The splitting fields and Generators of Shioda's elliptic surfaces $y^2=x^3 +t^{m} +1$ (I)
Sajad Salami, Arman Shamsi Zargar
TL;DR
This work determines the splitting fields ${\mathcal K}_m$ and explicit independent generators of the Mordell–Weil lattices for Shioda's elliptic surfaces ${\mathcal E}_m: y^2=x^3+t^m+1$ over ${\mathbb Q}(t)$ for $2\le m\le 12$. Building on Usui's rank results and Shioda–Tate theory, the authors construct concrete generators and identify the corresponding splitting fields, often via radical expressions and high-degree minimal polynomials, with detailed Gram matrices showing independence. They leverage Galois representations, specialization via fundamental polynomials, and base-change decompositions to relate lattices across different $m$, and provide extensive computational data (Maple, PARI/GP, Sage) to verify the structures. The results illuminate how explicit generators realize MW lattices of varying types (including $E_8^*$, $E_6^*$, $D_4^*$) and how splitting fields arise from the arithmetic of the fibers, contributing to a concrete understanding of the interaction between Galois theory and elliptic-surface lattices. Overall, the paper delivers a comprehensive, computationally backed account of the splitting fields and MW-generators for a family of Shioda elliptic surfaces, with explicit data for application in lattice computations and arithmetic geometry.
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 $1\leq m\leq 12$.