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

Abstract

Let $${\mathbb {E}}$$ be a quadratic algebra over 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: $$\begin{aligned} \int \limits _{Z_{{\mathbb {A}}}{\text {GL}}_{2}({\mathbb {F}})\backslash {\text {GL}}_{2}({\mathbb {A}}_{{\mathbb {F}}})} F(g)E(g,s) dg. \end{aligned}$$ 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, giving a criterion about when the integral is automatically zero, and otherwise the L-functions it represents. We will also calculate the local integrals at some ramified places, where the level of the ramification can be arbitrarily large.