Mod ℓ non-vanishing of self-dual Hecke L-values over CM fields and applications

2025/08/27 by Burungale, Ashay, He, Wei, Tian, Ye +1
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.19706

Abstract

Let λ be a self-dual Hecke character over a CM field K. Let \mathfrakp be a prime of the maximal totally real subfield F of K and Γ_\mathfrakp the Galois group of the maximal anticyclotomic ℤ_p^deg \mathfrakp-extension of K unramified outside \mathfrakp. We prove that L(1,λν)≠ 0 for all but finitely many finite order characters ν of Γ_\mathfrakp such that ε(λν)=+1. For an ordinary prime ℓ with respect to the CM quadratic extension K/F, we also determine the ℓ-adic valuation of the normalised Hecke L-values L^alg(1,λν). 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 ℓ analogue of the André--Oort conjecture. The previous results established the non-vanishing only for a Zariski dense subset of characters ν. 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.

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• Non-vanishing of Hecke L-values modulo p