## Algebra & Number Theory

### Tetrahedral elliptic curves and the local-global principle for isogenies

#### 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 $PGL2(Fl)$. 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

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

Dates
Revised: 25 March 2014
Accepted: 26 April 2014
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513730229

Digital Object Identifier
doi:10.2140/ant.2014.8.1201

Mathematical Reviews number (MathSciNet)
MR3263141

Zentralblatt MATH identifier
1303.11066

#### Citation

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

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