Algebra & Number Theory
- Algebra Number Theory
- Volume 8, Number 5 (2014), 1201-1229.
Tetrahedral elliptic curves and the local-global principle for isogenies
We study the failure of a local-global principle for the existence of -isogenies for elliptic curves over number fields . Sutherland has shown that over there is just one failure, which occurs for and a unique -invariant, and has given a classification of such failures when does not contain the quadratic subfield of the -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 . By constructing models of two modular curves, and , 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.
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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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. https://projecteuclid.org/euclid.ant/1513730229