Some mixed character sum identities of Katz II

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.

Certain product formulas and values of Gaussian hypergeometric series

“…Greene [13] found several transformation formulas satisfied by the Gaussian hypergeometric series analogous to those satisfied by the classical hypergeometric series. Since then many mathematicians have obtained finite field analogues of transformation and summation identities satisfied by the classical hypergeometric series (see for example [6,7,9,10,11,18,22]). Finite field hypergeometric series are known to be related to various arithmetic objects.…”

Some mixed character sum identities of Katz

“…In the case q ≡ 3 (mod 4), the sums V (j) have a more complex definition, in that they are sums over F q 2 [9, p. 228]. A direct proof of (1.4) for the case q ≡ 3 (mod 4) has been given by Evans and Greene [6].…”