The splitting fields and Generators of Shioda's elliptic surfaces $y^2=x^3 +t^{m} +1$ (I)

TL;DR

This paper determines the splitting fields and generators of the Mordell--Weil lattice for a family of elliptic surfaces defined by specific cubic equations over rational function fields, for positive integers up to 12.

Contribution

It explicitly computes the splitting fields and provides generators for the Mordell--Weil lattice of Shioda's elliptic surfaces with given parameters, extending understanding of their arithmetic structure.

Findings

Explicit splitting fields ${ m K}_m$ for 1 ≤ m ≤ 12.

Constructed linearly independent generators for the Mordell--Weil lattice.

Enhanced understanding of the arithmetic of these elliptic surfaces.

Abstract

The splitting field of an elliptic surface $\mathcal E$ defined over ${\mathbb Q}(t)$ is the smallest subfield $\mathcal K$ of $\mathbb C$ such that ${\mathcal E}({\mathbb C}(t))\cong {\mathcal E}({\mathcal K}(t))$. In this paper, we determine the splitting field ${\mathcal K}_m$ and a set of linearly independent generators for the Mordell--Weil lattice of Shioda's elliptic surface with generic fiber given by ${\mathcal E}_m: y^2=x^3 +t^{m} +1$ over ${\mathbb Q}(t)$ for positive integers $1\leq m\leq 12$.