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