Hypergeometric Distributions and Joint Families of Elliptic Curves

TL;DR

This paper investigates the limiting distribution of hypergeometric functions related to Frobenius traces of elliptic curves, extending previous work to pairs of curves and employing advanced algebraic and cohomological methods.

Contribution

It generalizes Michel's Sato-Tate law to pairs of elliptic curve families and connects hypergeometric values to algebraic geometry and monodromy group theory.

Findings

Established the limiting distribution for certain ${_4F_3}$ hypergeometric functions.

Showed that pairs of elliptic curve families exhibit independent Sato-Tate distributions.

Connected modular forms with étale cohomology in the context of hypergeometric functions.

Abstract

Recently, the first author as well as the second author with Ono, Pujahari, and Saikia determined the limiting distribution of values of certain finite field ${_2F_1}$ and ${_3F_2}$ hypergeometric functions. These hypergeometric values are related to Frobenius traces of elliptic curves and their limiting distribution is determined using connections to the theory of modular forms and harmonic Maass forms. Here we determine the limiting distribution of values of some ${_4F_3}$ hypergeometric functions which are sums of traces of Frobenius for a pair of elliptic curves. To obtain this result, we generalize Michel's work on Sato-Tate laws for families of elliptic curves to the setting of pairs of families, and we show that a generic pair admits an independent Sato-Tate distribution as the finite field grows. To this end, we use various results from the theory of \'etale cohomology, Deligne's work on the Weil conjectures, and the work of Katz on monodromy groups. In the cases previously studied using modular methods, we elucidate the connection between the modular forms that appear and the machinery of \'etale cohomology.