Anticyclotomic p-adic L-functions and Ichino's formula

TL;DR

This paper introduces a new method for constructing anticyclotomic p-adic L-functions for modular forms using Ichino's formula, enabling better analysis of congruences and residual reducibility.

Contribution

It provides a novel p-adic L-function construction leveraging Ichino's triple product formula and Hida's interpolation, with potential applications to congruences and residual reducibility.

Findings

Construction is compatible with congruences of modular forms.

Framework facilitates studying residual reducibility cases.

Method enhances understanding of p-adic L-functions in anticyclotomic settings.

Abstract

We give a new construction of a $p$-adic $L$-function $\mathcal{L}(f,\Xi)$, for $f$ a holomorphic newform and $\Xi$ an anticyclotomic family of Hecke characters of $\mathbb{Q}(\sqrt{-d})$. The construction uses Ichino's triple product formula to express the central values of $L(f,\xi,s)$ in terms of Petersson inner products, and then uses results of Hida to interpolate them. The resulting construction is well-suited for studying what happens when $f$ is replaced by a modular form congruent to it modulo $p$, and has future applications in the case where $f$ is residually reducible.