Distribution of values of Gaussian hypergeometric functions

In the 1980s, Greene defined hypergeometric functions over finite fields using Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to those of the classical hypergeometric series studied by Gauss and Kummer. These functions have played important roles in the study of Apéry-style supercongruences, the Eichler–Selberg trace formula, Galois representations, and zeta-functions of arithmetic varieties. We study the value distribution (over large finite fields) of natural families of these functions. For the $_2 F_1$ functions, the limiting distribution is semicircular (i.e. $SU(2)$), whereas the distribution for the $_3 F_2$ functions is the Batman distribution for the traces of the real orthogonal group $O_3$.

Keywords

Gaussian hypergeometric functions, distributions, elliptic curves

2010 Mathematics Subject Classification

11F11, 11F46, 11G20, 11T24, 33E50

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The first-named author thanks the Thomas Jefferson Fund and the NSF (DMS-2002265 and DMS-2055118) for their generous support, as well as the Kavli Institute grant NSF PHY-1748958.

The third-named author is grateful for the support of a Fulbright Nehru Postdoctoral Fellowship

Received 16 September 2021

Accepted 25 October 2022

Published 3 April 2023