??? all right, i think that i've got an emerging conjectural picture here ... that i should try to write down before i start forgetting it and/or the conjectures start falling part ...

?? some parts of the picture may be wrong ... other parts seem so obvious ... as conjectures to make if not as statements to prove ... that it seems annoying and strange for me not to have thought about them before ... ?? might even turn out that i _have_ thought about some of them before ...

?? might also be some overlap between above classes (wrong, obvious, thought about before ...) ... ???.....


?? and as usual i'm having some trouble figuring out where to start in trying to write this down... so will try some random places ... ???....

??? so there's the field of all "cyclotomic numbers" .... ???? .... which is the absolute abelian extension of the rationals ... which we're supposed to know some stuff about ...

??? then there's also, for any given elliptic curve x, the field of all "x-tomic numbers" ... ??? or at least, i hope that this makes some sense ... ??? .... ?? ... i also have the feeling that there's some conventional terminology here that i'm struggling to remember ... having to do with this idea of "dividing up into equal parts ..." that we're imagining doing to an algebraic group of some sort .... ???? ..... ??anyway, i'm hoping that it will eventually make some sense to think about this alleged big field f_x generated by the coordinates (?? in some sense ...) of "equal-division points" of x ... ???? and then i'm imagining that maybe in the special case where x "has complex multiplication", f_x will be the absolute abelian extension of the imaginary quadratic number field corresponding to x ...

(?? what am i claiming, that f_x = f_y when x and y are ..... ????? ...... ???? ....)

(?? what am i claiming about what sort of number the value of elliptic modular function at x is .... ??? .... ???? analogy of elliptic modular function to exponential function here ??? ..... ???? ..... ???? ..... ??? ....)

?? and then ... ??? maybe this "complex multiplication" case is supposed to serve as a stepping-stone to more general arbitrary elliptic curve case ... ?? where more complicated but perhaps eventually very interesting stuff may happen ... ??? f_x in this case as maybe not abelian extension in any very interesting way ?? ... ??? but interesting as non-abelian extension ... ???? ..... ???? ... no idea, really; just taking stupid guesses here ... ???? ....

??moduli stacks of "line bundles with extra local structure" ... ??? .... ??? ... "jacobian ..." ... ??? ..... ???? ..... ????..... ???something "diaconescuesque" going on here ??? ....

??? ... "descent" ... "descent" ... ???? ....