Abstract:
If K/Q is abelian, E/Q is an elliptic curve, and \chi is a primitive non-trivial Dirichlet character factoring through K, then Dokchiters-Evans-Wiersema attempted to obtain a BSD-type formula for the special value of L(E,\chi, s) at s=1. In particular, they extract an ``algebraic part'' of L(E,\chi,1) (say L that is an algebraic integer) and show its connection with \Sha(E/K)/\Sha(E/Q) assuming certain conjectures. They asked a question if the ideal factorization of the ideal (L) can be recovered from the action of Gal(K/Q) on \Sha(E/K)/\Sha(E/Q).
In this joint work with Celine Maistret (University of Bristol) we provide evidence for a positive answer to this question by developing a method to perform 11-descent on certain elliptic curves with complex multiplication over a quintic extension K/Q.