?? ... trying to figure out what pattern we think we're following here ... ???...

algebraic group gl(1) ... realized over finite field ... as actual abelian group ... acting on pth roots of unity ??? .... ???? ??? how ???? ....

???try to imitate for algebraic group given by elliptic curve with complex multiplication ... ???? ..... ????


??? "formal group law" ... ???? ....

?? "complex cobordism" ... ?? ... "orientation class" ... ???? ..... ?? "elliptic cohomology" ... ??? .....


?? "tradeoff between time-independence and space-independence, by considering inverse of solution-candidate map time -> space" ... ???maybe i should say "domain-independence and codomain-independence ... " ... ??? ...

vector field as infinitesimal generator of one-parameter group ... ???? ...

"elliptic integral" = antiderivative (locally ... ?? ...) of function x |-> 1/y where y^2 = x^3 - x (say for example ...) ... ???is that right ??? ....

(pun on "inverse" .... ???? ....)

"elliptic function" = inverse function of that ...



"one-parameter group" f : time -> space .... group homomorphism ... solution to domain-independent de .... f'(t) = g(f(t)) where [g(s)]^2 = s^3 - s ... ????....

h = f^[-1] ... solution to codomain-independent de h'(s) = 1/[g(s)] .... h'(s) = anti-derivative .... ?? of function s |-> 1/[g(s)] ... ??? ....

??? t |-> (f(t),f'(t)) .... ??? "formal group law" on first coordinate dimension alone, vs elliptic curve group structure on variety combining both coordinate dimensions ... ???..... ???? .....


???trying to get elliptic curve with complex multiplication (??corresponding to imaginary quadratic number field k), manifested as specific abelian group over finite k-field f (???? .... ???? .....) to act on values of elliptic function at .... ????? ......