Special Values of Anticyclotomic L-functions Modulo \lambda

TL;DR

This paper extends results on the special values of Rankin-Selberg L-functions in anticyclotomic extensions, focusing on their valuations modulo a prime, with implications for understanding L-functions in number theory.

Contribution

It generalizes Vatsal's results by analyzing the $l$-adic valuation of special L-values for Hilbert modular forms over totally real fields in anticyclotomic extensions.

Findings

Proves bounds on $l$-adic valuations of special L-values.

Establishes conditions under which valuations are controlled.

Extends previous work to more general settings.

Abstract

The purpose of this article is to generalize some results of Vatsal on studying the special values of Rankin-Selberg L-functions in an anticyclotomic $\mathbb{Z}_{p}$-extension. Let $g$ be a cuspidal Hilbert modular form of parallel weight (2,...,2) and level $\mathcal{N}$ over a totally real field F, and let K/F be a totally imaginary quadratic extension of relative discriminant $\mathcal{D}$. We study the $l$-adic valuation of the special values $L(g,\chi,\frac{1}{2})$ as \chi varies over the ring class characters of K of $\mathcal{P}$-power conductor, for some fixed prime ideal $\mathcal{P}$. We prove our results under the only assumption that the prime to $\mathcal{P}$ part of $\mathcal{N}$ is relatively prime to $\mathcal{D}$.