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arXiv · 2009.08622

Conjecture: 100% of elliptic surfaces over $\mathbb{Q}$ have rank zero

Alex Cowan
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Abstract

Based on an equation for the rank of an elliptic surface over $\mathbb{Q}$ which appears in the work of Nagao, Rosen, and Silverman, we conjecture that 100% of elliptic surfaces have rank $0$ when ordered by the size of the coefficients of their Weierstrass equations, and present a probabilistic heuristic to justify this conjecture. We then discuss how it would follow from either understanding of certain $L$-functions, or from understanding of the local behaviour of the surfaces. Finally, we make a conjecture about ranks of elliptic surfaces over finite fields, and highlight some experimental evidence supporting it.

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Alex Cowan. 2020-09-18. Conjecture: 100% of elliptic surfaces over $\mathbb{Q}$ have rank zero. https://arxiv.org/abs/2009.08622

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