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 an affine variety over K. 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) ∈ X(K) is equal to k. We denote by X the projective closure of X and for a fixed k by X k a variety defined by the same equation as X.In this paper, we try to understand what can the geometry of varieties X k , X and X tell us about the arithmetic of Diophantine triples.First, we prove that the variety X is birational to P 3 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 ∈ F × p in the prime field F p of odd characteristic, determined by an abelian surface which is a product of two elliptic curvesWe derive an explicit formula for N (p, k), the number of Diophantine triples over F p with the product of elements equal to k. Moreover, we show that the variety X admits a fibration by rational elliptic surfaces and from it we derive the formula for the number of points on X over an arbitrary finite field F q . Using it we reprove the formula for the number of Diophantine triples over F q from [DK21].Curiously, from the interplay of the two (K3 and rational) fibrations of 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 (Γ 0 (8)).Finally, in the Appendix, Luka Lasić defines circular Diophantine m-tuples, and describes the parametrization of these sets. For m = 3 this method provides an elegant parametrization of Diophantine triples.