Some mixed character sum identities of Katz. (English) Zbl 1418.11155
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. Such a proof has been given by the author and J. Greene [Res. Number Theory 3, Paper No. 8, 14 p. (2017; Zbl 1418.11156)] 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 \(p > 2\).
MSC:
| 11T24 | Other character sums and Gauss sums |
| 33C05 | Classical hypergeometric functions, \({}_2F_1\) |
Keywords:
hypergeometric \({}_2F_1\) functions over finite fields; Gauss sums; Jacobi sums; quadratic transformations; Hasse-Davenport relation; quantum physicsCitations:
Zbl 1418.11156References:
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