Generators and splitting fields of certain elliptic K3 surfaces

Abstract

Let $k\subset \mathbb{C}$ be a number field and $\mathcal{E}$ be an elliptic curve defined over $k(t)$, the rational function field of the projective line ${\mathbb{P}}_k^1$, is isomorphic to the generic fiber of an elliptic surface $π:= \Sc_\Ee \rightarrow {\mathbb P}^1_k$. For any subfield $\mathcal{K}\subseteq \mathbb{C}$ of $k$, the set $\mathcal{E}(\mathcal{K}(t))$ of $\mathcal{K}(t)$-rational points of $\mathcal{E}$ is known to be a finitely generated abelian group. The splitting field of $\mathcal{E}$ defined over $k(t)$ is the smallest finite extension $\mathcal{K}\subset \mathbb{C}$ of $k$ such that ${\mathcal E} ({\mathbb C} (t)) \iso {\mathcal E} ({\mathcal K}(t))$. In this paper, we consider the elliptic $K3$ surfaces defined over $k=\mathbb{Q}$ with the generic fiber given by the Weierstrass equation ${\mathcal{E}}_n:y^2=x^3+t^n+1/t^n$, $1\le n\le 6$, and determine the splitting field ${\mathcal{K}}_n$, and find an explicit set of independent generators for ${\mathcal{E}}_n({\mathcal{K}}_n(t))$ for $1\le n\le 6$.

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