On a Generalization of Motohashi's Formula: Non-archimedean Weight Functions

TL;DR

This paper extends Motohashi's formula to non-archimedean settings, providing bounds on weight functions for PGL groups and explaining the emergence of Katz's hypergeometric sums in related contexts.

Contribution

It generalizes Kwan's adelic formula to non-archimedean places and applies to any tempered PGL_3 representation, revealing structural reasons for hypergeometric sums.

Findings

Bounded weight functions at non-archimedean places for PGL groups

Applicable to any tempered PGL_3 representation

Explains the appearance of Katz's hypergeometric sums

Abstract

This is a continuation of the adelic version of Kwan's formula. At non-archimedean places we give a bound of the weight function on the mixed moment side, when the weight function on the $\mathrm{PGL}_3 \times \mathrm{PGL}_2$ side is nearly the characteristic function of a short family. Our method works for any tempered representation $\Pi$ of $\mathrm{PGL}_3$, and reveals the structural reason for the appearance of Katz's hypergeometric sums in a previous joint work with P.Xi.