Diophantine triples and K3 surfaces

A Diophantine $m$-tuple over $K$ 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 fibre over $k$. First, we prove that the variety $\overline{X}$ is rational which leads us to a new rational parametrization of the set of Diophantine triples. Next, specializing to finite fields, we find a Shioda-Inose structure of the K3 surface $X_k$ for a given $k\in\mathbb{F}_{p}^{\times}$ in the prime field $\mathbb{F}_{p}$ of odd characteristic, determined by an abelian surface which is a square $E_k\times E_k$ of an explicit elliptic curve. We derive an explicit formula for $N(p,k)$, the number of Diophantine triples over $\mathbb{F}_{p}$ with the product of elements equal to $k$. Moreover, 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). From the interplay of the two (K3 and rational) fibrations of $\overline{X}$, 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 Lasic defines circular Diophantine $m$-tuples and describes the parametrization of these sets.