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 $$p \not \mid N$$ 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 $$F \rightarrow \infty $$ with $$(N,F)=1$$ and $$p\not \mid F\phi (F)$$ . 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 $${{\mathbb {Z}}}_{\ell }$$ -extension of $${{\mathbb {Q}}}$$ for an odd prime $$\ell \ne p$$ with the corresponding critical L-value twists being p-indivisible.