#### Vol. 8, No. 5, 2014

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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 $ℚ$ 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 ${PGL}_{2}\left({\mathbb{F}}_{l}\right)$. By constructing models of two modular curves, ${X}_{s}\left(5\right)$ and ${X}_{{S}_{4}}\left(13\right)$, 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
Primary: 11G05
Secondary: 11G18