Frobenius trace distributions for Gaussian hypergeometric functions

Abstract: In the 1980's, Greene defined hypergeometric functions over finite fields using Jacobi sums. These functions possess many properties that are analogous to those of the classical hypergeometric series studied by Gauss, Kummer and others. These functions have played important roles in the study of supercongruences, the Eichler-Selberg trace formula, and zeta-functions of arithmetic varieties. We study the distribution (over large finite fields) of the values of certain families of these functions. For the $_2F_1$ functions, the limiting distribution is semicircular, whereas the distribution for the $_3F_2$ functions is the more exotic \it{Batman distribution.}

classical analysis and ODEscombinatoricsnumber theory

Audience: researchers in the topic

Special Functions and Number Theory seminar

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