Some mixed character sum identities of Katz. II. (English) Zbl 1418.11156
Summary: 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 [J. Number Theory 179, 17–32 (2017; Zbl 1418.11155)] gave such a proof of his identities when \(q \equiv 1 \pmod 4\), and this paper provides such a proof for the remaining case \(q \equiv 3 \pmod 4\). Our proofs are valid for all characteristics \(p>2\). Along the way we prove some elegant new character sum identities.
MSC:
| 11T24 | Other character sums and Gauss sums |
| 33C05 | Classical hypergeometric functions, \({}_2F_1\) |
Keywords:
hypergeometric \({}_2F_1\) character sums over finite fields; Gauss and Jacobi sums; norm-restricted Gauss and Jacobi sums; Eisenstein sums; Hasse-Davenport theorems; quantum physicsCitations:
Zbl 1418.11155References:
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