Quantitative non-vanishing of Dirichlet L-values modulo p

Abstract

Let p be an odd prime and k a non-negative integer. Let N be a positive integer such that p inverted iota N and lambda a Dirichlet character modulo N. We obtain quantitative lower bounds for the number of Dirichlet character chi modulo F with the critical Dirichlet L-value L(-k, lambda chi) being p-indivisible. Here F -> infinity with ( N, F) = 1 and p inverted iota F phi(F). We explore the indivisibility via an algebraic and a homological approach. The latter leads to a bound of the form F-1/2. The p-indivisibility yields results on the distribution of the associated p-Selmer ranks. We also consider an Iwasawa variant. It leads to an explicit upper bound on the lowest conductor of the characters factoring through the Iwasawa Z(l)-extension of Q for an odd prime l not equal p with the corresponding critical L-value twists being p-indivisible.