Determining explicitly the Mordell-Weil group of certain rational elliptic surfaces

TL;DR

This paper provides a shorter, more geometric proof for an algorithm that determines the Mordell-Weil group rank of certain rational elliptic surfaces, enhancing understanding of their rational points.

Contribution

It offers a new, concise geometric proof of an existing algorithm for computing ranks of specific elliptic surfaces, and discusses its applicability.

Findings

The proof confirms the correctness of the algorithm.

The geometric approach simplifies the understanding of the rank computation.

Potential applicability to broader classes of elliptic surfaces.

Abstract

Let $A,B$ be nonzero rational numbers. Consider the elliptic curve $E_{A,B}/\mathbb{Q}(t)$ with Weierstrass equation $y^2=x^3+At^6+B$. An algorithm to determine $\mathrm{rank } E_{A,B}(\mathbb{Q}(t))$ as a function of $(A,B)$ was presented in a recent paper by Desjardins and Naskrecki. We will give a different and shorter proof for the correctness of that algorithm, using a more geometric approach and discuss for which classes of examples this approach might be useful.