# Automorphic kernel functions in four variables

Research paper by Jayce R. Getz

Indexed on: 17 Mar '16Published on: 17 Mar '16Published in: Mathematics - Number Theory

#### Abstract

Let $F$ be a number field, let $f_1,f_2 \in C_c^\infty(A \backslash \mathrm{GL}_2(\mathbb{A}_F))$, and let $g_1,g_2,h_1,h_2 \in \mathrm{GL}_2(\mathbb{A}_F)$. We provide an absolutely convergent geometric expression for \begin{align*} \sum_{\pi} K_{\pi(f_1)}(g_1,g_2)K_{\pi^{\vee}(f_2)}(h_1,h_2), \end{align*} where the sum is over isomorphism classes of cuspidal automorphic representations $\pi$ of $A \backslash \mathrm{GL}_2(\mathbb{A}_F)$. Here $K_{\pi(f)}$ is the typical kernel function representing the action of a test function $f$ on the space of $\pi$.