# Non-vanishing modulo $p$ of Hecke $L$-values over imaginary quadratic fields

## TL;DR

This paper proves that under certain conditions, the algebraic parts of Hecke $L$-values over imaginary quadratic fields do not vanish modulo $p$ for a dense set of characters, revealing stability in their $p$-adic valuations.

## Contribution

It establishes non-vanishing modulo $p$ of Hecke $L$-values over imaginary quadratic fields for a Zariski dense set of characters, extending understanding of their $p$-adic properties.

## Key findings

- $p$-adic valuation of $L$-values is constant for a dense set of characters.
- When $j=0$, the valuation is zero under certain hypotheses.
- Results imply non-vanishing modulo $p$ for a broad class of Hecke $L$-values.

## Abstract

Let $p$ and $q$ be two distinct odd primes. Let $K$ be an imaginary quadratic field over which $p$ and $q$ are both split. Let $\Psi$ be a Hecke character over $K$ of infinity type $(k,j)$ with $0\le-j< k$. Under certain technical hypotheses, we show that for a Zariski dense set of finite-order characters $\kappa$ over $K$ which factor through the $\mathbb{Z}_q^2$-extension of $K$, the $p$-adic valuation of the algebraic part of the $L$-value $L(\overline{\kappa\Psi},k+j)$ is a constant independent of $\kappa$. In addition, when $j=0$ and certain technical hypothesis holds, this constant is zero.