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Hypergeometric functions over finite fields. (English) Zbl 1443.11254

Wood, David R. (ed.) et al., 2017 MATRIX annals. Cham: Springer. MATRIX Book Ser. 2, 461-466 (2019).
Summary: We discuss recent work of the authors in which we study the translation of classical hypergeometric transformation and evaluation formulas to the finite field setting.
Our approach is motivated by the desire for both an algorithmic type approach that closely parallels the classical case, and an approach that aligns with geometry. In light of these objectives, we focus on period functions in our construction which makes point counting on the corresponding varieties as straightforward as possible.
We are also motivated by previous work joint with [A. Deines et al., J. Number Theory 161, 175–203 (2016; Zbl 1336.33009)] in which we study generalized Legendre curves using periods to determine a condition for when the endomorphism algebra of the primitive part of the associated Jacobian variety contains a quaternion algebra over \(\mathbb Q\). In most cases this involves computing Galois representations attached to the Jacobian varieties using Greene’s finite field hypergeometric functions.
For the entire collection see [Zbl 1411.37003].

MSC:

11T24 Other character sums and Gauss sums
33C05 Classical hypergeometric functions, \({}_2F_1\)
11T23 Exponential sums

Citations:

Zbl 1336.33009
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References:

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