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