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2021 | OriginalPaper | Chapter

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

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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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Metadata
Title
Conjecture: 100% of Elliptic Surfaces Over have Rank Zero
Author
Alex Cowan
Copyright Year
2021
DOI
https://doi.org/10.1007/978-3-030-80914-0_10

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