Sato–Tate type distributions for matrix points on elliptic curves and some K3 surfaces

Abstract

Generalizing the problem of counting rational points on curves and surfaces over finite fields, we consider the setting of n×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.