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

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 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.