?? since we seem to be dealing with abelian extensions of imaginary quadratic number fields rather than of Q ... well, i guess it's not as though we never had any idea about that before .... but nevertheless ... ??? ideas associated with this .... ??? .... ?? instead of just single zeta function or single l-function giving information only about splitting behavior over Z-prime, getting complex of them giving finer information, about splitting behavior over primes of higher base.... ???? ......

??zeta function vs l-functions .... p-torsion galois representation .... ??extent to which get full information about whole maximal abelian extension .... ????

???isogenous elliptic curves .... p-torsion galois representations for each .... ?? putting together information coming from all of them .... ???? ......

??? level slips here .... base vs total .... ???? ..... ??? ... ??fiber ... ???? ....


???solutions over number fields vs solutions over finite fields ... ?? interplay ... ??? ... for varieties of various dimensions .... ???? ...... ?? geometric dimensions vs arithmetic dimension .... ???? ....