Self-points on elliptic curves

Wuthrich, Christian

doi:10.2140/ant.2009.3.283

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Self-points on elliptic curves

Christian Wuthrich

Vol. 3 (2009), No. 3, 283–315

DOI: 10.2140/ant.2009.3.283

Abstract

Let $E ∕ ℚ$ be an elliptic curve of conductor $N$ and let $p$ be a prime. We consider trace-compatible towers of modular points in the noncommutative division tower $ℚ (E [p^{\infty}])$ . Under weak assumptions, we can prove that all these points are of infinite order and determine the rank of the group they generate. Also, we use Kolyvagin’s construction of derivative classes to find explicit elements in certain Tate–Shafarevich groups.

Keywords

elliptic curves, modular point, modular curves

Mathematical Subject Classification 2000

Primary: 11G05

Secondary: 11G18, 11G40

Milestones

Received: 10 June 2008

Revised: 23 February 2009

Accepted: 24 February 2009

Published: 1 May 2009

Authors

Christian Wuthrich
School of Mathematical Sciences University of Nottingham Nottingham NG7 2RD United Kingdom

Vol. 3, No. 3, 2009