Rank growth of elliptic curves over S3 extensions with fixed quadratic resolvents

2025/02/08 by Daniel Keliher, Keliher, Daniel, Sun Woo Park +1 · 1 citation
Arts and Humanities · Mathematics · #11G05 #Algebraic Geometry and Number Theory #FOS: Mathematics #French Historical and Cultural Studies #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2502.05705

openalex publication_date 2025/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the probability with which an elliptic curve E/k, subject to some technical conditions, gains rank upon base extension to an S₃-cubic extension K/k with quadratic resolvent field F/k, all three fields of which are subject to some mild technical conditions. To do so, we determine the distribution (under a non-standard ordering) of Selmer ranks of an auxiliary abelian variety associated to E and S₃-cubic extensions K/k following ideas of Klagsbrun, Mazur, and Rubin. One corollary of this distribution is that E gains rank by at most one upon base extension to K with probability at least 31.95%.

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• Selmer stability for elliptic curves in Galois ℓ-extensions

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