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

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 $1\leq m\leq 12$.

Categories: math.NT math.AG