On $L$-functions of Hecke characters and anticyclotomic towers

TL;DR

This paper investigates the behavior of Hecke L-functions associated with anticyclotomic twists of a fixed Hecke character over an imaginary quadratic field, establishing their vanishing order is typically 0 or 1 depending on the root number.

Contribution

It generalizes Rohrlich's work by analyzing the vanishing order of Hecke L-functions for a broad class of anticyclotomic twists with prescribed ramification.

Findings

Vanishing order of L(s, χ) is 0 or 1 for almost all twists.

The vanishing order depends on the root number W(χ).

Results extend understanding of L-functions in anticyclotomic towers.

Abstract

In this paper, we generalize a work of Rohrlich. Let $K/\mathbb{Q}$ be an imaginary quadratic field and $\phi$ be a Hecke character of $K$ of infinite type (1,0) whose restriction to $\mathbb{Q}$ is the quadratic character corresponding to $K/\mathbb{Q}$. We consider a class of Hecke characters $\chi$, which are anticyclotomic twists of $\phi$ with ramification in a prescribed finite set of primes. We shall prove the central vanishing order of the Hecke $L$-function $L(s,\chi$) attached to each $\chi$ is 0 or 1 depending on the root number $W(\chi)$ for all but finitely many such $\chi$.