In the 1980's, Greene defined hypergeometric functions over finite fields using Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to those of the classical hypergeometric series studied by Gauss and Kummer. These functions have played important roles in the study of Apéry-style supercongruences, the Eichler-Selberg trace formula, Galois representations, and zeta-functions of arithmetic varieties. We study the value distribution (over large finite fields) of natural families of these functions. For the 2 F 1 functions, the limiting distribution is semicircular, whereas the distribution for the 3 F 2 functions is the more exotic Batman distribution.

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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.