# Automorphic kernel functions in four variables

Research paper by **Jayce R. Getz**

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

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### 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
$\pi$.