Mod $\ell$ non-vanishing of self-dual Hecke $L$-values over CM fields and applications

TL;DR

This paper proves the non-vanishing of certain self-dual Hecke L-values over CM fields for almost all characters, determines their $ ext{l}$-adic valuations, and applies these results to advance the CM Iwasawa main conjecture.

Contribution

It establishes the finiteness of characters with vanishing L-values, determines $ ext{l}$-adic valuations of normalized L-values, and completes Hsieh's proof of Eisenstein congruence divisibility.

Findings

Non-vanishing of $L(1,\lambda u)$ for all but finitely many characters $ u$

Explicit determination of $ ext{l}$-adic valuations of normalized L-values

Completion of Hsieh's proof of Eisenstein congruence divisibility

Abstract

Let $\lambda$ be a self-dual Hecke character over a CM field $K$. Let $\mathfrak{p}$ be a degree one prime of the maximal totally real subfield $F$ of $K$ and $\Gamma_{\mathfrak{p}}$ the Galois group of the anticyclotomic $\mathbb{Z}_p$-extension of $K$ unramified outside $\mathfrak{p}$. We prove that $$L(1,\lambda\nu)\neq 0$$ for all but finitely many finite order characters $\nu$ of $\Gamma_\mathfrak{p}$ such that $\varepsilon(\lambda\nu)=+1$. For an ordinary prime $\ell$ with respect to the CM quadratic extension $K/F$, we also determine the $\ell$-adic valuation of the normalised Hecke $L$-values $L^{alg}(1,\lambda\nu)$. As an application, we complete Hsieh's proof of Eisenstein congruence divisibility towards the CM Iwasawa main conjecture over $K$. Our approach and results complement the prior work initiated by Hida's ideas on the arithmetic of Hilbert modular Eisenstein series, studied via mod $\ell$ analogue of the Andr\'e--Oort conjecture. The previous results established the non-vanishing only for infinitely many characters $\nu$. Our approach is based on the arithmetic of a CM modular form on a Shimura set, studied via arithmetic of the CM field and Ratner's ergodicity of unipotent flows.