Thursday, June 23, 2011

?? 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" ??? .....

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