Friday, November 21, 2008

Modular degrees, congruence moduli, and the Heegner-Kolyvagin subgroup of Mordell-Weil

Today in the advanced number theory reading seminar, Robert Bradshaw talked about the algorithm of Delaunay-Watkins for computing modular degrees using L-functions. In particular, he talked about some of the background definitions involving the symmetric square L-function and the Petersson norm. Craig Citro (in the audience) kept bringing up the adjoint L-function, since it is very relevant to his Ph.D. thesis work. There is a formula for L(Sym^2(f),2) that involves the modular degree, and there is a formula that involves L(Ad(f),1) and the congruence modulus. There's a paper of Agashe-Ribet-Stein (me) that proves that the modular degree m divides the congruence modulus c for elliptic curves, and we conjecture that 2*ord_p(c/m) <= ord_p(N), where N is the conductor of the curve. Ken Ribet in fact proved this when ord_p(N) <= 1, but none of us have made any progress when ord_p(N) >= 2. I'm very curious if rephrasing these divisibilities as relations about L-functions yields any new insight either about L-functions or the conjecture? Or nothing? I have no idea yet.

I also thought some about various ways of defining subgroups of Mordell-Weil groups of elliptic curves of arbitrary rank. The result is that now I have I think three a priori completely different definitions of a subgroup V of E(K), where K is a Heegner quadratic imaginary field, and I suspect that all three definitions give the same group. The first definition is the subgroup of E(K) that is in the kernel of all the maps to component groups and to the dual of all groups Sha(E/K)(p^oo), where dual is defined by lifting to the Selmer group and using the cup product. The second definition is the intersection of the sums of the inverse images of all subgroups of E(K_lambda)/M E(K_lambda) generated by Kolyvagins points P_lambda. The third subgroup is the inverse image of the subgroups of J_0(N*lambda)(K)/M generated by Heegner points x_K in J_0(N*lambda)(K).

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