Algebra & Number Theory

Tetrahedral elliptic curves and the local-global principle for isogenies

Barinder Banwait and John Cremona

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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.

Article information

Algebra Number Theory, Volume 8, Number 5 (2014), 1201-1229.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G05: Elliptic curves over global fields [See also 14H52]
Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]

elliptic curves local-global isogeny exceptional modular curves


Banwait, Barinder; Cremona, John. Tetrahedral elliptic curves and the local-global principle for isogenies. Algebra Number Theory 8 (2014), no. 5, 1201--1229. doi:10.2140/ant.2014.8.1201.

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