Naslov
Diophantine triples and K3 surfaces

Autori
Kazalicki, Matija ; Naskręcki, Bartosz

Izvornik
Journal of number theory (0022-314X) 236 (2022); 41-70

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

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

Sažetak
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 $(x, y, z, k)\in X(K)$ 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 $X$. In this paper, we try to understand what can the geometry of varieties $X_k$, $X$ and $\overline{; ; ; ; ; ; X}; ; ; ; ; ; $ tell us about the arithmetic of Diophantine triples. First, 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 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 $\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}; ; ; ; ; ; $. Using it we reprove the formula for the number of Diophantine triples over $\Fq$ from \cite{; ; ; ; ; ; Dujella_Kazalicki_ANT}; ; ; ; ; ; . Curiously, 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 Lasi\'c defines circular Diophantine $m$-tuples, and describes the parametrization of these sets. For $m=3$ this method provides an elegant parametrization of Diophantine triples.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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