?? various attempted analogies between various groups and "one-parameter groups" and so forth ... in connection with jugendtraum, in part ... ?? possibly with some level slips ... ????....

gl(1) and cyclotomy ... p-torsion points on gl(1) ... ???? as forming 1d vector space over f_p .... ??? thus something about gl(1,f_p) showing up, though ... ?? not sure to what extent those two occurrences of "gl(1)" are "coincidence" ... ???? ....

?? then ... elliptic curve with complex multiplication .... p-torsion points on it ... as forming 2d vector space over f_p ... ??so sort of looks like gl(2,f_p) is involved here??? ..... ??? but then maybe ... ???actually something about ... ????algebraic subgroup of gl(2) ??? ..... ???? given by "multiplicative group of imaginary quadratic numbner field..." .... ????? (a,b) * (c,d) = ... ???multiply like (a+bx)*(c+dx) with x satisfying irreducible quadratic .... ????? ....

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

???i was going to say, go back and re-consider that idea we mentioned ... something about elliptic curve over adeles .... ????? ......

?? but .... ??? not sure that's the right way to say it ... ????? ...... ???? ....

??? look at elliptic curve over f_p ... ???compare to ... mult gp of imaginary quadratic number ring over f_p .... ????? ...... ?????? ......


?? was also going to say something about ... automorphism group of elliptic curve ... ??? ...



?? ... one-parameter groups .... "circle group" vs "gl(1)" ..... ??which leads me to other idea that i was going to put in completely separate post .... ???? ..... "weil restriction" of gl(1) from C to R .... ?? relationship to stuff people say / write about "d z-bar" where d is some typographically funny version of d and z-bar is supposed to be complex conjugate .... ???? ..... ??? "hodge ... " .... ???? ....