Ranks of Abelian varieties over infinite extensions of the rationals
type: Item
instance of: scholarly article
title: Ranks of Abelian varieties over infinite extensions of the rationals
zbMATH Open document ID: 1157.11024
author: Álvaro Lozano-Robledo
published in: Manuscripta Mathematica
publication date: 2008-10-16T00:00:00Z
review text: The author studies the rank of the Mordell-Weil groups for some abelian varieties over infinite extensions of \({\mathbb Q}.\) If \(L/{\mathbb Q}\) is finite then the well-known Mordell-Weil Theorem asserts that \(A(L)\) is of finite rank. For infinite extensions the rank of the corresponding Mordell-Weil group might be infinite (cf. [\textit{G. Frey, M. Jarden}, Proc. Lond. Math. Soc. (3) 28, 112--128 (1974; Zbl 0275.14021)] for the case of \(A({\overline{\mathbb Q}})\)). On the other hand the theorem of Kato, Ribet and Rohrlich asserts that if \(E/{\mathbb Q}\) is an elliptic curve, \(\Sigma\) a finite set of rational primes then the group \(E({\mathbb Q}^{ab}_{\Sigma})\) is finitely generated. \({\mathbb Q}^{ab}_{\Sigma}\) denotes here the maximal abelian extension of \({\mathbb Q}\) unramified outside \({\Sigma}\). Let \(S\) be an infinite set of primes of \({\mathbb Z},\) and \({\mathbb Q}^{ab}_{S}\) be the maximal abelian extension of \({\mathbb Q}\) unramified outside \(S\) and contained in \({\overline{\mathbb Q}}.\) For a prime \(p\geq 2\) let \({\mathbb Q}^{(p)}_{S}\) be the compositum of all extensions unramified outside \(S\) of the form \({\mathbb Q}({\mu}_{p}, {\root p \of d})\) for some \(d\in {\mathbb Q}^{\times}.\) Let \(F\subset {\overline{\mathbb Q}}\), \quad \((\sigma) = ({\sigma}_{1},\cdots ,{\sigma}_{n}) \in \text{Gal} ({\overline{\mathbb Q}}/{\mathbb Q})^{n}\) anf \(F^{({\sigma})}\) be the intersection of all fixed fields \(F^{({\sigma}_{i})}.\) The author proves the following: Theorem 1.2: Let \(E/{\mathbb Q}\) be an elliptic curve and let \(S\) be an infinite set of primes. (1) Suppose that \(\text{rank}_{\mathbb Z}(E(\mathbb Q))\) is odd. If the parity conjecture holds for all quadratic twists of \(E\) then the rank of \(E(({\mathbb Q}_{S}^{(2)})^{(\sigma)})\) is infinite for all \(n\geq 0\) and all \((\sigma)\in \text{Gal} ({\overline{\mathbb Q}}/{\mathbb Q})^{n}.\) Hence \(\text{rank}_{\mathbb Z}(E(({\mathbb Q}_{S}^{ab})^{(\sigma)}))\) is…
reviewed by: Piotr Krasoń
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14K15
Mathematics Subject Classification ID: 11G10
zbMATH DE Number: 5351817
zbMATH Keywords: abelian variety
zbMATH Keywords: infinite extension
zbMATH Keywords: Selmer group
zbMATH Keywords: rank
MaRDI profile type: Publication
full work available at URL: s00229-008-0189-4
OpenAlex ID: W2001997930
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DOI: 10.1007/S00229-008-0189-4
description: scientific article; zbMATH DE number 5351817