Mordell-Weil groups in procyclic extensions of a function field
type: Item
instance of: scholarly article
title: Mordell-Weil groups in procyclic extensions of a function field
zbMATH Open document ID: 0903.14006
DOI: 10.1215/S0012-7094-97-08911-0
author: Lisa A. Fastenberg
published in: Duke Mathematical Journal
publication date: 1997-11-16T00:00:00Z
review text: Let \(K\) be a number field and let \(L\) be its cyclotomic \({\mathbb{Z}}_p\)-extension. Let \(E/K\) be an elliptic curve with good reduction at \(p\). Then Mazur has conjectured that the group \(E(L)\) is finitely generated. The author considers an analogue of this conjecture. The number field \(K\) is replaced by \(\mathbb{C}(T)\) (the rational function field in one variable over the complex numbers), and \(L\) is replaced by its procyclic extension \(\bigcup_{r \geq 1} \mathbb{C}(T^{1/r})\). Let \(E/K\) be an elliptic curve that has good or multiplicative reduction outside \(\{0,\infty\}\). Then the author proves that \(E(L)\) is finitely generated, provided one or the other of two technical conditions on the reduction behaviour of \(E\) is satisfied. The proof is split into two parts. First it is shown that the rank of \(E(K(T^{1/r}))\) is bounded independently of \(r\). This part of the proof follows a method due to \textit{P. F. Stiller} [Pac. J. Math. 128, No. 1, 157-189 (1987; Zbl 0591.14022)] and takes up most of the paper. This is also the point where the assumption on the reduction behaviour comes into play. The second part is to control the torsion part in the quotient of the Mordell-Weil groups at different levels.
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 1061773
zbMATH Keywords: Mordell-Weil group
zbMATH Keywords: procyclic extension of rational function field
zbMATH Keywords: elliptic curves over function fields
reviewed by: Michael Stoll
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1061773