Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Invariant four-variable automorphic kernel functions   Let $F$ be a number field, let $\mathbb{A}_F$ be its ring of adeles, and let
$g_1,g_2,h_1,h_2 \in \mathrm{GL}_2(\mathbb{A}_F)$. Previously the author
provided an absolutely convergent geometric expression for the four variable
kernel function $$ \sum_{\pi} K_{\pi}(g_1,g_2)K_{\pi^{\vee}}(h_1,h_2)L(s,(\pi
\times \pi^{\vee})^S), $$ where the sum is over isomorphism classes of cuspidal
automorphic representations $\pi$ of $\mathrm{GL}_2(\mathbb{A}_F)$. Here
$K_{\pi}$ is the typical kernel function representing the action of a test
function on the space of the cuspidal automorphic representation $\pi$. In this
paper we show how to use ideas from the circle method to provide an alternate
expansion for the four variable kernel function that is visibly invariant under
the natural action of $\mathrm{GL}_2(F) \times \mathrm{GL}_2(F)$.

27
The categories of this paper are: comp-gas
4
The categories of this paper are: cond-mat
26
The categories of this paper are: bayes-an
37
The categories of this paper are: supr-con
25
The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
8
The categories of this paper are: hep-ex
3
The categories of this paper are: physics
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
10
The categories of this paper are: q-bio
31
The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org
34
The categories of this paper are: chem-ph
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg