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

Abstract

Let $\mathbb{E}$ be a quadratic extension of a number field $\mathbb{F}$. Let $E(g,s)$ be an Eisenstein series on $G{L}_2(\mathbb{E})$, and let $F$ be a cuspidal automorphic form on $G{L}_2(\mathbb{F})$. We will consider in this paper the following automorphic integral: $${\int }_{Z_{A}G{L}_2(\mathbb{F})\backslash G{L}_2({\mathbb{A}}_{\mathbb{F}})}F(g)E(g,s)dg.$$ 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.

arXiv commentThesis

Cited by in corpus (2)

• The subconvexity bound for triple product L-function in level aspect
• Triple product formula and mass equidistribution on modular curves of level N