Rank growth of elliptic curves over S3 extensions with fixed quadratic resolvents
Daniel Keliher, Sun Woo Park
TL;DR
The paper establishes a probabilistic framework for rank growth of elliptic curves E/k when base extending to S3-cubic extensions K/k with fixed quadratic resolvent F/k. It reduces rank growth to the study of Selmer groups of a constructed 4-dimensional abelian variety B_{K/k} via the 1−σ_K-Selmer group, and then analyzes these Selmer dimensions through a Markov-chain model guided by a non-standard fan structure. By parameterizing S3-cubic extensions and applying local-global Selmer theory with Chebotarev densities, the authors derive an explicit density distribution for dim_{F3} Sel_{1−σ_K}(B_{K/k}/k) and show that large rank-growth events decay quadratically-exponentially. A key corollary is that rk(E/K) increases by at most 1 with probability at least 31.95% under the fan-ordered sampling, with a computable parity-influenced distribution governed by a parameter ρ_E. The work extends previous Markov-model approaches (Klagsbrun–Mazur–Rubin, etc.) to S3-extensions and provides a detailed link between cubic extensions, 3-Selmer data, and rank growth phenomena in elliptic curves.
Abstract
We study the probability with which an elliptic curve $E/k$, subject to some technical conditions, gains rank upon base extension to an $S_3$-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 $S_3$-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 $31.95\%$.