Distribution of the Hessian values of Gaussian hypergeometric functions

TL;DR

This paper investigates the distribution of specific Gaussian hypergeometric function values linked to elliptic curves, proving they follow a semi-circular distribution consistent with the Sato-Tate conjecture using advanced modular form theory.

Contribution

It establishes the limiting distribution of hypergeometric function values associated with elliptic curves as semi-circular, connecting hypergeometric functions, elliptic curves, and harmonic Maass forms.

Findings

Distribution of values is semi-circular (SU(2)).

Confirms the Sato-Tate distribution in this context.

Uses harmonic Maass forms and mock modular forms.

Abstract

We consider a special family of Gaussian hypergeometric functions whose entries are cubic and trivial characters over finite fields. The special values of these functions are known to give the Frobenius traces of families of Hessian elliptic curves. Using the theory of harmonic Maass forms and mock modular forms, we prove that the limiting distribution of these values is semi-circular (i.e. $SU(2)$), confirming the usual Sato-Tate distribution in this setting.