Cuspidal part of an Eisenstein series restricted to an index 2 subfield

TL;DR

This paper studies a specific automorphic integral involving Eisenstein series and cuspidal forms over quadratic extensions, using Waldspurger's formula to analyze when it vanishes and to compute associated L-functions.

Contribution

It introduces a new automorphic integral related to Eisenstein series over quadratic fields and computes local integrals at ramified places with arbitrary level of ramification.

Findings

Identifies conditions under which the integral vanishes.

Expresses the integral in terms of L-functions.

Calculates local integrals at ramified places with high ramification levels.

Abstract

Let $\mathbb{E}$ be a quadratic extension of a number field $\mathbb{F}$. Let $E(g, s)$ be an Eisenstein series on $GL_2(\mathbb{E})$, and let $F$ be a cuspidal automorphic form on $GL_2(\mathbb{F})$. We will consider in this paper the following automorphic integral: $$\int_{Z_{A}GL_{2}(\mathbb{F})\backslash GL_{2}(\mathbb{A}_{\mathbb{F}})} F(g)E(g,s) dg.$$ This is in some sense the complementary case to the well-known Rankin-Selberg integral and the triple product formula. We will approach this integral by Waldspurger's formula. We will discuss when the integral is automatically zero, and otherwise the L-function it represents. We will calculate local integrals at some ramified places, where the level of the ramification can be arbitrarily large.