engleski

Diophantine triples and K3 surfaces

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 : (x2 -1)(y2 -1)(z2 -1) = k2, 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) is an element of X(K) is equal to k. We denote by X the projective closure of X and for a fixed k by Xk a variety defined by the same equation as X. In this paper, we try to understand what can the geometry of varieties Xk, X and X tell us about the arithmetic of Diophantine triples. First, we prove that the variety X is birational to P3 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 Xk for a given k is an element of Fpx in the prime field Fp of odd characteristic and an abelian surface which is a product of two elliptic curves Ek x Ek where Ek : y2 = x(k2(1 +k2)3+2(1 +k2)2x +x2). We derive an explicit formula for N(p, k), the number of Diophantine triples over Fp 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 Fq. Using it we reprove the formula for the number of Diophantine triples over Fq 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 Ek (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 S4(Gamma 0(8)). Finally, in the Appendix, Luka Lasic defines circular Diophantine m-tuples, and describes the parametrization of these sets. For m=3 this method provides an elegant parametrization of Diophantine triples.

Diophantine tuples ; elliptic curves ; K3 surfaces ; higher moments ; bias conjecture

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