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arXiv · 2506.19423

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

Remke Kloosterman
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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.

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Remke Kloosterman. 2025-06-24. Determining explicitly the Mordell-Weil group of certain rational elliptic surfaces. https://arxiv.org/abs/2506.19423

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