Diophantine triples and K3 surfaces

Abstract: A Diophantine $m$-tuple with elements in the field $K$ is a set of $m$ non-zero (distinct) elements of $K$ with the property that the product of any two distinct elements is one less than a square in $K$. Let $X: (x^2-1)(y^2-1)(z^2-1)=k^2,$ be a threefold. Its $K$-rational points parametrize Diophantine triples over $K$ such that the product of the elements of the triple that corresponds to the point $(x,y,z,k)\in X(K)$ is equal to $k$. We denote by $\overline{X}$ the projective closure of $X$ and for a fixed $k$ by $X_k$ a variety defined by the same equation as $X$. We prove that the variety $\overline{X}$ is birational to $\mathbb{P}^3$ which leads us to a new rational parametrization of the set of Diophantine triples. Next, specializing to finite fields, we find a correspondence between a K3 surface $X_k$ for a given $k\in\mathbb{F}{p}^{\times}$ in the prime field $\mathbb{F}{p}$ of odd characteristic and an abelian surface which is a product of two elliptic curves $E_k\times E_k$ where $E_k: y^2=x(k^2(1 + k^2)^3 + 2(1 + k^2)^2 x + x^2)$. We derive a formula for $N(p,k)$, the number of Diophantine triples over $\mathbb{F}{p}$ with the product of elements equal to $k$. We show that the variety $\overline{X}$ admits a fibration by rational elliptic surfaces and from it we derive the formula for the number of points on $\overline{X}$ over an arbitrary finite field $\mathbb{F}{q}$. We reprove the formula for the number of Diophantine triples over $\mathbb{F}{q}$ from Dujella-Kazalicki(2021). We derive the formula for the second moment of the elliptic surface $E_k$ (and thus confirming Steven J. Miller's Bias conjecture in this particular case) which we describe in terms of Fourier coefficients of a rational newform generating $S_4(\Gamma{0}(8))$. Finally, in the Appendix, Luka Lasi\'c defines circular Diophantine $m$-tuples, and describes the parametrization of these sets.