Some mixed character sum identities of Katz

TL;DR

This paper provides a direct proof for mixed character sum identities originally discovered by Katz, extending the validity to all characteristics greater than 2 and completing the proof for all cases.

Contribution

The paper offers a straightforward proof of Katz's mixed character sum identities for all prime characteristics greater than 2, covering all residue classes mod 4.

Findings

Identities hold for all prime characteristics p > 2

Proof completed for the case q ≡ 1 mod 4

Extended validity of Katz's identities to all cases

Abstract

A conjecture connected with quantum physics led N. Katz to discover some amazing mixed character sum identities over a field of $q$ elements, where $q$ is a power of a prime $p>3$. His proof required deep algebro-geometric techniques, and he expressed interest in finding a more straightforward direct proof. Such a proof has been given by Evans and Greene in the case $q \equiv 3 \pmod 4$, and in this paper we give a proof for the remaining case $q \equiv 1 \pmod 4$. Moreover, we show that the identities are valid for all characteristics $p >2$.