Vol. 8, No. 5, 2014

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Tetrahedral elliptic curves and the local-global principle for isogenies

Barinder S. Banwait and John E. Cremona

Vol. 8 (2014), No. 5, 1201–1229
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 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 PGL2(Fl). By constructing models of two modular curves, Xs(5) and XS4(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