ??? .... ??? maybe "stacky" and / or .... ???? ..... ??? maybe somewhat different versions depending on .... ????? ..... ????? ...... ??? "coarse vs fine" ??? .... ???? ..... ?? maybe "higher-affine" sometimes ... ??? ....
??well, so what about "multiplicative group of Z/n" as attempted algebraic group here ??? .... ??? with some sort of "correction" ... ??? ....
?? relationship between "homotopy fiber of dimensional functor" as occurring in artin reciprocity] and [... ?? "galois stack" ??? ....] ???? ....
?? algebraic homotopy-fiber ... geometric homotopy-cofiber ... "geometric homotopy-colimit" .... ???? ....
??? syntactic / semantic viewpoints here ... ??? .....
modules ...
models ...
?? ....
?? algebraic homotopy-fiber ... geometric homotopy-cofiber ... "geometric homotopy-colimit" .... ???? ....
??? syntactic / semantic viewpoints here ... ??? .....
modules ...
models ...
?? ....
?? algebraic integers -> [algebraic integers]/p ... ????
cyclotomic integers -> [cyclotomic integers]/p
??? ....
??? ??? adjoin to Z all algebraic integers whose defining equations are solvable over f_q .... ???? .... ???maybe we thought about this a pretty long time ago ??? ....
???adjoin to Z all cyclotomic integers whose defining equations are solvable over f_3 .... ??? ....
??? equivariance wrt galois groups .... ???? .....
cyclotomic integers -> [cyclotomic integers]/p
??? ....
??? ??? adjoin to Z all algebraic integers whose defining equations are solvable over f_q .... ???? .... ???maybe we thought about this a pretty long time ago ??? ....
???adjoin to Z all cyclotomic integers whose defining equations are solvable over f_3 .... ??? ....
??? equivariance wrt galois groups .... ???? .....
Monday, June 20, 2011
?? various attempted analogies between various groups and "one-parameter groups" and so forth ... in connection with jugendtraum, in part ... ?? possibly with some level slips ... ????....
gl(1) and cyclotomy ... p-torsion points on gl(1) ... ???? as forming 1d vector space over f_p .... ??? thus something about gl(1,f_p) showing up, though ... ?? not sure to what extent those two occurrences of "gl(1)" are "coincidence" ... ???? ....
?? then ... elliptic curve with complex multiplication .... p-torsion points on it ... as forming 2d vector space over f_p ... ??so sort of looks like gl(2,f_p) is involved here??? ..... ??? but then maybe ... ???actually something about ... ????algebraic subgroup of gl(2) ??? ..... ???? given by "multiplicative group of imaginary quadratic numbner field..." .... ????? (a,b) * (c,d) = ... ???multiply like (a+bx)*(c+dx) with x satisfying irreducible quadratic .... ????? ....
.................. ??????? .......
???i was going to say, go back and re-consider that idea we mentioned ... something about elliptic curve over adeles .... ????? ......
?? but .... ??? not sure that's the right way to say it ... ????? ...... ???? ....
??? look at elliptic curve over f_p ... ???compare to ... mult gp of imaginary quadratic number ring over f_p .... ????? ...... ?????? ......
?? was also going to say something about ... automorphism group of elliptic curve ... ??? ...
?? ... one-parameter groups .... "circle group" vs "gl(1)" ..... ??which leads me to other idea that i was going to put in completely separate post .... ???? ..... "weil restriction" of gl(1) from C to R .... ?? relationship to stuff people say / write about "d z-bar" where d is some typographically funny version of d and z-bar is supposed to be complex conjugate .... ???? ..... ??? "hodge ... " .... ???? ....
gl(1) and cyclotomy ... p-torsion points on gl(1) ... ???? as forming 1d vector space over f_p .... ??? thus something about gl(1,f_p) showing up, though ... ?? not sure to what extent those two occurrences of "gl(1)" are "coincidence" ... ???? ....
?? then ... elliptic curve with complex multiplication .... p-torsion points on it ... as forming 2d vector space over f_p ... ??so sort of looks like gl(2,f_p) is involved here??? ..... ??? but then maybe ... ???actually something about ... ????algebraic subgroup of gl(2) ??? ..... ???? given by "multiplicative group of imaginary quadratic numbner field..." .... ????? (a,b) * (c,d) = ... ???multiply like (a+bx)*(c+dx) with x satisfying irreducible quadratic .... ????? ....
.................. ??????? .......
???i was going to say, go back and re-consider that idea we mentioned ... something about elliptic curve over adeles .... ????? ......
?? but .... ??? not sure that's the right way to say it ... ????? ...... ???? ....
??? look at elliptic curve over f_p ... ???compare to ... mult gp of imaginary quadratic number ring over f_p .... ????? ...... ?????? ......
?? was also going to say something about ... automorphism group of elliptic curve ... ??? ...
?? ... one-parameter groups .... "circle group" vs "gl(1)" ..... ??which leads me to other idea that i was going to put in completely separate post .... ???? ..... "weil restriction" of gl(1) from C to R .... ?? relationship to stuff people say / write about "d z-bar" where d is some typographically funny version of d and z-bar is supposed to be complex conjugate .... ???? ..... ??? "hodge ... " .... ???? ....
wikipedia search on "field of moduli" gives basically just this hit:
Dessin d'enfant
The two Belyi functions ƒ 1 and ƒ 2 of this example are defined over the field of moduli, but there exist dessins for which the field of ...
23 KB (3,621 words) - 02:40, 15 September 2010
?? which is definitely interesting .... although i admit that i'm not sure to what extent i actually remember what prompted me to search on this ...
Dessin d'enfant
The two Belyi functions ƒ 1 and ƒ 2 of this example are defined over the field of moduli, but there exist dessins for which the field of ...
23 KB (3,621 words) - 02:40, 15 September 2010
?? which is definitely interesting .... although i admit that i'm not sure to what extent i actually remember what prompted me to search on this ...
?? "weil restriction" ... ???... ?? does this ever take complex projective varieties to real affine ones ??? .....
?? ... confusion about this ... ???? ??? relationship of weil restriction to idea of associating to complex projective variety real affine variety of nice projection operators canonically associated to points of projective space ... ??? does this actually make any sense ??, is there actually such a canonical association ?? ... ??? ....
?? vs ... ???confusion between real affine and real projective varies which superficially look similar ... "circle" vs "projective line" ... ????... ??? understood this to some extent but seems hazy now ... ?? as does previous paragraph ?? .... ??? ....
?? ... confusion about this ... ???? ??? relationship of weil restriction to idea of associating to complex projective variety real affine variety of nice projection operators canonically associated to points of projective space ... ??? does this actually make any sense ??, is there actually such a canonical association ?? ... ??? ....
?? vs ... ???confusion between real affine and real projective varies which superficially look similar ... "circle" vs "projective line" ... ????... ??? understood this to some extent but seems hazy now ... ?? as does previous paragraph ?? .... ??? ....
?? in discussion with kenji yesterday (unfortunately windowa journall malfunctioned and the notes were lost) ... kenji pointed out (ess ...) that "halving" structure on 4-point set amounts to "square" structure .... as the corners (??or equivalently sides??) of a square ... ?? the two halves being the two opposite diagonals ....
?? consider tetrahedron as "join" (is that what it's called?) of two opposite edges ... mapping down to interval with fiber over midpoint being sqaure ... ??? .. holding the tetrahedron so that that fiber is square in your visual field, so to speak ... ?? then where/how do the corners lie in visual field ... observer at infinity ... ??? .... ??? hmmm, i guess that they're lying at the midpoints of the sides (of an enlarged version of the square ...) .... ??? so maybe better to think of the halves as the "opposite-side pairs" than as the "diagonals" ....
?? because you can think of the "opposite-side pairs" as the two "dimensions" ("height" and "width", say ...) of the square ... ??which is sort of what's going with the "tetrahedron as join of two mid-line axises of middle square" picture ... ?????....
??switching between "diagonals" and "mid-lines" pictures as corresponding to an outer automorphism here ???..... .... ???just curious; does that outer automorphism become inner in 4! ??? .... ?? perhaps not, since seems to be order 2 outer automorphism ... ???? ..... ?? so what _is_ the whole outer automorphism group ??? ... ??? getting confused with "g / center(g)" ... ???conceptual interpretation of which is _what_ ??? ...
?? dihedral group always has "side/corner duality" outer involution ??? .... ??actually confuses me a bit at moment ... semi-direct product ... ??? .... ???
??is this square maybe of some interest in connection with quartic formula ??? .....
??came as bit of a cryptomorphism to me ... ??? maybe look for similar such ??
?? maybe this involved me failing to sufficiently appreciate "one person's decoration as another's graffiti" ??? .... which i thought of mentioning to kenji and probably should have but didn't ...
??where for example is ... ??does it seem like the quaternion unit group ought to be represented faithfully on a 4-elt set ??? ..... ??or no, maybe it's sort of clear that it can't ... "downward normalization" of subgroups as identity process here since famously all subgroups are nprmal here ... ?? vaguely reminds me of something baez sometimes says about lie algebra e8 (i think ...) ... "can't be faithfully represented on anything smaller than itself" ... ??? ....
?? consider tetrahedron as "join" (is that what it's called?) of two opposite edges ... mapping down to interval with fiber over midpoint being sqaure ... ??? .. holding the tetrahedron so that that fiber is square in your visual field, so to speak ... ?? then where/how do the corners lie in visual field ... observer at infinity ... ??? .... ??? hmmm, i guess that they're lying at the midpoints of the sides (of an enlarged version of the square ...) .... ??? so maybe better to think of the halves as the "opposite-side pairs" than as the "diagonals" ....
?? because you can think of the "opposite-side pairs" as the two "dimensions" ("height" and "width", say ...) of the square ... ??which is sort of what's going with the "tetrahedron as join of two mid-line axises of middle square" picture ... ?????....
??switching between "diagonals" and "mid-lines" pictures as corresponding to an outer automorphism here ???..... .... ???just curious; does that outer automorphism become inner in 4! ??? .... ?? perhaps not, since seems to be order 2 outer automorphism ... ???? ..... ?? so what _is_ the whole outer automorphism group ??? ... ??? getting confused with "g / center(g)" ... ???conceptual interpretation of which is _what_ ??? ...
?? dihedral group always has "side/corner duality" outer involution ??? .... ??actually confuses me a bit at moment ... semi-direct product ... ??? .... ???
??is this square maybe of some interest in connection with quartic formula ??? .....
??came as bit of a cryptomorphism to me ... ??? maybe look for similar such ??
?? maybe this involved me failing to sufficiently appreciate "one person's decoration as another's graffiti" ??? .... which i thought of mentioning to kenji and probably should have but didn't ...
??where for example is ... ??does it seem like the quaternion unit group ought to be represented faithfully on a 4-elt set ??? ..... ??or no, maybe it's sort of clear that it can't ... "downward normalization" of subgroups as identity process here since famously all subgroups are nprmal here ... ?? vaguely reminds me of something baez sometimes says about lie algebra e8 (i think ...) ... "can't be faithfully represented on anything smaller than itself" ... ??? ....
??? from reading about some of this stuff .... ?? seems like situation where ... jugendtraum grew out of analogy of gl(1) with [elliptic curve with complex multiplication] ... ??? ... ??? but ... in pretty highly generalized jugendtraum, original example gl(1) maybe didn't really fit .....???? .....
Sunday, June 19, 2011
?? shimura suggests that "fields of moduli" of elliptic curves related via ... ????corresponding to different ideal classes of same imaginary quadratic number field ??? .... may be different class fields part of one big abelian extension ??? .... ??
?? well, or something vaguely like that ... i may have completely misread ... ??? maybe something about higher-dim abelian varieties ??? .....
?? well, or something vaguely like that ... i may have completely misread ... ??? maybe something about higher-dim abelian varieties ??? .....
?? relationship between "jugendtraum" and "elliptic modular function as machine for turning ideal numbers into actual numbers" ... ???? maybe as guide to relationship between "elliptic" stuff and "modular" stuff in general ??? .... ????? ....
?? _can_ jugendtraum as we're currently trying to think of it (meshing with artin reciprocity ... ???? ...) be construed as some version of "... machine for turning ideal numbers into actual numbers ..." ??? .... ????? .....
?? hmmm... well, inputs to modular functions ( / forms ??? ...) can reasonably be interpreted "as" ideals, it seems .... but what about inputs to elliptic functions ??? ... ?? especially torsion point inputs ... ??? ..... ???? "ideal" vs "ideal number" ???? ..... ????? ....
?? interpreting value of modular function at nice elliptic curve with complex multiplication as value of certain elliptic function at certain torsion point somehow .... ????? .....
?? _can_ jugendtraum as we're currently trying to think of it (meshing with artin reciprocity ... ???? ...) be construed as some version of "... machine for turning ideal numbers into actual numbers ..." ??? .... ????? .....
?? hmmm... well, inputs to modular functions ( / forms ??? ...) can reasonably be interpreted "as" ideals, it seems .... but what about inputs to elliptic functions ??? ... ?? especially torsion point inputs ... ??? ..... ???? "ideal" vs "ideal number" ???? ..... ????? ....
?? interpreting value of modular function at nice elliptic curve with complex multiplication as value of certain elliptic function at certain torsion point somehow .... ????? .....
?? from wpa on "weil restriction" ... ??seems interesting for various reasons ... ????...
Restriction of scalars over a finite extension of fields takes group schemes to group schemes. In particular, the torus
\mathbb{S} := \mathrm{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{G}_m
where Gm denotes the multiplicative group, plays a significant role in Hodge theory, since the Tannakian category of real Hodge structures is equivalent to the category of representations of S. The real points have a Lie group structure isomorphic to \mathbb{C}^\times. See Mumford–Tate group.
??? wpa on "hodge atructure" ...
One can define a notion of a morphism of mixed Hodge structures, which has to be compatible with the filtrations F and W and prove the following theorem.
Mixed Hodge structures form an abelian category. The kernels and cokernels in this category coincide with the usual kernels and cokernels in the category of vector spaces, with the induced filtrations.
Moreover, the category of (mixed) Hodge structures admits a good notion of tensor product, corresponding to the product of varieties, as well as related concepts of inner Hom and dual object, making it into a Tannakian category. By Tannaka-Krein philosophy, this category is equivalent to the category of finite-dimensional representations of a certain group, which Deligne has explicitly described.
??????
The machinery based on the notions of Hodge structure and mixed Hodge structure forms a part of still largely conjectural theory of motives envisaged by Alexander Grothendieck. Arithmetic information for nonsingular algebraic variety X, encoded by eigenvalue of Frobenius elements acting on its l-adic cohomology, has something in common with the Hodge structure arising from X considered as a complex algebraic variety. Sergei Gelfand and Yuri Manin remarked around 1988 in their Methods of homological algebra, that unlike Galois symmetries acting on other cohomology groups, the origin of "Hodge symmetries" is very mysterious, although formally, they are expressed through the action of the fairly uncomplicated group R_{\mathbf {C/R}}{\mathbf C}^* on the de Rham cohomology. Since then, the mystery has deepened with the discovery and mathematical formulation of mirror symmetry.
???????
??? ... back to "weil restriction" ... ???....
???? confusion here (just me, i mean ....) about ... ???... analogy to "total space of bundle" ... ???? .... ??preservation of cartesian products ??? .... ???? ....
?? some other analogy ... ??? .... ?? that we used to think about ??? ... ???? ....
From the standpoint of sheaves of sets, restriction of scalars is just a pushforward along the morphism Spec L \to Spec k and is right adjoint to fiber product, so the above definition can be rephrased in much more generality. In particular, one can replace the extension of fields by any morphism of ringed topoi, and the hypotheses on X can be weakened to e.g. stacks. This comes at the cost of having less control over the behavior of the restriction of scalars.
???????
Restriction of scalars over a finite extension of fields takes group schemes to group schemes. In particular, the torus
\mathbb{S} := \mathrm{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{G}_m
where Gm denotes the multiplicative group, plays a significant role in Hodge theory, since the Tannakian category of real Hodge structures is equivalent to the category of representations of S. The real points have a Lie group structure isomorphic to \mathbb{C}^\times. See Mumford–Tate group.
??? wpa on "hodge atructure" ...
One can define a notion of a morphism of mixed Hodge structures, which has to be compatible with the filtrations F and W and prove the following theorem.
Mixed Hodge structures form an abelian category. The kernels and cokernels in this category coincide with the usual kernels and cokernels in the category of vector spaces, with the induced filtrations.
Moreover, the category of (mixed) Hodge structures admits a good notion of tensor product, corresponding to the product of varieties, as well as related concepts of inner Hom and dual object, making it into a Tannakian category. By Tannaka-Krein philosophy, this category is equivalent to the category of finite-dimensional representations of a certain group, which Deligne has explicitly described.
??????
The machinery based on the notions of Hodge structure and mixed Hodge structure forms a part of still largely conjectural theory of motives envisaged by Alexander Grothendieck. Arithmetic information for nonsingular algebraic variety X, encoded by eigenvalue of Frobenius elements acting on its l-adic cohomology, has something in common with the Hodge structure arising from X considered as a complex algebraic variety. Sergei Gelfand and Yuri Manin remarked around 1988 in their Methods of homological algebra, that unlike Galois symmetries acting on other cohomology groups, the origin of "Hodge symmetries" is very mysterious, although formally, they are expressed through the action of the fairly uncomplicated group R_{\mathbf {C/R}}{\mathbf C}^* on the de Rham cohomology. Since then, the mystery has deepened with the discovery and mathematical formulation of mirror symmetry.
???????
??? ... back to "weil restriction" ... ???....
???? confusion here (just me, i mean ....) about ... ???... analogy to "total space of bundle" ... ???? .... ??preservation of cartesian products ??? .... ???? ....
?? some other analogy ... ??? .... ?? that we used to think about ??? ... ???? ....
From the standpoint of sheaves of sets, restriction of scalars is just a pushforward along the morphism Spec L \to Spec k and is right adjoint to fiber product, so the above definition can be rephrased in much more generality. In particular, one can replace the extension of fields by any morphism of ringed topoi, and the hypotheses on X can be weakened to e.g. stacks. This comes at the cost of having less control over the behavior of the restriction of scalars.
???????
?? hmmm, so ... shimura and taniyama in that preface say lots of interesting stuff, it seems ...
in particular... well, they're just suggesting to me stuff about ... zeta function of arbitrary elliptic curve, and how it relates to zeta functions of imaginary quadratic number fields ... ?? in particular in complex multplication case, but perhaps suggesting generalizations beyond that ... ???? .....
?? hmmm... this _does_ relate to stuff that we've thought about before... but ... it's really seeming like we have a much better view of what's really going on now ... ???? .... ???? .... from "combining artin reciprocity with jugendtraum" ... ?? it'd be interesting though if there's some other way of thinking about this stuff that we've secretly already known about ... ???? ....
?? elliptic curve with complex multiplication .... ???as (?? ...) y^2 = f(x) ... ??? analogous to imaginary quadratic number field y^2 = ?? .... ??? that it corresponds to ..... ????? ..... ???? .... ???? .....
in particular... well, they're just suggesting to me stuff about ... zeta function of arbitrary elliptic curve, and how it relates to zeta functions of imaginary quadratic number fields ... ?? in particular in complex multplication case, but perhaps suggesting generalizations beyond that ... ???? .....
?? hmmm... this _does_ relate to stuff that we've thought about before... but ... it's really seeming like we have a much better view of what's really going on now ... ???? .... ???? .... from "combining artin reciprocity with jugendtraum" ... ?? it'd be interesting though if there's some other way of thinking about this stuff that we've secretly already known about ... ???? ....
?? elliptic curve with complex multiplication .... ???as (?? ...) y^2 = f(x) ... ??? analogous to imaginary quadratic number field y^2 = ?? .... ??? that it corresponds to ..... ????? ..... ???? .... ???? .....
Saturday, June 18, 2011
?? so what _about_ whether "jugendtraum" might be generalized to cover full extent of artin reciprocity ?? ... ??? ?? seems like maybe it could be / is ??? ... ??? "abelian variety with complex multiplication" .... ????? ..... ?????......
?? "local ..." .... ???? ....
?? "shimura variety" ???? .....
?? "local ..." .... ???? ....
?? "shimura variety" ???? .....
??? so ... ??? we should pick a "ramification index" x over the gaussian integers ... ??? ...
(?? should x be invariant under the aut gp of the gaussian integers?? ...)
?? and then ... ???we should try to calculate according to artin reciprocity theorem what's the galois group of the x-maximal abelian extension of the gaussian integers, ... ?? and then we should try to connect that up with jugendtraum ideas, by identifying that extension as being generated by the values of certain lemniscatic functions at certain torsion points of the lemniscate ... and identifying how that galois group acts on those values .... ???? ......
(?? should x be invariant under the aut gp of the gaussian integers?? ...)
?? and then ... ???we should try to calculate according to artin reciprocity theorem what's the galois group of the x-maximal abelian extension of the gaussian integers, ... ?? and then we should try to connect that up with jugendtraum ideas, by identifying that extension as being generated by the values of certain lemniscatic functions at certain torsion points of the lemniscate ... and identifying how that galois group acts on those values .... ???? ......
?? let x be the category as follows:
p
^
|
s
||
vv
m
and consider x -> _presheaf(x)_ agreeing with yoneda embedding y on full subcategory containing s and m, but taking p to pushout of y(s -> p) along y("left" : s -> m) and taking s -> p to ... ??? certain presheaf morphism ... ???...
p = "point"
s = "set"
m = "map"
?? "walking point-displaced-by-map" ... ????.....
p
^
|
s
||
vv
m
and consider x -> _presheaf(x)_ agreeing with yoneda embedding y on full subcategory containing s and m, but taking p to pushout of y(s -> p) along y("left" : s -> m) and taking s -> p to ... ??? certain presheaf morphism ... ???...
p = "point"
s = "set"
m = "map"
?? "walking point-displaced-by-map" ... ????.....
Friday, June 17, 2011
?? trying to actually see, for example, where 4th roots of integers live in "jugendtraum" picture of elliptic function on lemniscate ... ???..... ?? or are we not guaranteed that they'll actually show up as specific values of the function, as opposed to as just some sort of weird algebraic combinations (?? "gauss sums" ???) of such values ???? .....
hmmm ... octagonal / "dihedral" orbit ... ????.....
hmmm ... octagonal / "dihedral" orbit ... ????.....
?? "homotopy fiber" .... of dimensional functor from dimensional category associated with "ramification index" to dimensional category of all fractional ideals .... ????....
???algebraic homotopy fiber .... geometric homotopy co-fiber .... ??? "anchored bundle" ... "anchoring" = trivialization (?? connection ... ??? .... macro vs micro ... ??? ...) over (?? generalized ...) basepoint ... ??? "line bundle with connection jet" .... ???? ...... ??? "trivialization jet" ... ???? .....
??? _why_ isomorphism classes of such thing should behave as _galois group_ .... ????? ...... ??? "frobenius" .... ????? ..... ???? ..... jugendtraum ... ??? .....
???duality between "ramification index" and "complement" thereof ??? .... ?? but problematicness of ... "multiplicity" in "negative" picture ???? ....
???dimensional algebra where dimensions are fractional ideals and quantities are ... ?? _some_ of their elements ??? .... ?????with .... perhaps some previously invertible elements no longer invertible because their inverses are now missing ??? ... ??? .... ??or maybe more likely (???) _both_ the element and its inverse are now missing ??? ..... ?????.....
???but ... ???this as suggesting (?? ...) that ... ?? dealing here with ... ??equipping quantities with not merely property, but structure .... ????.... level-bleeding .... ???? .... from object to morphism level ... ????....
???algebraic homotopy fiber .... geometric homotopy co-fiber .... ??? "anchored bundle" ... "anchoring" = trivialization (?? connection ... ??? .... macro vs micro ... ??? ...) over (?? generalized ...) basepoint ... ??? "line bundle with connection jet" .... ???? ...... ??? "trivialization jet" ... ???? .....
??? _why_ isomorphism classes of such thing should behave as _galois group_ .... ????? ...... ??? "frobenius" .... ????? ..... ???? ..... jugendtraum ... ??? .....
???duality between "ramification index" and "complement" thereof ??? .... ?? but problematicness of ... "multiplicity" in "negative" picture ???? ....
???dimensional algebra where dimensions are fractional ideals and quantities are ... ?? _some_ of their elements ??? .... ?????with .... perhaps some previously invertible elements no longer invertible because their inverses are now missing ??? ... ??? .... ??or maybe more likely (???) _both_ the element and its inverse are now missing ??? ..... ?????.....
???but ... ???this as suggesting (?? ...) that ... ?? dealing here with ... ??equipping quantities with not merely property, but structure .... ????.... level-bleeding .... ???? .... from object to morphism level ... ????....
?? "modified ideal class group" .... ?? "modified jacobian" .... ????
?? comma construction here ??? .... ??? in 2-category of dimensional categories ??? .... ??homotopy fiber construction ???? ...... ????
?? hmm, return here to issue of ... ?? how "abelian-like" and/or "abelian-unlike" 2-cat of dimensional categories may be .... ????.....
??? any possibility that progression from abelian class field theory ("artin reciprocity") to non-abelian ("langlands reciprocity") might have to do with moving comma construction / homotopy fiber idea from dimensional doctrine to AG doctrine ???? ..... ???? .... ???? hmmm, possibility of generalized "ramification index" here ???? ......
?? possibility of .... ?? understanding stuff about archimedean ramification in terms of ... ?? extending of "differential calculus" / "blow-up" (???? ... ???relationship and/or non-relationship to homotopy fiber idea ... ??? ... ??? bit about ... ????_(cartier) divisor_ as already blown-up ... ???? .....) from AG to "AG without -1" doctrine ??? ..... ????......
??stuff that todd was trying to tell me about .... trying to unify archimedean with non-archimedean factors of zeta function .... ??? .... ??? "gaussian as self-dual under fourier transform" ... ????? ..... (?? relationship to "poisson summation" ??? ..... ????? ......) ..... "tate's thesis" ... ???...
?? non-archimedean "gaussian" as ... ??? characteristic function of algebraic integers among algebraic numbers ???? .... ???? ....
?? alternation between which of "archimedean" / "non-archimedean" qualifies as "exotic" / "mundane" ... ???? ....
??? ?? "discrete vs continuous" ?? ... ??? ?? "quantum vs classical" ?????? .....
?? input vs output of function subject to fourier transform ... ??? .... ???? function vs measure .... ???? .... ???? ..... ???? ......... ??? ....
?? approaches to class field theory .... ????mentioned by lang (?? see book "algebraic number theory" ... ?? ... ?? "part 2 on class field theory ... about 5 pages of historical overview somewhere ... " ... ?? ...) ??? ...
1 ?? "cohomological" ...
2 ?? "zeta function" .... ??? "tannakian duality" ???? .... (decategorified vs categorified .... ???? ....) ...
3 ?? artin reciprocity ??? ....
4 ?? jugendtraum .... ????? ....
???or something like that ???
(order of listing here as probably completely random ?? ...)
?? comma construction here ??? .... ??? in 2-category of dimensional categories ??? .... ??homotopy fiber construction ???? ...... ????
?? hmm, return here to issue of ... ?? how "abelian-like" and/or "abelian-unlike" 2-cat of dimensional categories may be .... ????.....
??? any possibility that progression from abelian class field theory ("artin reciprocity") to non-abelian ("langlands reciprocity") might have to do with moving comma construction / homotopy fiber idea from dimensional doctrine to AG doctrine ???? ..... ???? .... ???? hmmm, possibility of generalized "ramification index" here ???? ......
?? possibility of .... ?? understanding stuff about archimedean ramification in terms of ... ?? extending of "differential calculus" / "blow-up" (???? ... ???relationship and/or non-relationship to homotopy fiber idea ... ??? ... ??? bit about ... ????_(cartier) divisor_ as already blown-up ... ???? .....) from AG to "AG without -1" doctrine ??? ..... ????......
??stuff that todd was trying to tell me about .... trying to unify archimedean with non-archimedean factors of zeta function .... ??? .... ??? "gaussian as self-dual under fourier transform" ... ????? ..... (?? relationship to "poisson summation" ??? ..... ????? ......) ..... "tate's thesis" ... ???...
?? non-archimedean "gaussian" as ... ??? characteristic function of algebraic integers among algebraic numbers ???? .... ???? ....
?? alternation between which of "archimedean" / "non-archimedean" qualifies as "exotic" / "mundane" ... ???? ....
??? ?? "discrete vs continuous" ?? ... ??? ?? "quantum vs classical" ?????? .....
?? input vs output of function subject to fourier transform ... ??? .... ???? function vs measure .... ???? .... ???? ..... ???? ......... ??? ....
?? approaches to class field theory .... ????mentioned by lang (?? see book "algebraic number theory" ... ?? ... ?? "part 2 on class field theory ... about 5 pages of historical overview somewhere ... " ... ?? ...) ??? ...
1 ?? "cohomological" ...
2 ?? "zeta function" .... ??? "tannakian duality" ???? .... (decategorified vs categorified .... ???? ....) ...
3 ?? artin reciprocity ??? ....
4 ?? jugendtraum .... ????? ....
???or something like that ???
(order of listing here as probably completely random ?? ...)
?? so far my attempt to explain to todd my attempt to connect "artin reciprocity" with "jugendtraum" ideas didn't get much further than this :
?? artin reciprocity has to do with fractional ideals in imaginary quadratic number field ...
?? jugendtraum has to do with torsion points on elliptic curve ....
?? and we're looking at this stuff in context where elliptic curve arises as complex numbers modulo algebraic integers in imaginary quadratic number field ...
?? ... so ... in this context, those fractional ideals and those torsion points are closely related to each other ... ????? ....
?? artin reciprocity has to do with fractional ideals in imaginary quadratic number field ...
?? jugendtraum has to do with torsion points on elliptic curve ....
?? and we're looking at this stuff in context where elliptic curve arises as complex numbers modulo algebraic integers in imaginary quadratic number field ...
?? ... so ... in this context, those fractional ideals and those torsion points are closely related to each other ... ????? ....
?? (any??) picture of "wave" (?? -function ??? ... ?? eigen- ??? ....???) as secretly picture of ... ??? harmonic representative of homotopy class .... ???? .... ???? .... ???? ....
?? relationship to "side view of helix for observer at infinity" .... ??? ....
xkcd .... ??? ....
?? general theme of ... ?? "hidden dimension" ... thing whose behavior appears more natural (??) from higher-dimensional (?? ... ?? vs "co-higher" ??? ....) perspective ...
??? ??cross-dimensional application of cavalieri's principle ...
?? manin ... gyroscope ... ??? .....
?? relationship to "side view of helix for observer at infinity" .... ??? ....
xkcd .... ??? ....
?? general theme of ... ?? "hidden dimension" ... thing whose behavior appears more natural (??) from higher-dimensional (?? ... ?? vs "co-higher" ??? ....) perspective ...
??? ??cross-dimensional application of cavalieri's principle ...
?? manin ... gyroscope ... ??? .....
Thursday, June 16, 2011
?? elliptic curve with complex multiplication ... ?? ... over adeles ... ?? ... ?? of associated imaginary quadratic number field ???? .... ... ???? ....
??? ....
?? hmm... ?? or maybe what i'm really looking for here is just ... ideles over imaginary quadratic number field .... ????? .....
??? so what am i vaguely imagining here ??? .... that ... ?? pair x,y of elliptic functions wraps complex numbers around elliptic curve "y^2 = f(x)" for monic depressed cubic polynomial f ... ???.... ??with the ring of algebraic integers as the "kernel of the wrapping" ... ??? but then besides the wrapping there's also "folding" ... (??? relationship to "ramification" ???? .... ????? .....) .... ?? with folding of the imaginary quadratic number field wrt its automorphism group resonating with folding of the elliptic curve wrt .... ??? well, "remembering just the x coordinate and forgetting the y coordinate" ...
(???hmmm, or ... ???getting theta functions involved ... ??those map the complex numbers into cone of projective embedding of the elliptic curve ???? ??? in some funny way ?? ... ?????)
?? ideles acting .... ??? ...
???well, there certainly is something going on here about ... ??? bit about ... ??? not sure exactly how to say it ... analogy between ... imaginary quadratic number field as quadratic extension of field of rational numbers, and field of meromorphic functions over particular elliptic curve as quadratic extension of field of meromorphic functions over riemann sphere .... ???? .... ??? but ... ??? maybe some funny level slips here ??? .... ?? felt like i was grasping something .... ???? .....
?? level slip .... ????composition of extensions .... ????action of complex conjugation on elements in maximal abelian extension of gaussian field ... combined galois group ... 2-step solvable ... ????.....
???weil conjectures ... "arithmetic dimension" .... ???? .....
?? cyclotomic case .... ?? "exponential map takes rational numbers to roots of unity .... but exponential map is homomorphism _from_ addition _to_ multiplication, whereas ... ???.... ??we're interested somehow in _multiplication_ of the inputs, because ... that's the gl(1) operation ... that becomes galois group operation .... ????? ......
??? ... so far seemingly somewhat ad hoc "geometric" solutions of the "explicitness problem ... ???? .... for maximal abelian extension of rationals, and of imaginary quadratic number field .... ???? ....
??? "frobenius ..." .... ?????
??? ....
?? hmm... ?? or maybe what i'm really looking for here is just ... ideles over imaginary quadratic number field .... ????? .....
??? so what am i vaguely imagining here ??? .... that ... ?? pair x,y of elliptic functions wraps complex numbers around elliptic curve "y^2 = f(x)" for monic depressed cubic polynomial f ... ???.... ??with the ring of algebraic integers as the "kernel of the wrapping" ... ??? but then besides the wrapping there's also "folding" ... (??? relationship to "ramification" ???? .... ????? .....) .... ?? with folding of the imaginary quadratic number field wrt its automorphism group resonating with folding of the elliptic curve wrt .... ??? well, "remembering just the x coordinate and forgetting the y coordinate" ...
(???hmmm, or ... ???getting theta functions involved ... ??those map the complex numbers into cone of projective embedding of the elliptic curve ???? ??? in some funny way ?? ... ?????)
?? ideles acting .... ??? ...
???well, there certainly is something going on here about ... ??? bit about ... ??? not sure exactly how to say it ... analogy between ... imaginary quadratic number field as quadratic extension of field of rational numbers, and field of meromorphic functions over particular elliptic curve as quadratic extension of field of meromorphic functions over riemann sphere .... ???? .... ??? but ... ??? maybe some funny level slips here ??? .... ?? felt like i was grasping something .... ???? .....
?? level slip .... ????composition of extensions .... ????action of complex conjugation on elements in maximal abelian extension of gaussian field ... combined galois group ... 2-step solvable ... ????.....
???weil conjectures ... "arithmetic dimension" .... ???? .....
?? cyclotomic case .... ?? "exponential map takes rational numbers to roots of unity .... but exponential map is homomorphism _from_ addition _to_ multiplication, whereas ... ???.... ??we're interested somehow in _multiplication_ of the inputs, because ... that's the gl(1) operation ... that becomes galois group operation .... ????? ......
??? ... so far seemingly somewhat ad hoc "geometric" solutions of the "explicitness problem ... ???? .... for maximal abelian extension of rationals, and of imaginary quadratic number field .... ???? ....
??? "frobenius ..." .... ?????
?? ... trying to figure out what pattern we think we're following here ... ???...
algebraic group gl(1) ... realized over finite field ... as actual abelian group ... acting on pth roots of unity ??? .... ???? ??? how ???? ....
???try to imitate for algebraic group given by elliptic curve with complex multiplication ... ???? ..... ????
??? "formal group law" ... ???? ....
?? "complex cobordism" ... ?? ... "orientation class" ... ???? ..... ?? "elliptic cohomology" ... ??? .....
?? "tradeoff between time-independence and space-independence, by considering inverse of solution-candidate map time -> space" ... ???maybe i should say "domain-independence and codomain-independence ... " ... ??? ...
vector field as infinitesimal generator of one-parameter group ... ???? ...
"elliptic integral" = antiderivative (locally ... ?? ...) of function x |-> 1/y where y^2 = x^3 - x (say for example ...) ... ???is that right ??? ....
(pun on "inverse" .... ???? ....)
"elliptic function" = inverse function of that ...
"one-parameter group" f : time -> space .... group homomorphism ... solution to domain-independent de .... f'(t) = g(f(t)) where [g(s)]^2 = s^3 - s ... ????....
h = f^[-1] ... solution to codomain-independent de h'(s) = 1/[g(s)] .... h'(s) = anti-derivative .... ?? of function s |-> 1/[g(s)] ... ??? ....
??? t |-> (f(t),f'(t)) .... ??? "formal group law" on first coordinate dimension alone, vs elliptic curve group structure on variety combining both coordinate dimensions ... ???..... ???? .....
???trying to get elliptic curve with complex multiplication (??corresponding to imaginary quadratic number field k), manifested as specific abelian group over finite k-field f (???? .... ???? .....) to act on values of elliptic function at .... ????? ......
algebraic group gl(1) ... realized over finite field ... as actual abelian group ... acting on pth roots of unity ??? .... ???? ??? how ???? ....
???try to imitate for algebraic group given by elliptic curve with complex multiplication ... ???? ..... ????
??? "formal group law" ... ???? ....
?? "complex cobordism" ... ?? ... "orientation class" ... ???? ..... ?? "elliptic cohomology" ... ??? .....
?? "tradeoff between time-independence and space-independence, by considering inverse of solution-candidate map time -> space" ... ???maybe i should say "domain-independence and codomain-independence ... " ... ??? ...
vector field as infinitesimal generator of one-parameter group ... ???? ...
"elliptic integral" = antiderivative (locally ... ?? ...) of function x |-> 1/y where y^2 = x^3 - x (say for example ...) ... ???is that right ??? ....
(pun on "inverse" .... ???? ....)
"elliptic function" = inverse function of that ...
"one-parameter group" f : time -> space .... group homomorphism ... solution to domain-independent de .... f'(t) = g(f(t)) where [g(s)]^2 = s^3 - s ... ????....
h = f^[-1] ... solution to codomain-independent de h'(s) = 1/[g(s)] .... h'(s) = anti-derivative .... ?? of function s |-> 1/[g(s)] ... ??? ....
??? t |-> (f(t),f'(t)) .... ??? "formal group law" on first coordinate dimension alone, vs elliptic curve group structure on variety combining both coordinate dimensions ... ???..... ???? .....
???trying to get elliptic curve with complex multiplication (??corresponding to imaginary quadratic number field k), manifested as specific abelian group over finite k-field f (???? .... ???? .....) to act on values of elliptic function at .... ????? ......
?? "putting some transcendental structure on the complex numbers and seeing what sort of (??"finitary" ?? ...) structure this puts on the algebraic numbers ..." .... ????? .....
?? schanuel's conjecture ... ???? ..... ?? does this say anything at all interesting about cyclotomic numbers ... ????? .....
?? exponential function evaluated at rational number .... ????
???exponential function evaluated at algebraic number ... ???? .....
?? hmmm, wpa on schanuel's conjecture _does_ say that it's implied by some motivic thing ... ?? ...
?? schanuel's conjecture ... ???? ..... ?? does this say anything at all interesting about cyclotomic numbers ... ????? .....
?? exponential function evaluated at rational number .... ????
???exponential function evaluated at algebraic number ... ???? .....
?? hmmm, wpa on schanuel's conjecture _does_ say that it's implied by some motivic thing ... ?? ...
Wednesday, June 15, 2011
?? talking to prasad ... representation theory of simple lie algebra g tensored with laurent polynomials .... ????
??representation obtained by ... ???? ... ??? ... not sure ...
?? supposed to be some fd rep not completely reducible here ??? ...
?? well, there is ... homomorphism to g tensored with study numbers, for example ... which has fd non-[completely reducible] reps ... but i thought that prasad was describing something else .... ???? .... ??? ...
??reps that can be extended to larger groups where the elements are still maps from the circle (??...) to the group (??...) but with less smoothness required ... ?? ...
??? tannaka-krein philosophy ... ?? ...
??representation obtained by ... ???? ... ??? ... not sure ...
?? supposed to be some fd rep not completely reducible here ??? ...
?? well, there is ... homomorphism to g tensored with study numbers, for example ... which has fd non-[completely reducible] reps ... but i thought that prasad was describing something else .... ???? .... ??? ...
??reps that can be extended to larger groups where the elements are still maps from the circle (??...) to the group (??...) but with less smoothness required ... ?? ...
??? tannaka-krein philosophy ... ?? ...
Tuesday, June 14, 2011
??so ... ?? are we claiming that all cube roots of integers are rationally definable in terms of equal-division points of the eisenstein elliptic curve, and all 4th roots in terms of equal-division points of the lemniscate ??? .... ???? ....
??relationship to way ... lemniscate traditionally presented in terms of quartic ... ???? ...... ?? maybe also eisenstein in terms of cubic ??? .... ????? .....
?? all square roots in terms of equal-division points of gl(1) ... ??? ....
?? that bit about ... ??? "solving arbitrary polynomial equation in one variable in terms of modular functions" ... ???? .... ???? .....
?? dimensional algebra of modular forms ... vs dimensional algebra of fractional ideals ... ?? ....
?? "alternation between randomness and order" ... ???? ....
?? diamond and shurman ... preface ... analogy ... ??? .... ??? can i find some of my notes on that stuff ... trying to figure out what they were really saying ... system of eigenvalues ..... ???? .... ???? .....
?? cubic reciprocity and ... ???... maybe obtaining cube root of n from nth equal-division points of eisenstein curve ??? ... ?? and similarly for biquadratic reciprocity ??? ....
?? cubic (?? ...) numbers obtainable from nth equal-division points of gl(1) ... ??? ....
??vague memories of something ... ?? maybe vaguely like one of above ... ??? ...
??relationship to way ... lemniscate traditionally presented in terms of quartic ... ???? ...... ?? maybe also eisenstein in terms of cubic ??? .... ????? .....
?? all square roots in terms of equal-division points of gl(1) ... ??? ....
?? that bit about ... ??? "solving arbitrary polynomial equation in one variable in terms of modular functions" ... ???? .... ???? .....
?? dimensional algebra of modular forms ... vs dimensional algebra of fractional ideals ... ?? ....
?? "alternation between randomness and order" ... ???? ....
?? diamond and shurman ... preface ... analogy ... ??? .... ??? can i find some of my notes on that stuff ... trying to figure out what they were really saying ... system of eigenvalues ..... ???? .... ???? .....
?? cubic reciprocity and ... ???... maybe obtaining cube root of n from nth equal-division points of eisenstein curve ??? ... ?? and similarly for biquadratic reciprocity ??? ....
?? cubic (?? ...) numbers obtainable from nth equal-division points of gl(1) ... ??? ....
??vague memories of something ... ?? maybe vaguely like one of above ... ??? ...
Monday, June 13, 2011
Sunday, June 12, 2011
?? talking to alex ... bad audio connection, but ...
??? "hecke group" ... ??? ... ??any relationship to that picture with the mysterious tiny region ?? ... ??? maybe not, because doesn't that have a bigon or unigon or something ??? .... ??? ....
?? "schwarz triangle ..." .... ??? ??? "overlapping ... " ... ??? .... ??? ....
??alex somehow got into this from thinking about flag-geometric random-jump operators on projective plane ... ????....
??? lie algebra sl(3,reals) as acting on 2-sphere ??? .... ??? pulling back vector field along covering map ...
alex said some stuff around here that sounded maybe sort of interesting but i didn't quite follow ... something about ... sheaf ... etale space ... ??? .... ?? ???? ....
?? just curious : what's the group like that you get by combining the lie group here (?? what's the universal simply connected group like for the lie algebra sl(3,reals) ??? ... ??? ...) with the conformal group ??? .... ???? ... ??? general technique (pretty un-subtle ... ??? ...) we had for ... sort of describing structure given as "infimum" of two structures ... ??? " .... up to ... " ... ????? ..... ??? .... ??connection with "invariant higher-order distribution" ??? ....
?? stuff quinn said ... ?? about history of manifold theory .... ???? ..... ??? status of objects ... ??? as somehow fundamental vs later as sets with structure ... ... ??? "disillusionment" ... ??? .... ??"pathology" ... ??? ... ??? "concrete problems" ... ??? ....
??? "hecke group" ... ??? ... ??any relationship to that picture with the mysterious tiny region ?? ... ??? maybe not, because doesn't that have a bigon or unigon or something ??? .... ??? ....
?? "schwarz triangle ..." .... ??? ??? "overlapping ... " ... ??? .... ??? ....
??alex somehow got into this from thinking about flag-geometric random-jump operators on projective plane ... ????....
??? lie algebra sl(3,reals) as acting on 2-sphere ??? .... ??? pulling back vector field along covering map ...
alex said some stuff around here that sounded maybe sort of interesting but i didn't quite follow ... something about ... sheaf ... etale space ... ??? .... ?? ???? ....
?? just curious : what's the group like that you get by combining the lie group here (?? what's the universal simply connected group like for the lie algebra sl(3,reals) ??? ... ??? ...) with the conformal group ??? .... ???? ... ??? general technique (pretty un-subtle ... ??? ...) we had for ... sort of describing structure given as "infimum" of two structures ... ??? " .... up to ... " ... ????? ..... ??? .... ??connection with "invariant higher-order distribution" ??? ....
?? stuff quinn said ... ?? about history of manifold theory .... ???? ..... ??? status of objects ... ??? as somehow fundamental vs later as sets with structure ... ... ??? "disillusionment" ... ??? .... ??"pathology" ... ??? ... ??? "concrete problems" ... ??? ....
?? "pth roots of unity" in elliptic curve as forming _2_d f_p vector space ... ??? ....
??special stuff about 2th roots of unity here ... and dimensional algebra of theta functions ... ??? ....
??? f_[p^2] .... ?????? ...... ???"complex multiplication" .... ??? ???gl(1,f_[p^2]) ?? ... ????? ....
??aspects of field associated to elliptic curve which maybe don't depend too much on the particular elliptic curve, vs those that do .... ???? ....
??? relationships among various fields associated to an elliptic curve ... ???? ....
???? field generated by coordinates of equal-division pointe of _generic_ elliptic curve ?? .... ??? .....
?? relationships among ... ??? elliptic curves associated to fractional ideals (of given imaginary quadratic number field ... ) in different classes ... ???? ....
??special stuff about 2th roots of unity here ... and dimensional algebra of theta functions ... ??? ....
??? f_[p^2] .... ?????? ...... ???"complex multiplication" .... ??? ???gl(1,f_[p^2]) ?? ... ????? ....
??aspects of field associated to elliptic curve which maybe don't depend too much on the particular elliptic curve, vs those that do .... ???? ....
??? relationships among various fields associated to an elliptic curve ... ???? ....
???? field generated by coordinates of equal-division pointe of _generic_ elliptic curve ?? .... ??? .....
?? relationships among ... ??? elliptic curves associated to fractional ideals (of given imaginary quadratic number field ... ) in different classes ... ???? ....
?? ... non-invertible morphisms between (underlying modules of) fractional ideals .... ??? vs ... ???? non-invertible morphisms between elliptic curves (?? with complex multiplication ... ??? ...) ... ???? ??very unclear ??? ....
???hall algebra (???for elliptic curves ??? ....) vs hecke operator ... ??? .....
???kernel of non-invertible morphism between elliptic curves (?? for example with complex multiplication ...) ... ??? as "equal-division points" .... ??? ....
??? "fomral group law" for elliptic curve ... differential equation ... "tradeoff between space-independence and time-independence" ... ??? ...
?? "theta function" ...
?? peirce quincunxial projection ... ???....
???hall algebra (???for elliptic curves ??? ....) vs hecke operator ... ??? .....
???kernel of non-invertible morphism between elliptic curves (?? for example with complex multiplication ...) ... ??? as "equal-division points" .... ??? ....
??? "fomral group law" for elliptic curve ... differential equation ... "tradeoff between space-independence and time-independence" ... ??? ...
?? "theta function" ...
?? peirce quincunxial projection ... ???....
Saturday, June 11, 2011
??? all right, i think that i've got an emerging conjectural picture here ... that i should try to write down before i start forgetting it and/or the conjectures start falling part ...
?? some parts of the picture may be wrong ... other parts seem so obvious ... as conjectures to make if not as statements to prove ... that it seems annoying and strange for me not to have thought about them before ... ?? might even turn out that i _have_ thought about some of them before ...
?? might also be some overlap between above classes (wrong, obvious, thought about before ...) ... ???.....
?? and as usual i'm having some trouble figuring out where to start in trying to write this down... so will try some random places ... ???....
??? so there's the field of all "cyclotomic numbers" .... ???? .... which is the absolute abelian extension of the rationals ... which we're supposed to know some stuff about ...
??? then there's also, for any given elliptic curve x, the field of all "x-tomic numbers" ... ??? or at least, i hope that this makes some sense ... ??? .... ?? ... i also have the feeling that there's some conventional terminology here that i'm struggling to remember ... having to do with this idea of "dividing up into equal parts ..." that we're imagining doing to an algebraic group of some sort .... ???? ..... ??anyway, i'm hoping that it will eventually make some sense to think about this alleged big field f_x generated by the coordinates (?? in some sense ...) of "equal-division points" of x ... ???? and then i'm imagining that maybe in the special case where x "has complex multiplication", f_x will be the absolute abelian extension of the imaginary quadratic number field corresponding to x ...
(?? what am i claiming, that f_x = f_y when x and y are ..... ????? ...... ???? ....)
(?? what am i claiming about what sort of number the value of elliptic modular function at x is .... ??? .... ???? analogy of elliptic modular function to exponential function here ??? ..... ???? ..... ???? ..... ??? ....)
?? and then ... ??? maybe this "complex multiplication" case is supposed to serve as a stepping-stone to more general arbitrary elliptic curve case ... ?? where more complicated but perhaps eventually very interesting stuff may happen ... ??? f_x in this case as maybe not abelian extension in any very interesting way ?? ... ??? but interesting as non-abelian extension ... ???? ..... ???? ... no idea, really; just taking stupid guesses here ... ???? ....
??moduli stacks of "line bundles with extra local structure" ... ??? .... ??? ... "jacobian ..." ... ??? ..... ???? ..... ????..... ???something "diaconescuesque" going on here ??? ....
??? ... "descent" ... "descent" ... ???? ....
?? some parts of the picture may be wrong ... other parts seem so obvious ... as conjectures to make if not as statements to prove ... that it seems annoying and strange for me not to have thought about them before ... ?? might even turn out that i _have_ thought about some of them before ...
?? might also be some overlap between above classes (wrong, obvious, thought about before ...) ... ???.....
?? and as usual i'm having some trouble figuring out where to start in trying to write this down... so will try some random places ... ???....
??? so there's the field of all "cyclotomic numbers" .... ???? .... which is the absolute abelian extension of the rationals ... which we're supposed to know some stuff about ...
??? then there's also, for any given elliptic curve x, the field of all "x-tomic numbers" ... ??? or at least, i hope that this makes some sense ... ??? .... ?? ... i also have the feeling that there's some conventional terminology here that i'm struggling to remember ... having to do with this idea of "dividing up into equal parts ..." that we're imagining doing to an algebraic group of some sort .... ???? ..... ??anyway, i'm hoping that it will eventually make some sense to think about this alleged big field f_x generated by the coordinates (?? in some sense ...) of "equal-division points" of x ... ???? and then i'm imagining that maybe in the special case where x "has complex multiplication", f_x will be the absolute abelian extension of the imaginary quadratic number field corresponding to x ...
(?? what am i claiming, that f_x = f_y when x and y are ..... ????? ...... ???? ....)
(?? what am i claiming about what sort of number the value of elliptic modular function at x is .... ??? .... ???? analogy of elliptic modular function to exponential function here ??? ..... ???? ..... ???? ..... ??? ....)
?? and then ... ??? maybe this "complex multiplication" case is supposed to serve as a stepping-stone to more general arbitrary elliptic curve case ... ?? where more complicated but perhaps eventually very interesting stuff may happen ... ??? f_x in this case as maybe not abelian extension in any very interesting way ?? ... ??? but interesting as non-abelian extension ... ???? ..... ???? ... no idea, really; just taking stupid guesses here ... ???? ....
??moduli stacks of "line bundles with extra local structure" ... ??? .... ??? ... "jacobian ..." ... ??? ..... ???? ..... ????..... ???something "diaconescuesque" going on here ??? ....
??? ... "descent" ... "descent" ... ???? ....
?? "graffiti" .... "decoration" ... ???? .... "break symmetry" .... ??? ....
( ?? ... " ... shoes to drop ... " ... ?? ...)
??? "one person's graffiti as another's decoration" ..... ?????......
??? this way of interpreting "orientation" as decoration on simplex .... ???? ....
?? "recursive" ... ??? ... "put orientation on each face, fitting together ..." ...
???vague feeling about "induced ..." ??? .... ????.... also ... ??? barycentric and related subdivision processes ... "model ..." ... pun... orbiplex .... "breadcrumbs" / "fairy dust" .... ???? ....
???? "orientation" vs "orientation" ( = "frame" ??? ...) here ???? ...... ?? pre-sheaf topos here ??? ..... ???? .....
??? orbiplex .... ???presheaf topos ... ??? ... ???? classical logic vs ... ??? "geometric" logic ... ???? .... ???? .... .... "kan condition" ... "beck-chevalley condition" ... ???? ..... ???? ..... ???? .... ??? "marked moduli complex" .... ?????? ........ ????? .......
??? "crackpot tensor" .... ????? ...... ???? .....
??? experience of ... ?? finding any sort of example of ... "part of nature modeled by some mathematical model" ... (??further pun ... ??? ....) ... ???vs ... ??? .. ???case of _all_ of nature ... ???? ....
??? "nature" ... "being" ... "becoming" ... ??? .... "ser" / "estar" ... ???? ....
?? ... "ramsey" ... ?? .... ??? ....
?? ... ??? ... "skolemization" ... ?????... "partial model" ... "double negation topology" ... "forcing" .... ??? .....
( ?? ... " ... shoes to drop ... " ... ?? ...)
??? "one person's graffiti as another's decoration" ..... ?????......
??? this way of interpreting "orientation" as decoration on simplex .... ???? ....
?? "recursive" ... ??? ... "put orientation on each face, fitting together ..." ...
???vague feeling about "induced ..." ??? .... ????.... also ... ??? barycentric and related subdivision processes ... "model ..." ... pun... orbiplex .... "breadcrumbs" / "fairy dust" .... ???? ....
???? "orientation" vs "orientation" ( = "frame" ??? ...) here ???? ...... ?? pre-sheaf topos here ??? ..... ???? .....
??? orbiplex .... ???presheaf topos ... ??? ... ???? classical logic vs ... ??? "geometric" logic ... ???? .... ???? .... .... "kan condition" ... "beck-chevalley condition" ... ???? ..... ???? ..... ???? .... ??? "marked moduli complex" .... ?????? ........ ????? .......
??? "crackpot tensor" .... ????? ...... ???? .....
??? experience of ... ?? finding any sort of example of ... "part of nature modeled by some mathematical model" ... (??further pun ... ??? ....) ... ???vs ... ??? .. ???case of _all_ of nature ... ???? ....
??? "nature" ... "being" ... "becoming" ... ??? .... "ser" / "estar" ... ???? ....
?? ... "ramsey" ... ?? .... ??? ....
?? ... ??? ... "skolemization" ... ?????... "partial model" ... "double negation topology" ... "forcing" .... ??? .....
Friday, June 10, 2011
?? galois ... artin ... langlands ... ??? ....
??? given a group in "tannakian" form ... ??? and an action of it that we want to understand ... on some commutative algebra .... ?? how does this manifest ??? ... ???? ....
?? "concretely identify reps of gl(1,Z/20) with functors from "twentieth-root-of-unity completions of Q" to vector spaces" ... ??? ....
??pretty obvious to get a gl(1,Z/20)-torsor from a field with 20t roots of unity ... ??? ....
??? field with ... ??what's that word that i'm trying to think of ... ???some sort of analog of "cyclotomy", but more general ??? .... ?? points of certain order on certain elliptic curve ... ???.... ??? .....
??is the explicitness with which we think we understand absolute abelian galois group of Q not matched for more general absolute abelian galois groups ?? .... ??? ...
????so _do_ we get a dimensional theory from a "level of ramification" ??? .... ???? .... ???? ....
??? "gauss sum" ... ??? ....
?? "frobenius" idea in connection with joyalesque interpretation of zeta function ... ????? .....
??? "invertible module equipped with frame over ... " ... ??? ...
??? given a group in "tannakian" form ... ??? and an action of it that we want to understand ... on some commutative algebra .... ?? how does this manifest ??? ... ???? ....
?? "concretely identify reps of gl(1,Z/20) with functors from "twentieth-root-of-unity completions of Q" to vector spaces" ... ??? ....
??pretty obvious to get a gl(1,Z/20)-torsor from a field with 20t roots of unity ... ??? ....
??? field with ... ??what's that word that i'm trying to think of ... ???some sort of analog of "cyclotomy", but more general ??? .... ?? points of certain order on certain elliptic curve ... ???.... ??? .....
??is the explicitness with which we think we understand absolute abelian galois group of Q not matched for more general absolute abelian galois groups ?? .... ??? ...
????so _do_ we get a dimensional theory from a "level of ramification" ??? .... ???? .... ???? ....
??? "gauss sum" ... ??? ....
?? "frobenius" idea in connection with joyalesque interpretation of zeta function ... ????? .....
??? "invertible module equipped with frame over ... " ... ??? ...
Thursday, June 9, 2011
?? so ... dimensional category of fractional ideals as having lots of non-invertible morphisms ... ??? non-generators of fractional ideals ... ???.....
?? fractional ideal as ... ??? "holomorphic structure on standard meromorphic line bundle" ... ??? ?? this as yet another (?? and more or less best ?? ... ... when cleaned up a bit ...) definition of "fractional ideal" ... .... ??? sort-of answer for one of todd's questions ... ???.... ?? bit about "strictification by faithful embedding ..." ... ?? ... (?? full-and-faithful as in yoneda (...?? ...) case as overkill ??? ... ??or do i actually understand how this (...??...) case is supposed to work ??? .... ???level slip ?? .... ?? ...)
?? fractional ideal as ... ??? "holomorphic structure on standard meromorphic line bundle" ... ??? ?? this as yet another (?? and more or less best ?? ... ... when cleaned up a bit ...) definition of "fractional ideal" ... .... ??? sort-of answer for one of todd's questions ... ???.... ?? bit about "strictification by faithful embedding ..." ... ?? ... (?? full-and-faithful as in yoneda (...?? ...) case as overkill ??? ... ??or do i actually understand how this (...??...) case is supposed to work ??? .... ???level slip ?? .... ?? ...)
?? in discussion with todd this morning, we found that we had slightly different approaches ... which however we could pretty straightforwardly relate to each other ... todd taking a more "legendre symbol" approach, with me taking a more "galois correspondence" approach ... ?? but so then there are various ideas which seem particularly obvious over on todd's side of the bridge, which it seems like it would be good to understand how they relate to my side of the bridge ... legendre symbols as forming character of (?? galois ir)rep ... tensor product of those irreps ... "artin reciprocity" .... ????.... representation-theoretic approach to galois group ... ???? lambda ring structure of character ring, and galois snake eating its own tail ?? ... ??? .... ?? zeta function and l-functions ... ?? factors (?? ...) corresponding to ideal classes ... ????? .... ??? .... ??? character of regular rep factoring into factors for irreps ... ???? .... ???? ....
?? also, seems like it would be a really good idea if i could get a working copy of mathematica real soon, for use with todd and summer course students ... ??? ....
x^2 + 5*y^2 ....
0 1 4 9 16 25 36 49 64 81 100
5 6 9 14 21 30 41 54 69 86 105
20 21 24 29 36 45 56 69 84 101 120
45 46 49 54 61 70 81 94 109 126 145
80 81 84 89 96 105 116 129 144 161 180
125 126 129 134 141 150 161 174 189 206 325
180 181 184 189 196 205 216 229 244 261 280
245 246 249 254 241 270 281 294 309 326 345
320 321 324 329 336 345 356 369 384 401 420
405 406 409 414 421 430 441 454 469 486 505
500 501 504 509 516 525 536 549 564 581 600
?? "congruence subgroup" ... ??? gl(1) vs sl(2) ... ??? ..... ???? ... ??elements of finite order in affine group gl(1) vs in "elliptic curve" .... ???? ..... ????....
fearless symmetry ... ??? .... ash & gross ... ?? ....
??? free AG theory on dimensional theory of fractional ideals ... vs AG theory with syntactic category given by modules ... ??? ....
"zeta vs theta" .... ???? ..... ????? ......
???any chance of .... ???derived category interpretation of l-function to fit with (joyalesque ...) "combinatorial" interpretation of zeta function ??? ... ?? ... "modulis stack ... " ... ???? .... ??? .....
?? trying to understand "analytic class number formula" (?? saying something about zeta function rather than l-function, at least naively perhaps ?? .... ??though presumably interesting to relate to l-functions ... ??? ....) in terms of ... ??? joyalesque stuff ... ??? ..... ???? .....
?? also, seems like it would be a really good idea if i could get a working copy of mathematica real soon, for use with todd and summer course students ... ??? ....
x^2 + 5*y^2 ....
0 1 4 9 16 25 36 49 64 81 100
5 6 9 14 21 30 41 54 69 86 105
20 21 24 29 36 45 56 69 84 101 120
45 46 49 54 61 70 81 94 109 126 145
80 81 84 89 96 105 116 129 144 161 180
125 126 129 134 141 150 161 174 189 206 325
180 181 184 189 196 205 216 229 244 261 280
245 246 249 254 241 270 281 294 309 326 345
320 321 324 329 336 345 356 369 384 401 420
405 406 409 414 421 430 441 454 469 486 505
500 501 504 509 516 525 536 549 564 581 600
?? "congruence subgroup" ... ??? gl(1) vs sl(2) ... ??? ..... ???? ... ??elements of finite order in affine group gl(1) vs in "elliptic curve" .... ???? ..... ????....
fearless symmetry ... ??? .... ash & gross ... ?? ....
??? free AG theory on dimensional theory of fractional ideals ... vs AG theory with syntactic category given by modules ... ??? ....
"zeta vs theta" .... ???? ..... ????? ......
???any chance of .... ???derived category interpretation of l-function to fit with (joyalesque ...) "combinatorial" interpretation of zeta function ??? ... ?? ... "modulis stack ... " ... ???? .... ??? .....
?? trying to understand "analytic class number formula" (?? saying something about zeta function rather than l-function, at least naively perhaps ?? .... ??though presumably interesting to relate to l-functions ... ??? ....) in terms of ... ??? joyalesque stuff ... ??? ..... ???? .....
Wednesday, June 8, 2011
?? concerning recent (?...) idea of "underlying module of ideal as embodying blow-up concept", i mentioned how i wasn't sure to fit it with the idea of "forcing ideal to become invertible as module" ... ??? ... ??? but now it seems to me that it sort of ftis in some ways ... ??? ???ideal as "lumpy" (at corresponding subvariety ...), and smoothing out the lumpiness as corresponding to blowing-up the base-space to compensate ... ??? ... ?? ....
??this as suggesting ... ?? that to some extent this (...) idea makes sense in connection with ... ?? forcing arbitrary module to become invertible?? .... ???? "symmetric algebra of module" ??? ....
?? possibility that that's about as far as that goes without "anchor map" ??... ??rest of story ... "scaling deformation" ... ??? as maybe relying on that ?? ... ??? .... ??? ...
(??? vague feeling here about ... ???bit about "forcing ring r to become local doesn't really work but forcing topos in which it lives to become such that r is local does" ... ?? "forcing module to become invertible" as really forcing its ring to do something ... ??? ..... ??? ....)
??this as suggesting ... ?? that to some extent this (...) idea makes sense in connection with ... ?? forcing arbitrary module to become invertible?? .... ???? "symmetric algebra of module" ??? ....
?? possibility that that's about as far as that goes without "anchor map" ??... ??rest of story ... "scaling deformation" ... ??? as maybe relying on that ?? ... ??? .... ??? ...
(??? vague feeling here about ... ???bit about "forcing ring r to become local doesn't really work but forcing topos in which it lives to become such that r is local does" ... ?? "forcing module to become invertible" as really forcing its ring to do something ... ??? ..... ??? ....)
?? concerning idea that "underlying module of ideal embodies concept of blow-up" ... (maybe ... ??....) ... ??? connecting this with "zariski tangent space" ... ???"zariski nomral bundle" ???? .... ?? of course we already knew to do this, sort of ... but ... want to think about it fresh, in light of the "embody" idea which is maybe somewhat new to us ...
"ideal" ... "normal cone bundle" ... ?? "frontier" ?? ... ??? ..... ???? .... "glueing" ... ??? .... ??? .....
1/x
x/x^2
x^2/x^3
.
.
.
?? ....
"ideal" ... "normal cone bundle" ... ?? "frontier" ?? ... ??? ..... ???? .... "glueing" ... ??? .... ??? .....
1/x
x/x^2
x^2/x^3
.
.
.
?? ....
?? "ramification" ... "assistance" ... ??? .... ???? ...
?? "ramification at the infinite prime" ... ???? .... ??? .... ??? ....
?? "ramification" vs "localization" ... ??? ....
?? "galois stack" ... ???
?? "gaussian module with anti-involution" .... ???? .... ??? .... ?? is this coming out different from the way that i thought that i remembered it ?? ...
??? "free gaussian module with anti-involution on a gaussian module" .... ????? .... ?????? .....
?? "galois representation" ... ???? .... ???? ....
?? "ramification at the infinite prime" ... ???? .... ??? .... ??? ....
?? "ramification" vs "localization" ... ??? ....
?? "galois stack" ... ???
?? "gaussian module with anti-involution" .... ???? .... ??? .... ?? is this coming out different from the way that i thought that i remembered it ?? ...
??? "free gaussian module with anti-involution on a gaussian module" .... ????? .... ?????? .....
?? "galois representation" ... ???? .... ???? ....
"frobenius ... " .... ???? ....
?? certain regularity in table that we struggled to understand ...
artin reciprocity ...??? ....
??? more to making artin reciprocity "explicit" / "canonical" than just working with subgroup lattice of absolute abelian galois group ... ?? working with the group itself ... ???? .....
?? "weil reciprocity ... " ....???
?? certain regularity in table that we struggled to understand ...
artin reciprocity ...??? ....
??? more to making artin reciprocity "explicit" / "canonical" than just working with subgroup lattice of absolute abelian galois group ... ?? working with the group itself ... ???? .....
?? "weil reciprocity ... " ....???
??so ... ??given a number ring x ... ??? .... consider it's ideal class group ... ??and also consider its "splitting rule" ... ????then ... does the ideal class group sort of "fit into the splitting rule" ??? ... ???in a certain sense ??? .... ??? ....
??so what do i think i mean by that ?? ... that primes of x are somewhat organized according to "splitting rule" of x (??for case of x abelian extension of something ... ??so maybe i'm just talking about that case for now ... ??? ... ??? ...) ... ?? then maybe using this organization in describing ideal class group .... ???? ....
?? might i be getting at "easy part of ideal class group" or something here ???? .....
??ideal class group of x as relating to unramified extensions _of_ x ... ????....
(??then also "idele ..." .... ???? ..... ???? .... ramified extensions ...)
?? so is it at least true that ... "previously principal ideals don't suddenly become non-principal without splitting ... " ??? .... ???? ....
??so what do i think i mean by that ?? ... that primes of x are somewhat organized according to "splitting rule" of x (??for case of x abelian extension of something ... ??so maybe i'm just talking about that case for now ... ??? ... ??? ...) ... ?? then maybe using this organization in describing ideal class group .... ???? ....
?? might i be getting at "easy part of ideal class group" or something here ???? .....
??ideal class group of x as relating to unramified extensions _of_ x ... ????....
(??then also "idele ..." .... ???? ..... ???? .... ramified extensions ...)
?? so is it at least true that ... "previously principal ideals don't suddenly become non-principal without splitting ... " ??? .... ???? ....
??? stuff that i thought that i sort of understood at one time, about ... ?? "ramification up to a certain point" ... ??? ?? "invertible module with local structure ... " .... ??? .... ???? ..... ??? revisiting from some of our more recent viewpoints ... ???? .....
??? "naming an abelian exteniosn by its splitting rule" ... ??? ....
??? "naming an abelian exteniosn by its splitting rule" ... ??? ....
Tuesday, June 7, 2011
notes for next discussion with todd
?? maybe take very scattered, ambitious approach ... ??? ... ??with alleged excuse of preparing for summer course, in part ?? ...
?? 2 in some way related themes??? :
1 ideal with invertible underlying module .....
2 forcing ideal to have invertible underlying module ...
(??then even more general, invertibleness of modules not associated with ideals ... ??? .... and renormalization ... ?? ... ???)
??? so maybe... try to concentrate on theme 1, though try to _briefly_ mention theme 2 ...
?? ideal class groups of quadratic number fields ... relating to modular forms ... ??? ...
?? also ideal class groups of cyclotomic number fields ... ??relating to toric stuff ??? ....
?? ideal class groups of affine curves ??? .... ?? getting into geometric interpretation and theme 2 a bit ... ??? so maybe back off a bit ??? .... ???
or maybe don't ... ???? ....
???base change stuff ???? ....
?? dimensional theory of fractional ideals ... ???toric analogs?? ....
???? "galois stack" ... ????.....
?? 2 in some way related themes??? :
1 ideal with invertible underlying module .....
2 forcing ideal to have invertible underlying module ...
(??then even more general, invertibleness of modules not associated with ideals ... ??? .... and renormalization ... ?? ... ???)
??? so maybe... try to concentrate on theme 1, though try to _briefly_ mention theme 2 ...
?? ideal class groups of quadratic number fields ... relating to modular forms ... ??? ...
?? also ideal class groups of cyclotomic number fields ... ??relating to toric stuff ??? ....
?? ideal class groups of affine curves ??? .... ?? getting into geometric interpretation and theme 2 a bit ... ??? so maybe back off a bit ??? .... ???
or maybe don't ... ???? ....
???base change stuff ???? ....
?? dimensional theory of fractional ideals ... ???toric analogs?? ....
???? "galois stack" ... ????.....
?? so ... this idea that "underlying module of an ideal" more or less embodies whole "blow-up" concept still feels mostly right-track to me ... (?? though .... ?? extent to which underlying module really knows about ideal power filtration ... idealpowers vs tensor powers ... filtration .... ??? ... ??? ....) ... but there is at least one huge key aspect that i was overlooking in a couple of recent posts, evidently responsible for a certain proportion of the confusion there ...
?? ... this key aspect ... idea that "co-domension 1 subvarieties can't really be blown up" ... vs obvious fact that spectrum of symmetric algebra of invertible module has extra fiber dimension ... so it must be that it's really important to _projectivize_ to cancel out this extra dimension ... ????.....
???so symmetric algebras of different invertible modules are _not_ all the same .... and relate (of course ... ??? ...) to different projective embeddings of the "vacuous blow-up" ...
(??maybe though i should still wonder a little about what happens if you forget the grading of the symmetric algebra ... ??probably you can canonically reconstruct it though ??? ... ??? ....)
??anyway, this is probably helping in connecting recent emphasis on "underlying module of ideal as embodying blow-up" vs ... ??emphasis at other times on ... ideal power filtration and associated graded stuff and "stack" (...) interpretation ("renormalization group" ...) of all that stuff ... in understanding "blow-up" ... ??? .....
?? "inherently projective aspect of blow-up" ... ?? ... "introducing one new line bundle" ... ?? as tending to move to foreground in case where more purely blow-up aspect of blow-up is vacuous ... ???....
???some confusion here about ... ???projectivization wrt different dimensions ... ??? ..... ??? ...
???how does idea of "global sections of line bundle as forming ambient vector space of projective embedding" fit in here ??? ....
??? global sections of _new_ line bundle as ..... ????? .... ??? ....
"renormalization" .... "associated graded ..." .... "normal cone" .... ??in singular case ... ??? "conicalness of singularity" ... ????... ... ???? ..... "rees ..." .... ??? ..... deformation ...... ????? .....
??????????????? ......
???normal cone as "thing being ("scalingly" ... ???) deformed" ?? .... (or more or less equivalently but better ... ??? ... "re-scaling limit" ...) .... whole deformation thing, and "rees ..." ... ??? .....
?? "differential calculus as special case of renormalization group ...." ... ??? ... ?? "dimensional analysis" ...
?? ... this key aspect ... idea that "co-domension 1 subvarieties can't really be blown up" ... vs obvious fact that spectrum of symmetric algebra of invertible module has extra fiber dimension ... so it must be that it's really important to _projectivize_ to cancel out this extra dimension ... ????.....
???so symmetric algebras of different invertible modules are _not_ all the same .... and relate (of course ... ??? ...) to different projective embeddings of the "vacuous blow-up" ...
(??maybe though i should still wonder a little about what happens if you forget the grading of the symmetric algebra ... ??probably you can canonically reconstruct it though ??? ... ??? ....)
??anyway, this is probably helping in connecting recent emphasis on "underlying module of ideal as embodying blow-up" vs ... ??emphasis at other times on ... ideal power filtration and associated graded stuff and "stack" (...) interpretation ("renormalization group" ...) of all that stuff ... in understanding "blow-up" ... ??? .....
?? "inherently projective aspect of blow-up" ... ?? ... "introducing one new line bundle" ... ?? as tending to move to foreground in case where more purely blow-up aspect of blow-up is vacuous ... ???....
???some confusion here about ... ???projectivization wrt different dimensions ... ??? ..... ??? ...
???how does idea of "global sections of line bundle as forming ambient vector space of projective embedding" fit in here ??? ....
??? global sections of _new_ line bundle as ..... ????? .... ??? ....
"renormalization" .... "associated graded ..." .... "normal cone" .... ??in singular case ... ??? "conicalness of singularity" ... ????... ... ???? ..... "rees ..." .... ??? ..... deformation ...... ????? .....
??????????????? ......
???normal cone as "thing being ("scalingly" ... ???) deformed" ?? .... (or more or less equivalently but better ... ??? ... "re-scaling limit" ...) .... whole deformation thing, and "rees ..." ... ??? .....
?? "differential calculus as special case of renormalization group ...." ... ??? ... ?? "dimensional analysis" ...
summer course ...
"power of observation" (asso0ciated games ... ??? ...) vs "decoration" .... ??? .... ??? .... ???? ....
cubic reciprocity mathematica program ... ??
?? hyperlemniscate ...
20th roots of unity ... table of galois groups ... ??? ...
?? maybe try "square lattice" cheating proof of 2 squares theorem ... challenge to find flaw ... ??? ... after trying to actually trick them ?? ....
???? escher ... ????? .....
?? "double coset" interpretation of certain moduli spaces ... ???? ..... (and / or stacks ... ????)
?? "sprinkle fairy dust in kaleidoscope" and ... "theta series" ..... ????? ..... ?????? ...... ??? "gauss sum" .... ????? .....
?? "covering space" idea as arguably mere way of putting more fundamental idea in "geometric" form ??? ....
??? my experience ... mostly getting only chance to talk about stuff i _don't_ understand, and really wanting chance to talk about stuff i _do_ understand .... continued pressure even here (... ??? ...) to lean in direction in which i don't understand .... tension .... ?? perhaps see some mixture of both tendencies ... not sure which mixture ... more in one direction or another .... ???? ....
?? also my own uncertainty about what is it that i understand .... ?? actual fluctuation (at the very least) in both directions .....
?? tying in these uncertainties with general uncertainties about ... ?? lots of other stuff??? .... ???aiming too high vs aiming too low .... ????? relationship to possibilities of just completely flopping in all sorts of ways ....
??? still viewing big punchline (?? ...) of course (pun?) as ... ??? bit about ... "lagrange extrapolation" as "say anything", justifying "converse" aspect of "galois theory" .... ??? ..... ???? no idea if it'll actually work out that way though .... ???? ....
??? puzzle ... can you define this in terms of that ?? ....
this = less than on N ; that = addition on N ..
this = less than on N ; that = mult on N ... ??? ....
and so forth ....
??? "there's not enough information in multiplication of natural numbers to tell you about the (...) order structure on the natural numbers .... it can't tell you, because _it doesn't know_ !!!!!! ......" ....
??? "more powerful power of observation" =?= "more desctructive (??? ...) graffiti" .... ????? ... ??? "vandalism" .... ???? .... ??? ....
cubic reciprocity mathematica program ... ??
?? hyperlemniscate ...
20th roots of unity ... table of galois groups ... ??? ...
?? maybe try "square lattice" cheating proof of 2 squares theorem ... challenge to find flaw ... ??? ... after trying to actually trick them ?? ....
???? escher ... ????? .....
?? "double coset" interpretation of certain moduli spaces ... ???? ..... (and / or stacks ... ????)
?? "sprinkle fairy dust in kaleidoscope" and ... "theta series" ..... ????? ..... ?????? ...... ??? "gauss sum" .... ????? .....
?? "covering space" idea as arguably mere way of putting more fundamental idea in "geometric" form ??? ....
??? my experience ... mostly getting only chance to talk about stuff i _don't_ understand, and really wanting chance to talk about stuff i _do_ understand .... continued pressure even here (... ??? ...) to lean in direction in which i don't understand .... tension .... ?? perhaps see some mixture of both tendencies ... not sure which mixture ... more in one direction or another .... ???? ....
?? also my own uncertainty about what is it that i understand .... ?? actual fluctuation (at the very least) in both directions .....
?? tying in these uncertainties with general uncertainties about ... ?? lots of other stuff??? .... ???aiming too high vs aiming too low .... ????? relationship to possibilities of just completely flopping in all sorts of ways ....
??? still viewing big punchline (?? ...) of course (pun?) as ... ??? bit about ... "lagrange extrapolation" as "say anything", justifying "converse" aspect of "galois theory" .... ??? ..... ???? no idea if it'll actually work out that way though .... ???? ....
??? puzzle ... can you define this in terms of that ?? ....
this = less than on N ; that = addition on N ..
this = less than on N ; that = mult on N ... ??? ....
and so forth ....
??? "there's not enough information in multiplication of natural numbers to tell you about the (...) order structure on the natural numbers .... it can't tell you, because _it doesn't know_ !!!!!! ......" ....
??? "more powerful power of observation" =?= "more desctructive (??? ...) graffiti" .... ????? ... ??? "vandalism" .... ???? .... ??? ....
?? so we've got this idea now that .... the idea of "blowing up a sub-variety" is actually very tightly connected with the whole idea of interpreting a sub-variety as an ideal (or more generally... that is in the non-affine case ... ??well, simply sub-variety ??? ... ??? ...) in the first place ... or perhaps i should emphasize more with taking the _underlying module (??or more generally line bundle?? ...)_ of the ideal ... in the sense that this module is "a lot like the unit module away from the sub-variety, but blown-up over the sub-variety" ...
(??? idea of "blowing up a module at an ideal" ??? .... ??? does anything like this parse??? hmmm, maybe just tensoring the module with the ideal ??? .... ??relationship to tensoring the module with the symmetric algebra of the ideal ??? .... ????? ....)
?? but ... ??something a bit funny here ... worth understanding most likely ... ???that in the case where the module is invertible, the effect of blowing-up is supposed to be vacuous, right?? ... whereas ... notoriously not all such invertible modules are equivalent as modules to the unit module ... so ... seems to be something like ... ??all invertible modules have equivalent symmetric algebra, though are not themselves all equivalent ... ??? .... ??so there should be an interesting variety of ways in which you can take the symmetric algebra of the unit module and find inside of it alternative free generating submodules .... ????? ??am i saying this right ???
???underlying affine scheme bundle of line bundle as always trivial ??? .... .... ???putting line bundle structure back on it as thus interesting?? ... ???or as in contrast of course (??) just trivial ... ???
?? so now it seems like i've got some obvious paradox, so probably screwed up somewhere ... so definitely try to straighten this out ... ???
???hmmm, well ... there is obvious fact that ... of course the "linear" structure _can_ be modified ... namely by changing origin ... ????so is _that_ what's going on here ?????? ...... hmmmm ...... ?????...... ??seems like attractive idea, but ... ???? in danger of believing that choosing section of trivial line bundle amounts to ..... ?????? ... ???choosing non-trivial line bundle ???? ..... sounds ... almost but not quite right-track ??? .... ???? ..... ???? ....
??relationship between "underlying affine scheme bundle as trivial" and "underlying meromorphic line bundle as trivial" ??? .... ???? .... ???? ....
??vague memory about .... ????coherent sheaf cohomology ... affine space as short exact sequence ... ???????..... ?? h^2 ... ??? ....
(??? idea of "blowing up a module at an ideal" ??? .... ??? does anything like this parse??? hmmm, maybe just tensoring the module with the ideal ??? .... ??relationship to tensoring the module with the symmetric algebra of the ideal ??? .... ????? ....)
?? but ... ??something a bit funny here ... worth understanding most likely ... ???that in the case where the module is invertible, the effect of blowing-up is supposed to be vacuous, right?? ... whereas ... notoriously not all such invertible modules are equivalent as modules to the unit module ... so ... seems to be something like ... ??all invertible modules have equivalent symmetric algebra, though are not themselves all equivalent ... ??? .... ??so there should be an interesting variety of ways in which you can take the symmetric algebra of the unit module and find inside of it alternative free generating submodules .... ????? ??am i saying this right ???
???underlying affine scheme bundle of line bundle as always trivial ??? .... .... ???putting line bundle structure back on it as thus interesting?? ... ???or as in contrast of course (??) just trivial ... ???
?? so now it seems like i've got some obvious paradox, so probably screwed up somewhere ... so definitely try to straighten this out ... ???
???hmmm, well ... there is obvious fact that ... of course the "linear" structure _can_ be modified ... namely by changing origin ... ????so is _that_ what's going on here ?????? ...... hmmmm ...... ?????...... ??seems like attractive idea, but ... ???? in danger of believing that choosing section of trivial line bundle amounts to ..... ?????? ... ???choosing non-trivial line bundle ???? ..... sounds ... almost but not quite right-track ??? .... ???? ..... ???? ....
??relationship between "underlying affine scheme bundle as trivial" and "underlying meromorphic line bundle as trivial" ??? .... ???? .... ???? ....
??vague memory about .... ????coherent sheaf cohomology ... affine space as short exact sequence ... ???????..... ?? h^2 ... ??? ....
?? so ... given a dedekind domain ... ???.... consider abelian group of fractional ideals, and commutative monoid over it, where an element over fractional ideal x is an element y of the field of fractions such that ... ??? .... "y gives a module homomomorphism from the unit module to the fractional ideal" ... ?? ... ??in other words, simply y is an element of x .... ??? .....
??wait, did i say that right? ... ???maybe i should have said, y is a _non-fractional_ element of x ... ????..... ...confusion ... ???
?? well, the ideal class group is finite, according to what we've heard ... so
certainly individual elements are of finite order, so ... ?? we should expect that any fractional ideal has morphisms from the unit object if we expect that sufficiently extreme (perhaps in the negative direction) powers of it do ... because the finite order guarantees that it's itself equivalent to one of htose sufficiently extreme powers... ???right ??? ... so ... it shouldn't bug us so much then that all fractional ideals have elements, in contrast to the sort of situation that we're more familiar with, where the fibers outside a certain cone are trivial ... ??? ...
?? so then can we prove that invertible = principal here ?? .... actually i don't quite see it yet ... ??? ....
??so let x be a fractional ideal, with y in x and y' inverse to it (thus in x^[-1]) ... ???.....
???well, so what _is_ the inverse of a fractional ideal x, concretely ??? ... ???fractions which "can be expressed with any denominator taken from x" ??? ??well, perhaps that's an ok way of saying it for the special case of the inverse fractional ideal of a non-fractional ideal ... which perhaps is all we really need here ....
??? "residual ... " ??? .... ???? "residue" ??? ??? ... ???
?? actually let me try saying it more straightforwardly ... for fractional ideal x, fraction y is in x^[-1] iff x1*y integeral for any x1 in x ... ?????
(??which could perhaps be rephrased as "has integeral numerator when expressed with any denominator taken from x" ... ???? .....)
??so then suppose that y in x and that x1*y^[-1] is integeral for any x1 in x ... ??or in other words, x1 = integeral elt * y for any x1 in x ....
??so maybe a more relevant rephrasing here would have been "every x1 in x is an integeral multiple of y^[-1]" ???
??so "y is in x and y^[-1] is in x^[-1]" is ess "y is in x and every x1 in x is an integeral multiple of y" ??? ..... ?? which is pretty much exactly saying that y generates x / y witnesses the principalness of x ... ???
(??? in general, the naive idea of the inverse of a fractional ideal as being formed from the inverses of its elements as wrong ... ???but in the case of principal elements it maybe sort of ends up being true??? ... ??is that sort of what we're saying here ??? ....)
?? hmm, so ... ??? to say that y is in x is to say that the principal ideal <= y, and to say that y^[-1] is in x^[-1] is to say that y <= ... ??? ...
??? ideal class group as "k-group" ... ???forgot about weirdness of this ... tensor product of modules vs direct sum ... requiring invertibility ahead of time vs compelling it afterwards / granting it .... (??ways of thinking about right vs left adjoint ??? .... ????..... ?? "before / after" ... ??? ...) ... ???whether that k-group is _always_ (??for any commutative ring??? ...??? ???....) the group of iso classes of invertible modules, or only in the dedekind case ??? .... ???? ....
??stuff baez said about ... ??? euler characteristic valued in k-group ... ????...
??wait, did i say that right? ... ???maybe i should have said, y is a _non-fractional_ element of x ... ????..... ...confusion ... ???
?? well, the ideal class group is finite, according to what we've heard ... so
certainly individual elements are of finite order, so ... ?? we should expect that any fractional ideal has morphisms from the unit object if we expect that sufficiently extreme (perhaps in the negative direction) powers of it do ... because the finite order guarantees that it's itself equivalent to one of htose sufficiently extreme powers... ???right ??? ... so ... it shouldn't bug us so much then that all fractional ideals have elements, in contrast to the sort of situation that we're more familiar with, where the fibers outside a certain cone are trivial ... ??? ...
?? so then can we prove that invertible = principal here ?? .... actually i don't quite see it yet ... ??? ....
??so let x be a fractional ideal, with y in x and y' inverse to it (thus in x^[-1]) ... ???.....
???well, so what _is_ the inverse of a fractional ideal x, concretely ??? ... ???fractions which "can be expressed with any denominator taken from x" ??? ??well, perhaps that's an ok way of saying it for the special case of the inverse fractional ideal of a non-fractional ideal ... which perhaps is all we really need here ....
??? "residual ... " ??? .... ???? "residue" ??? ??? ... ???
?? actually let me try saying it more straightforwardly ... for fractional ideal x, fraction y is in x^[-1] iff x1*y integeral for any x1 in x ... ?????
(??which could perhaps be rephrased as "has integeral numerator when expressed with any denominator taken from x" ... ???? .....)
??so then suppose that y in x and that x1*y^[-1] is integeral for any x1 in x ... ??or in other words, x1 = integeral elt * y for any x1 in x ....
??so maybe a more relevant rephrasing here would have been "every x1 in x is an integeral multiple of y^[-1]" ???
??so "y is in x and y^[-1] is in x^[-1]" is ess "y is in x and every x1 in x is an integeral multiple of y" ??? ..... ?? which is pretty much exactly saying that y generates x / y witnesses the principalness of x ... ???
(??? in general, the naive idea of the inverse of a fractional ideal as being formed from the inverses of its elements as wrong ... ???but in the case of principal elements it maybe sort of ends up being true??? ... ??is that sort of what we're saying here ??? ....)
?? hmm, so ... ??? to say that y is in x is to say that the principal ideal
??? ideal class group as "k-group" ... ???forgot about weirdness of this ... tensor product of modules vs direct sum ... requiring invertibility ahead of time vs compelling it afterwards / granting it .... (??ways of thinking about right vs left adjoint ??? .... ????..... ?? "before / after" ... ??? ...) ... ???whether that k-group is _always_ (??for any commutative ring??? ...??? ???....) the group of iso classes of invertible modules, or only in the dedekind case ??? .... ???? ....
??stuff baez said about ... ??? euler characteristic valued in k-group ... ????...
??so consider, for example, say, Z[x]/(x^37-1)/(x-1) ...
???prime ideals here ... ????..... ???mapping down to prime ideals of Z ...
???should be pretty clear what these prime ideals are, in general ... ????....
???but then what about principalness of such ideals ???? ....
??compare to examples that we've thought more about ... ??? .....
?? prime p with 37 dividing p-1 ... ???? p = 1 mod 37 ....
1 38 75 112 149 186 223 260 297 334 371 408
149 223 ...
Z[x]/(x^37-1)/(x-1) -> Z/149
1 2 4 8 16 32 64 128 107 65 130 111 73 146 143 137 125 101 53 106 63 126 103 57 114 79 9 18 36 72 144 139 129 109 69 138 127 105 61 122 95 41 82 15 30 60 120 91
33 66 132 115 81 13 26 52 104 59 118 87 25 50 100 51 102 55 110 71 142 135 121 93 37 74 148 147 145 141 133 117 85 21 42 84 19 38 76 3 6 12 24 48 96 43 86 23 46 92 35 70 140 131 113 77 5 10 20 40 80 11 22 44 88 27 54 108 67 134 119 89 29 58 116 83 17 34 68 136 123 97 45 90 31 62 124 99 49 98 47 94 39 78 7 14 28 56 112 75
x |-> 16
x |-> 107
x |-> 73
x |-> 125
.
.
.
x |-> 28
???so for example the ideal given as kernel of x |-> 16 here ... ??? what are some ways of thinking abut whether it's principal ?? .... ??? ...
?? well, perhaps there's a naive direct way ... ???
x-16 is in the kernel ...
x^2-107 is as well ...
x^3-73
.
.
.
hmm, well, also 149 is in the kernel, right ?? ....
??well, let's try testing how close x-16 comes to generating the ideal ...
so, we're setting x to 16 ... ?so basically we're taking Z and forcing (16^37-1)/(16-1) to be zero ?? .... hmmm ...
??? by the way i'm assuming for the moment that the algebraic integers here are the obvious ones ... i think todd said that that's what he thought they were in this case ...
???prime ideals here ... ????..... ???mapping down to prime ideals of Z ...
???should be pretty clear what these prime ideals are, in general ... ????....
???but then what about principalness of such ideals ???? ....
??compare to examples that we've thought more about ... ??? .....
?? prime p with 37 dividing p-1 ... ???? p = 1 mod 37 ....
1 38 75 112 149 186 223 260 297 334 371 408
149 223 ...
Z[x]/(x^37-1)/(x-1) -> Z/149
1 2 4 8 16 32 64 128 107 65 130 111 73 146 143 137 125 101 53 106 63 126 103 57 114 79 9 18 36 72 144 139 129 109 69 138 127 105 61 122 95 41 82 15 30 60 120 91
33 66 132 115 81 13 26 52 104 59 118 87 25 50 100 51 102 55 110 71 142 135 121 93 37 74 148 147 145 141 133 117 85 21 42 84 19 38 76 3 6 12 24 48 96 43 86 23 46 92 35 70 140 131 113 77 5 10 20 40 80 11 22 44 88 27 54 108 67 134 119 89 29 58 116 83 17 34 68 136 123 97 45 90 31 62 124 99 49 98 47 94 39 78 7 14 28 56 112 75
x |-> 16
x |-> 107
x |-> 73
x |-> 125
.
.
.
x |-> 28
???so for example the ideal given as kernel of x |-> 16 here ... ??? what are some ways of thinking abut whether it's principal ?? .... ??? ...
?? well, perhaps there's a naive direct way ... ???
x-16 is in the kernel ...
x^2-107 is as well ...
x^3-73
.
.
.
hmm, well, also 149 is in the kernel, right ?? ....
??well, let's try testing how close x-16 comes to generating the ideal ...
so, we're setting x to 16 ... ?so basically we're taking Z and forcing (16^37-1)/(16-1) to be zero ?? .... hmmm ...
??? by the way i'm assuming for the moment that the algebraic integers here are the obvious ones ... i think todd said that that's what he thought they were in this case ...
Monday, June 6, 2011
??so consider .... ????.... random jump operator on circle where .... ???probability density of destination is given by inner product with origin ... ??? .... ???then ... from this construct random jump operators on space of "ordered orthonormal basises of euclidean n-space" in hopefully sort of obvious way .... ????then whether these satisfy some sort of braid relation ??? ...
??hmm, maybe not ... ??pretty unclear to me at the moment ...
??? archimedes's method for picking random point on surface of sphere ... random longitude and random ... ?? inner product with north pole ... ????? ....
?? gauss ... -ian ... ??? ....
?? idea that it's no good if both choices are by longitude or both by inner product with north pole; you need one one way and the other the other; and that screws things up ?? ... ??? ....
?? sas vs asa ... ????
a,b,c
(a.b)(aXb . bXc)(b.c) ???
??? vs ... ???
(aXb . aXc)(a.c)(aXc . bXc) ???
????? .....
(a.b)(c'.a')(b.c) =?= (c'.b')(a.c)(b'.a') ???
???or ... ??? maybe some reciprocals somewhere ... ??? ... ????....
1 a b
a 1 c
b c 1
1 ? ?
? 1 ?
? ? 1
??hmm, maybe not ... ??pretty unclear to me at the moment ...
??? archimedes's method for picking random point on surface of sphere ... random longitude and random ... ?? inner product with north pole ... ????? ....
?? gauss ... -ian ... ??? ....
?? idea that it's no good if both choices are by longitude or both by inner product with north pole; you need one one way and the other the other; and that screws things up ?? ... ??? ....
?? sas vs asa ... ????
a,b,c
(a.b)(aXb . bXc)(b.c) ???
??? vs ... ???
(aXb . aXc)(a.c)(aXc . bXc) ???
????? .....
(a.b)(c'.a')(b.c) =?= (c'.b')(a.c)(b'.a') ???
???or ... ??? maybe some reciprocals somewhere ... ??? ... ????....
1 a b
a 1 c
b c 1
1 ? ?
? 1 ?
? ? 1
Sunday, June 5, 2011
notes for next discussion with kenji
?? introduce "frame" concept ... ???in part for understanding coset partition as just special case of "coarsening partition" of forgetful projection from richer to poorer features ... ?????.... this partly as motivation for me to get to 4-elt set example ... ??? .... catalog of all feature types ... ???...
??maybe bit about history / etymology of "group" here ... ??? ...
?? idea that conjugation amounts to "straightforward action on graph of permutation" ... ?? but / and also ... ??demonstration via ... "string diagram" and "replacing both input and output symbols ... " ... ??? ....
??maybe bit about history / etymology of "group" here ... ??? ...
?? idea that conjugation amounts to "straightforward action on graph of permutation" ... ?? but / and also ... ??demonstration via ... "string diagram" and "replacing both input and output symbols ... " ... ??? ....
?? ok, so ... we've got some conflicting if vague ideas about what the underlying module of a higher codimension subvariety is typically like ... so it seems like we should (??re- ... ???)visit the example of the ideal in k[x,y] ... as somewhat ultratypical ... ?? ....
??so ... we've got two generators x,y ... but i guess that we should give them different names to try to avoid too much confusion with the original x,y ... say a,b, i guess ...
so then ... a*y = b*x .... and that looks like pretty much it for the relators ... ?? ...
?? so for example let's think about what this says about "fibers" ... ??? ....
?? well, so away from the origin it seems to be saying about what you'd expect, that we're putting in 1 relator, cancelling out one of the 2 generators ... leaving fiber of net dimension 1 ...
?? but at the origin, the relator seems to become trivial ... ???so the fiber seems to be higher(namely 2)-dimensional ... ??? ... ??which i think fits our vague memories, but conflicts with recent idle imagining about ... ideal being trivially braided module ... conflict being that it seems pretty safe that AG morphisms preserve trivially braided objects, whereas a 2d vector space is not trivially braided ... ?? so presumably the idea that trivial-braiding is inherited by submodules (and that ideals would thus be trivially braided) is wrong ... but it would probably be good to see the non-trivial braiding here nice and concretely ... ??? ...
??so ... tensor square ... 4 generators aa,ab,ba,bb ... 4 relators aa y = ba x, ab y = bb x, aa y = ab x, ba y = bb x ... ???? then perhaps we're expecting the cokernel of the switch map here to be skyscraper at the origin ??? ...
??? so ... idea about ... it being interesting to develop doctrine of "symmetric monoidal (??V-)categories where every object is trivially braided" as seeming much iffier now ?? (?? ... ??functorial operations preserving property of being trivially braided ... ??? tensor product ??? .... ?? cokernel ???? .... ??? ??kernel ???? ...... ????? ..... ???? ideal _as_ kernel of map between trivially braided objects??? .... ..... ??? .... ??? being trivially braided as "smallness" condition ??? ....)
?? and also ... ?? idea of universally compelling underlying module of ideal to become line object seems somewhat trickier now, as we probably have to actually consciously think about the aspect of compelling it to become trivially braided, as opposed to merely invertible ... ???? .....
?? vague feeling about ... ?? we were already expecting that "blowing-up of subvariety" relates to doing certain stuff with corresponding ideal ... (?? universally compelling its underlying module to become invertible ... ??? ... "rees ..." .... ???? ...) ... ???but .... ???vague feeling that the underlying module of the ideal already embodies idea of "blowing up" in some sense .... ????? ..... ??? the rees (??? ...) stuff (??which i seem to be trying to shoe-horn in now as ess just usual way of "turning vector space into space (or algebraic counterpart of space)" ... ??) then simply transports the already blown-up situation from living in the module world to living in the commutative algebra world .... ????? ....
??idea that from module perspective, what's being "blown-up" is the unit module, and the inclusion of the ideal into the unit module is the algebraic (=?= contravariant ... ???) manifestation of a geometric "blowing-back-down" comparison map ?? ... ??or is this completely backwards / screwed up ?? ....
??maybe not screwed up ??? .... ???really true that included submodule (= ideal) here really does correspond to something geometrically (co-)bigger ... and the process of bringing this out is ess just the process of looking at the blowing-back-down comparison map contravariantly induced by the submodule inclusion map wrt certain hopefully obvious contravariant functor _module_ -> _relative variety_ ...
?? so ... ??? this seems like a pretty nice, almost lowbrow perspective on blowing-up ... ?? but good to develop those other highbrow perspectives as well ... ?? not sure at the moment how to fit them all together ... ??? .... ?? "universally compelling object to become line object" ... ??? "renormalization" ... ??? having trouble seeing how these relate now ... ?? they don't seem particularly visible yet from this lowbrow perspective... ?? because the lowbrowness is in part in the way it seems like you're just doing something very straightforward, namely inducing something from ideal inclusion ... ??maybe stuff is hidden in ideal inclusion ??? ... ??? .... ???? maybe it's something about ... relationship between "symmetric algebra" construction and line bundles .... ???? ... ??wasn't there some idea ... ????.....
??? "apparent inclusion where domain is actually morally bigger" here vaguely reminds me of other situations, maybe of same kind?? ... ?? "localization" ... ???.... ??any interesting relationships here ???.... (??vaguely imagining something about ... localization ... at vs away from ... ?? "completion" ... ??? ... ?? "filtered colimit" .... ???? ...... jets ... ??? ... ??? ... might be phantoms ... ??? .....) ???localization map as (algebraicially ...) blatantly mono and subtly epi ... ??ideal inclusion as blatantly mono ... ??subtly epi in any way ???? .... well, it's clearly not actually epi, right ?? ... being in abelian category ... (!! look at it's cokernel !! ... ??? ...) ??? .... ????but maybe it does induce epi .... ??????.....
??so ... we've got two generators x,y ... but i guess that we should give them different names to try to avoid too much confusion with the original x,y ... say a,b, i guess ...
so then ... a*y = b*x .... and that looks like pretty much it for the relators ... ?? ...
?? so for example let's think about what this says about "fibers" ... ??? ....
?? well, so away from the origin it seems to be saying about what you'd expect, that we're putting in 1 relator, cancelling out one of the 2 generators ... leaving fiber of net dimension 1 ...
?? but at the origin, the relator seems to become trivial ... ???so the fiber seems to be higher(namely 2)-dimensional ... ??? ... ??which i think fits our vague memories, but conflicts with recent idle imagining about ... ideal being trivially braided module ... conflict being that it seems pretty safe that AG morphisms preserve trivially braided objects, whereas a 2d vector space is not trivially braided ... ?? so presumably the idea that trivial-braiding is inherited by submodules (and that ideals would thus be trivially braided) is wrong ... but it would probably be good to see the non-trivial braiding here nice and concretely ... ??? ...
??so ... tensor square ... 4 generators aa,ab,ba,bb ... 4 relators aa y = ba x, ab y = bb x, aa y = ab x, ba y = bb x ... ???? then perhaps we're expecting the cokernel of the switch map here to be skyscraper at the origin ??? ...
??? so ... idea about ... it being interesting to develop doctrine of "symmetric monoidal (??V-)categories where every object is trivially braided" as seeming much iffier now ?? (?? ... ??functorial operations preserving property of being trivially braided ... ??? tensor product ??? .... ?? cokernel ???? .... ??? ??kernel ???? ...... ????? ..... ???? ideal _as_ kernel of map between trivially braided objects??? .... ..... ??? .... ??? being trivially braided as "smallness" condition ??? ....)
?? and also ... ?? idea of universally compelling underlying module of ideal to become line object seems somewhat trickier now, as we probably have to actually consciously think about the aspect of compelling it to become trivially braided, as opposed to merely invertible ... ???? .....
?? vague feeling about ... ?? we were already expecting that "blowing-up of subvariety" relates to doing certain stuff with corresponding ideal ... (?? universally compelling its underlying module to become invertible ... ??? ... "rees ..." .... ???? ...) ... ???but .... ???vague feeling that the underlying module of the ideal already embodies idea of "blowing up" in some sense .... ????? ..... ??? the rees (??? ...) stuff (??which i seem to be trying to shoe-horn in now as ess just usual way of "turning vector space into space (or algebraic counterpart of space)" ... ??) then simply transports the already blown-up situation from living in the module world to living in the commutative algebra world .... ????? ....
??idea that from module perspective, what's being "blown-up" is the unit module, and the inclusion of the ideal into the unit module is the algebraic (=?= contravariant ... ???) manifestation of a geometric "blowing-back-down" comparison map ?? ... ??or is this completely backwards / screwed up ?? ....
??maybe not screwed up ??? .... ???really true that included submodule (= ideal) here really does correspond to something geometrically (co-)bigger ... and the process of bringing this out is ess just the process of looking at the blowing-back-down comparison map contravariantly induced by the submodule inclusion map wrt certain hopefully obvious contravariant functor _module_ -> _relative variety_ ...
?? so ... ??? this seems like a pretty nice, almost lowbrow perspective on blowing-up ... ?? but good to develop those other highbrow perspectives as well ... ?? not sure at the moment how to fit them all together ... ??? .... ?? "universally compelling object to become line object" ... ??? "renormalization" ... ??? having trouble seeing how these relate now ... ?? they don't seem particularly visible yet from this lowbrow perspective... ?? because the lowbrowness is in part in the way it seems like you're just doing something very straightforward, namely inducing something from ideal inclusion ... ??maybe stuff is hidden in ideal inclusion ??? ... ??? .... ???? maybe it's something about ... relationship between "symmetric algebra" construction and line bundles .... ???? ... ??wasn't there some idea ... ????.....
??? "apparent inclusion where domain is actually morally bigger" here vaguely reminds me of other situations, maybe of same kind?? ... ?? "localization" ... ???.... ??any interesting relationships here ???.... (??vaguely imagining something about ... localization ... at vs away from ... ?? "completion" ... ??? ... ?? "filtered colimit" .... ???? ...... jets ... ??? ... ??? ... might be phantoms ... ??? .....) ???localization map as (algebraicially ...) blatantly mono and subtly epi ... ??ideal inclusion as blatantly mono ... ??subtly epi in any way ???? .... well, it's clearly not actually epi, right ?? ... being in abelian category ... (!! look at it's cokernel !! ... ??? ...) ??? .... ????but maybe it does induce epi .... ??????.....
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