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arXiv · 2308.02683

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

Avalon Blaser·Molly Bradley·Daniel Vargas·Kathy Xing
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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.

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Avalon Blaser, Molly Bradley, Daniel Vargas, Kathy Xing. 2023-08-04. Sato-Tate Type Distributions for Matrix Points on Elliptic Curves and Some $K3$ Surfaces. https://arxiv.org/abs/2308.02683

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