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 $\mbox{PGL}2(\mathbb{F}_l)$. By constructing models of two modular curves, $X{\text{s}}(5)$ and $X_{S_4}(13)$, 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.