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

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 ${\mathrm{P}\mathrm{G}\mathrm{L}}_3\times {\mathrm{P}\mathrm{G}\mathrm{L}}_2$ side is nearly the characteristic function of a short family. Our method works for any tempered representation $\mathrm{\Pi }$ of ${\mathrm{P}\mathrm{G}\mathrm{L}}_3$, and reveals the structural reason for the appearance of Katz’s hypergeometric sums in a previous joint work with P.Xi.