?? 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" ??? ....

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 ?? ... ???....

?? 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" .... ????? .....

?? 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 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 .... ???? .....

?? 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" ?????? ...... ????? ....

?? 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 .... ???? ....

?? 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 ??? .....

?? 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" ??? .....

?? 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 ... ??? .....

?? 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 ... ???? .....)

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

?? 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" .... ???? .... ??? .....

?? .... 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 .... ???? ....

?? 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 ??? .... ????? .....

?? 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" ... ??? ....

?? vague memory ... ?? fermat (??) ... dismissive of easy results following "directly from congruences considerations" .... ???? relationship to .... ??? "congruences as only seeing abelian aspect" (?? in some sense ??? ....) ??? ... ???? .... ??? ....

?? 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 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 .... ???? .....

?? structure type "semi-simple commutative ring structure" ... ?? whether after linearization (?? ...) might be equivalent to some other .... ??? ....

?? suggestion from todd about joyal using coarse, "unnatural" equivalence ... ??? ....