Vanishing of $L$-functions and ranks of Selmer groups

Abstract

This paper connects the vanishing at the central critical value of the L-functions of certain polarized regular motives with the positivity of the rank of the associated p-adic (Bloch–Kato) Selmer groups. For the motives studied it is shown that vanishing of the L-value implies positivity of the rank of the Selmer group. It is further shown that if the order of vanishing is positive and even then the Selmer group has rank at least two. The proofs make extensive use of families of p-adic modular forms. Additionally, the proofs assume the existence of Galois representations associated to holomorphic eigenforms on unitary groups over an imaginary quadratic field.