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