Invariant four variable automorphic kernel functions
Let F be a number field, let AF be its ring of adeles, and let g1,g2,h1,h2∈GL2(AF). Previously the author provided an absolutely convergent geometric expression for the four variable kernel function
where the sum is over isomorphism classes of cuspidal automorphic representations π of GL2(AF). Here Kπ is the typical kernel function representing the action of a test function on the space of the cuspidal automorphic representation π. 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 GL2(F)×GL2(F).