Tetrahedral elliptic curves and the local-global principle for isogenies

Banwait, Barinder; Cremona, John

doi:10.2140/ant.2014.8.1201

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Tetrahedral elliptic curves and the local-global principle for isogenies

Barinder S. Banwait and John E. Cremona

Vol. 8 (2014), No. 5, 1201–1229

DOI: 10.2140/ant.2014.8.1201

Abstract

We study the failure of a local-global principle for the existence of $l$ -isogenies for elliptic curves over number fields $K$ . Sutherland has shown that over $ℚ$ there is just one failure, which occurs for $l = 7$ and a unique $j$ -invariant, and has given a classification of such failures when $K$ does not contain the quadratic subfield of the $l$ -th cyclotomic field. In this paper we provide a classification of failures for number fields which do contain this quadratic field, and we find a new “exceptional” source of such failures arising from the exceptional subgroups of ${PGL}_{2} (F_{l})$ . By constructing models of two modular curves, $X_{s} (5)$ and $X_{S_{4}} (13)$ , we find two new families of elliptic curves for which the principle fails, and we show that, for quadratic fields, there can be no other exceptional failures.

Keywords

elliptic curves, local-global, isogeny, exceptional modular curves

Mathematical Subject Classification 2010

Primary: 11G05

Secondary: 11G18

Milestones

Received: 3 September 2013

Revised: 25 March 2014

Accepted: 26 April 2014

Published: 16 September 2014

Authors

Barinder S. Banwait
Institut de Mathématiques de Bordeaux Université de Bordeaux I 351 Cours de la Libération 33405 Talence France

John E. Cremona
Mathematics Institute University of Warwick Zeeman Building Coventry CV4 7AL United Kingdom

Vol. 8, No. 5, 2014