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.