Explicit Sato-Tate type distribution for a family of $K3$ surfaces

TL;DR

This paper explicitly characterizes the distribution of Frobenius traces for a family of K3 surfaces, showing convergence to an O(3) distribution with precise error bounds, extending classical Sato-Tate results.

Contribution

It provides an explicit proof of the limiting Frobenius trace distribution for K3 surfaces and quantifies the convergence rate with error bounds.

Findings

Distribution converges to O(3) distribution with explicit bounds

Error term decreases as p^{−1/4}

Determines field size needed for histogram accuracy

Abstract

In the 1960's, Birch proved that the traces of Frobenius for elliptic curves taken at random over a large finite field is modeled by the semicircular distribution (i.e. the usual Sato-Tate for non-CM elliptic curves). In analogy with Birch's result, a recent paper by Ono, the author, and Saikia proved that the limiting distribution of the normalized Frobenius traces $A_{\lambda}(p)$ of a certain family of $K3$ surfaces $X_\lambda$ with generic Picard rank $19$ is the $O(3)$ distribution. This distribution, which we denote by $\frac{1}{4\pi}f(t),$ is quite different from the semicircular distribution. It is supported on $[-3,3]$ and has vertical asymptotes at $t=\pm1.$ Here we make this result explicit. We prove that if $p\geq 5$ is prime and $-3\leq a<b\leq 3,$ then $$ \left|\frac{\#\{\lambda\in\mathbb{F}_p :A_{\lambda}(p)\in[a,b]\}}{p}-\frac{1}{4\pi}\int_a^b f(t)dt\right|\leq \frac{110.84}{p^{1/4}}. $$ As a consequence, we are able to determine when a finite field $\mathbb{F}_p$ is large enough for the discrete histograms to reach any given height near $t=\pm1.$ To obtain these results, we make use of the theory of Rankin-Cohen brackets in the theory of harmonic Maass forms.