Arithmetic Information of Rational Elliptic Surfaces, and Shioda's Rank 68 Elliptic Surface

TL;DR

This paper introduces an algorithm to determine the arithmetic properties of rational elliptic surfaces, including their field of definition, and applies it to a high-rank elliptic surface, revealing its complex field of definition.

Contribution

The paper presents a new algorithm implemented in Magma for computing the arithmetic information of rational elliptic surfaces, including their field of definition.

Findings

The algorithm successfully determines the field of definition for any rational elliptic surface.

Applied to Shioda's rank 68 elliptic surface, the field of definition has degree 829,440.

The method advances understanding of the arithmetic complexity of high-rank elliptic surfaces.

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 $829,440$.