Hecke $L$-values, definite Shimura sets and Mod $\ell$ non-vanishing

Abstract

Let $\hat{I}»$ be a self-dual Hecke character over an imaginary quadratic field $K$ of infinity type $(1,0)$. Let $\ell$ and $p$ be primes which are coprime to $6N_{K/\mathbb{Q}}(\mathrm{c}ond(\hat{I}»))$. We determine the $\ell$-adic valuation of Hecke $L$-values $L(1,\hat{I}»\ddot{I}‡)/\hat{I}{\text{c◯}}_K$ as $\ddot{I}‡$ varies over $p$-power order anticyclotomic characters over $K$. As an application, for $p$ inert in $K$, we prove the vanishing of the $\hat{I}¼$-invariant of Rubin's $p$-adic $L$-function, leading to the first results on the $\hat{I}¼$-invariant of imaginary quadratic fields at non-split primes. Our approach and results complement the work of Hida and Finis. The approach is rooted in the arithmetic of a CM form on a definite Shimura set.The application to Rubin's $p$-adic $L$-function also relies on the proof of his conjecture. Along the way, we present an automorphic view on Rubin's theory.

71 pages