Arithmetic Information of Rational Elliptic Surfaces, and Shioda's Rank 68 Elliptic Surface
Blair Butler, Andreas-Stephan Elsenhans
TL;DR
This work develops a Magma-based algorithm to determine the field of definition of the Mordell-Weil group for rational elliptic surfaces and related arithmetic data, enabling explicit computation of splitting fields, automorphism groups, generating sections, and Galois representations. It applies the method to Shioda's rank-$68$ elliptic surface $y^2 = x^3 + t^{360} + 1$, proving the field of definition is a number field of degree $829{,}440$ and decomposing the problem via 11 constituent surfaces, including a $K3$ piece defined over a degree-$192$ field. The approach relies on Galois-group computations, resolvent constructions, interpolation, and Newton lifting to reconstruct roots and automorphisms, and uses the Shioda–Tate philosophy to relate rank to the Picard lattice and singular fibres through the formula $r = \lambda - 2 - \sum_{v\in \mathbb{P}^1}(m_v - 1)$. The results provide a concrete computational framework for determining arithmetic invariants of high-rank elliptic surfaces and pave the way for systematic treatment of Delsarte and related families. This advances practical understanding of Mordell-Weil groups, splitting fields, and Galois representations in the arithmetic of elliptic surfaces, with potential impact on constructing high-rank families via specialization.
Abstract
The field of definition of the Mordell-Weil group of an elliptic surface $E/\mathbb{Q}$ is the smallest number field $k$ such that all of its $\mathbb{Q}(t)$-rational points are defined over $k(t)$. In this paper, we present an algorithm, implemented in Magma, which can determine the arithmetic information, including the field of definition, associated to any rational elliptic surface. As an application of this, we also demonstrate that the field of definition of Shioda's rank $68$ elliptic surface given by $y^2 = x^3 + t^{360} + 1$ is a number field of degree $829,440$.