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Published in:

2021 | OriginalPaper | Chapter

# Conjecture: 100% of Elliptic Surfaces Over $$\mathbb {Q}$$ have Rank Zero

Author : 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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