Some mixed character sum identities of Katz II

TL;DR

This paper provides a direct proof of certain mixed character sum identities originally discovered by Katz, applicable for all characteristics p > 2, and introduces new character sum identities.

Contribution

It offers a straightforward proof of Katz's identities for all cases p > 2, expanding understanding of character sums with new identities.

Findings

Proof of Katz's identities for q ≡ 3 (mod 4)

New elegant character sum identities

Valid for all characteristics p > 2

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. The first author recently gave such a proof of his identities when q = 1 (mod 4), and this paper provides such a proof for the remaining case q = 3 (mod 4). Our proofs are valid for all characteristics p > 2. Along the way we prove some elegant new character sum identities.