?? "elliptic curve with complex multiplication" as "elliptic curve on which it makes good sense to draw pictures of quotient rings of the ring of integers in the corresponding imaginary quadratic number field" ..... ?????

?? "makes sense to draw pictures of quotient rings of Z in/on gl(1)" ???? .....

??? "orders" in number fields ????

?? for an elliptic curve with complex multiplication we have the option of getting from its p-torsion points a representation of the absolute abelian galois group of the corresponding imaginary quadratic number field, or a representation of the non-abelian extension including also complex conjugation ... ??? .... ?? while for one without complex multiplication, don't seem to have such option; no number field in plain view so only latter alternative seems available ... ??? ..... in the complex multiplication case, choosing the "abelian" alternative, seems like two perspectives on .... ???? 1d-ness of rep .... ??? first, as just consequence (??...) of abelianness, vs second, as linked with rep living over the imaginary quadratic field, or over _some_ thing connected with that field ... ??? adeles or something ???? ....


?? golden elliptic curve ... ??????

?? elliptic curve one hecke operator away from having complex multiplication .... ???? ......

???? frobenius .... ???? relationship between this idea that we seem to be groping towards now of "making quotient field (?? ... or more generally ring ?? .... ) of integer ring of field k into representation of absolute abelian galois group of k ... " and .... "frobenius ..." idea .... ????? ...... ??????? .......... ????? .....

??? "frobenius ... " in connection with groupoid cardinality interpretation of zeta functions ....

??? idea about ... ??? interaction between source and target galois interactions on homomorphisms from number rings to finite fields .... .... ??? "k-fields" ???? .... schematic / intuitive picture in terms of "map between fundamental groups induced by map between spaces ... " .... ?? "ramification" .... ????

??? "frobenius ..." in "non-abelian" case .... ?????.....

??? any interesting overlap between "complex multiplication" and "being a jacobian variety" (??? ...) in higher genus case ??? ... ???....

??? are we claiming that you should think of points on elliptic curve with complex multiplication over finite field as having ring structure ???? .... ????? .... ??? but .... ??? non-square numbers here ... ??? ... ??? maybe ok ?? ....

?? gl(1,Z/m) vs ... ?? Z/n .... ???? .... ????? ....