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

2013/09/28 by Yueke Hu, Hu, Yueke
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1309.7467

Thesis

openalex publication_date 2013/09/28 · arxiv created 2013/11/12 · arxiv updated 2013/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let 𝔼 be a quadratic extension of a number field \mathbbF. Let E(g, s) be an Eisenstein series on GL₂(𝔼), and let F be a cuspidal automorphic form on GL₂(\mathbbF). We will consider in this paper the following automorphic integral: ∫_Z_AGL₂(\mathbbF)\backslash GL₂(\mathbbA_\mathbbF) 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.

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