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 $\mathbb{Q}$ 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.