Counting matrix points on certain varieties over finite fields

TL;DR

This paper extends finite field hypergeometric functions to count matrix points on certain algebraic varieties, deriving formulas for matrix elliptic curves and K3 surfaces, and analyzing their statistical distributions.

Contribution

It introduces methods to count matrix points on algebraic varieties over finite fields using hypergeometric functions, extending previous scalar point counting techniques.

Findings

Formulas for counting matrix points on elliptic curves and K3 surfaces.

Proof of Sato-Tate distribution for matrix point count errors.

Extension of Greene's hypergeometric functions to matrix settings.

Abstract

Classical hypergeometric functions are well-known to play an important role in arithmetic algebraic geometry. These functions offer solutions to ordinary differential equations, and special cases of such solutions are periods of Picard-Fuchs varieties of Calabi-Yau type. Gauss' $_2F_1$ includes the celebrated case of elliptic curves through the theory of elliptic functions. In the 80s, Greene defined finite field hypergeometric functions that can be used to enumerate the number of finite field points on such varieties. We extend some of these results to count finite field ``matrix points." For example, for every $n\geq 1,$ we consider the matrix elliptic curves $$ B^2 = A(A-I_n)(A-a I_n), $$ where $(A,B)$ are commuting $n\times n$ matrices over a finite field $\mathbb{F}_q$ and $a\neq 0,1$ is fixed. Our formulas are assembled from Greene's hypergeometric functions and $q$-multinomial coefficients. We use these formulas to prove Sato-Tate distributions for the error terms for matrix point counts for these curves and some families of $K3$ surfaces.