Explicit Sato–Tate type distribution for a family of K ⁢ 3 K3 surfaces

Abstract

Abstract In the 1960s, 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 λ ⁢ ( p ) {A_{\lambda}(p)} of a certain family of K ⁢ 3 {K3} surfaces X λ {X_{\lambda}} with generic Picard rank 19 is the O ⁢ ( 3 ) {O(3)} distribution. This distribution, which we denote by 1 4 ⁢ π ⁢ f ⁢ ( t ) {\frac{1}{4\pi}f(t)} , is quite different from the semicircular distribution. It is supported on [ - 3 , 3 ] {[-3,3]} and has vertical asymptotes at t = ± 1 {t=\pm 1} . Here we make this result explicit. We prove that if p ≥ 5 {p\geq 5} is prime and - 3 ≤ a < b ≤ 3 {-3\leq a<b\leq 3} , then | # ⁢ { λ ∈ 𝔽 p : A λ ⁢ ( p ) ∈ [ a , b ] } p - 1 4 ⁢ π ⁢ ∫ a b f ⁢ ( t ) ⁢ 𝑑 t | ≤ 98.28 p 1 / 4 . \biggl{\lvert}\frac{\#\{\lambda\in\mathbb{F}_{p}:A_{\lambda}(p)\in[a,b]\}}{p}-% \frac{1}{4\pi}\int_{a}^{b}f(t)\,dt\biggr{\rvert}\leq\frac{98.28}{p^{1/4}}. As a consequence, we are able to determine when a finite field 𝔽 p {\mathbb{F}_{p}} is large enough for the discrete histograms to reach any given height near t = ± 1 {t=\pm 1} . To obtain these results, we make use of the theory of Rankin–Cohen brackets in the theory of harmonic Maass forms.