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2014 Tetrahedral elliptic curves and the local-global principle for isogenies
Barinder Banwait, John Cremona
Algebra Number Theory 8(5): 1201-1229 (2014). 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 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.

Citation

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Barinder Banwait. John Cremona. "Tetrahedral elliptic curves and the local-global principle for isogenies." Algebra Number Theory 8 (5) 1201 - 1229, 2014. https://doi.org/10.2140/ant.2014.8.1201

Information

Received: 3 September 2013; Revised: 25 March 2014; Accepted: 26 April 2014; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1303.11066
MathSciNet: MR3263141
Digital Object Identifier: 10.2140/ant.2014.8.1201

Subjects:
Primary: 11G05
Secondary: 11G18

Rights: Copyright © 2014 Mathematical Sciences Publishers

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