Number Theory Seminar
Let F be a cuspidal Hida family, and let F_k run over the arithmetic specializations of F of even weight. A deep conjecture of Greenberg predicts that, expect for finitely many k, the cyclotomic p-adic L-function attached to F_k should vanish to order exactly 0 or 1 at the center, depending on the generic sign of F_k. When this sign is +1, by the work of Skinner-Urban the non-vanishing of central values predicted by Greenberg's conjecture follows from the torsion-ness of a certain Selmer group attached to F. In this talk, I'll describe an analogous result when the generic sign is -1. Based on a joint work with Xin Wan.