Sato-Tate Type Distributions for Matrix Points on Elliptic Curves and Some $K3$ Surfaces

TL;DR

This paper extends the Sato-Tate distribution framework to matrix points on elliptic curves and K3 surfaces over finite fields, providing exact formulas and limiting distributions for deviations from expected point counts.

Contribution

It introduces exact formulas for matrix point counts on elliptic curves and K3 surfaces at supersingular primes and establishes their limiting distributions, generalizing Sato-Tate type results.

Findings

Exact formulas for matrix point counts on elliptic curves and K3 surfaces.

Determination of limiting distributions for deviations from expected counts.

Explicit results for non-CM elliptic curves with square-free conductor.

Abstract

Generalizing the problem of counting rational points on curves and surfaces over finite fields, we consider the setting of $n \times n$ matrix points with finite field entries. We obtain exact formulas for matrix point counts on elliptic curves and certain $K3$ surfaces for "supersingular" primes. These exact formulas, which involve partitions of integers up to $n$, essentially coincide with the expected value for the number of such points. Therefore, in analogy with the Sato-Tate conjecture, it is natural to study the distribution of the deviation from the expected values for all primes. We determine the limiting distributions for elliptic curves and a family of $K3$ surfaces. For non-CM elliptic curves with square-free conductor, our results are explicit.