Distribution of the Hessian values of Gaussian hypergeometric functions

TL;DR

This work extends the distributional study of Gaussian hypergeometric values over finite fields to the Hessian family, linking $_2F_1(\lambda)_q$ values to the Frobenius traces of Hessian elliptic curves via $q\,{}_2F_1(3/\lambda)_q = -a_q^{\text{Hes}}(\lambda)$. Using harmonic Maass forms, Rankin-Cohen brackets, and holomorphic projection, the authors express weighted class-number sums as Fourier coefficients of modular objects and derive precise asymptotics for the Frobenius-trace moments in two residue classes $q\equiv1\pmod{3}$ and $q\equiv2\pmod{3}$. The even moments converge to the Catalan numbers $\frac{(2n)!}{n!(n+1)!}$ while odd moments vanish, implying that the normalized values $p^{-r/2}\,a_q^{\text{Hes}}(\lambda)$ (or equivalently $\sqrt{q}\,{}_2F_1(\lambda)_q$) obey a semicircular distribution on $[-2,2]$, i.e. the Sato–Tate law in this Hessian family. These results demonstrate a universal semicircular limiting behavior for a non-CM family of elliptic curves and illustrate a powerful framework connecting finite-field hypergeometric values, Hurwitz class numbers, and harmonic Maass form theory to statistical distributions of Frobenius traces.

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.

Theorems & Definitions (44)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Remark
• Corollary 1.4
• Remark
• Theorem 2.1
• proof
• Proposition 2.2
• Lemma 2.3