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

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 ${\mathrm{\Gamma }}_{\mathfrak{p}}$ the Galois group of the anticyclotomic ${\mathbb{Z}}_p$-extension of $K$ unramified outside $\mathfrak{p}$. We prove that $$L(1,\lambda \nu )\mathrm{\ne }0$$ for all but finitely many finite order characters $\nu$ of ${\mathrm{\Gamma }}_{\mathfrak{p}}$ such that $\epsilon (\lambda \nu )=+1$. For an ordinary prime $\mathrm{\ell }$ with respect to the CM quadratic extension $K\mathrm{/}F$, we also determine the $\mathrm{\ell }$-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 $\mathrm{\ell }$ analogue of the André–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.