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 $\bar{X}$ 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 $\bar{X}$ tell us about the arithmetic of Diophantine triples.

First, we prove that the variety $\bar{X}$ 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 ${\mathbb{F}}_p$ with the product of elements equal to k. Moreover, we show that the variety $\bar{X}$ admits a fibration by rational elliptic surfaces and from it we derive the formula for the number of points on $\bar{X}$ over an arbitrary finite field ${\mathbb{F}}_q$. Using it we reprove the formula for the number of Diophantine triples over ${\mathbb{F}}_q$ from [DK21].

Curiously, from the interplay of the two (K3 and rational) fibrations of $\bar{X}$, 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 $S_4({\mathrm{\Gamma }}_0(8))$.

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.

域K中具有元素的丢番图m元组是K的m个非零（不同）元素的集合，其属性是任何两个不同元素的乘积比K中的平方小一。让$X：(X^2-1)({\mathrm{是的}}^2-1)(z^2-1)={ķ}^2$, 是K上的仿射变体。它的K有理点参数化了丢番图对K的三元组，使得对应于该点的三元组元素的乘积$(X,\mathrm{是的},z,ķ)\in X(ķ)$等于k ​​。我们表示$\bar{X}$X的射影闭包，对于一个固定的k$X_{ķ}$由与X相同的方程定义的变体。

在本文中，我们试图了解品种的几何形状$X_{ķ}$, X和$\bar{X}$告诉我们丢番图三元组的算术。

首先，我们证明多样性$\bar{X}$是双理性的${\mathbb{磷}}^3$这导致我们对丢番图三元组进行新的理性参数化。

接下来，专门研究有限域，我们找到了 K3 曲面之间的对应关系$X_{ķ}$对于给定的$ķ\in {\mathbb{F}}_p^{\times }$在主要领域${\mathbb{F}}_p$具有奇特征和由两条椭圆曲线乘积的阿贝尔曲面${乙}_{ķ}\times {乙}_{ķ}$在哪里${乙}_{ķ}：{\mathrm{是的}}^2=X({ķ}^2{(1+{ķ}^2)}^3+2{(1+{ķ}^2)}^{2}X+X^2)$. 我们推导出一个明确的公式$ñ(p,ķ)$，丢番图的数量增加了三倍${\mathbb{F}}_p$元素的乘积等于k ​​。此外，我们证明了多样性$\bar{X}$允许有理椭圆表面的纤维化，并由此推导出上点数的公式$\bar{X}$在任意有限域上${\mathbb{F}}_q$. 使用它，我们重新验证了丢番图三倍数的公式${\mathbb{F}}_q$来自 [DK21]。

奇怪的是，从两个（K3 和有理）纤维的相互作用$\bar{X}$，我们推导出椭圆曲面二阶矩的公式${乙}_{ķ}$（并因此在这种特殊情况下证实了 Steven J. Miller 的偏差猜想），我们用有理新形式生成的傅立叶系数来描述它${\mathrm{小号}}_4({\mathrm{\Gamma }}_0(8))$.

最后，在附录中，Luka Lasić 定义了圆形丢番图m元组，并描述了这些集合的参数化。为了$米=3$此方法提供了丢番图三元组的优雅参数化。