Generators and splitting fields of certain elliptic K3 surfaces

TL;DR

This paper investigates the splitting fields and rational points of specific elliptic K3 surfaces over Q, explicitly determining their splitting fields and generators for the cases where n ranges from 1 to 6.

Contribution

It explicitly computes the splitting fields and generators of the Mordell-Weil groups for a family of elliptic K3 surfaces with given Weierstrass equations.

Findings

Determined splitting fields for n=1 to 6.

Constructed explicit generators for each case.

Analyzed the structure of Mordell-Weil groups.

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 ${\mathbb P}^1_k$, is isomorphic to the generic fiber of an elliptic surface $\pi:= \Sc_\Ee \rightarrow {\mathbb P}^1_k$. 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 ${\mathcal E}_n: \displaystyle y^2=x^3 + t^n + 1/t^n$, $1\leq n\leq 6$, 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 $1\leq n \leq 6$.