Some mixed character sum identities of Katz II

摘要

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 $$p 3$$ 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 $$q equiv 1 pmod 4$$ q ≡ 1 ( mod 4 ) , and this paper provides such a proof for the remaining case $$q equiv 3 pmod 4$$ q ≡ 3 ( mod 4 ) . Our proofs are valid for all characteristics $$p2$$ p 2 . Along the way we prove some elegant new character sum identities.