Diophantine triples and K3 surfaces

Matija Kazalicki,Bartosz Naskręcki

doi:10.1016/j.jnt.2021.07.009

Matija Kazalicki, Bartosz Naskręcki

Open Access

https://doi.org/10.1016/j.jnt.2021.07.009

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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:(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)∈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∈Fp× in the prime field Fp of odd characteristic and an abelian surface which is a product of two elliptic curves Ek×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(Γ0(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.

Highlights

We prove that the variety X is birational to P3 which leads us to a new rational parametrization of the set of Diophantine triples
A Diophantine m-tuple with elements in a commutative unital ring R is a set of m non-zero elements of R with the property that the product of any two of its distinct elements is one less than a square
We focus on Diophantine triples over the field K of odd characteristic that we describe as follows

Summary

Introduction

A Diophantine m-tuple with elements in a commutative unital ring R is a set of m non-zero (distinct) elements of R with the property that the product of any two of its distinct elements is one less than a square. Diophantine m-tuples over the finite fields were first studied in [DK21] where the authors derived a formula for the number of Diophantine triples and quadruples over Fp. In Section 3.1, we construct a fibration on the threefold X whose fibers are rational elliptic surfaces. We can rephrase the formula (1.3) in terms of the second moment sum associated to the elliptic surface Ek. In general, let Fk be a 1-parametric family of elliptic curves over Q, i.e. a generic fiber of an elliptic surface F → P1 defined over Q with a section. A function S3(p) = O(1) depends only on the places of bad reduction of Fk, S2(p) = −#{∆(Fk) = 0} is the number of Fp points of the minimal discriminant of Fk. S1 = −tr(Frobp | Hc1(Up, Sym2(Gp))) where Up is a certain open Fp-curve computed from Fk and Gp is a rank 2 lisse sheaf on Up built from the relative family Fk, [Mic95, §3]. 0, the highest order lower term of the second moment with a non-vanishing average is f2(p) term and its average μ2 is equal to −3, the bias conjecture holds

Parametrization of the triples

Diophantine triples over finite fields