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∤N and λ a Dirichlet character modulo N. We obtain quantitative lower bounds for the number of Dirichlet character χ modulo F with the critical Dirichlet L-value L(−k,λ_χ) being p-indivisible. Here F→∞ with (N,F)=1 and p∤Fϕ(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≠p with the corresponding critical L-value twists being p-indivisible.

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