Tetrahedral Elliptic Curves and the local-global principle for Isogenies

TL;DR

This paper classifies failures of the local-global principle for $l$-isogenies of elliptic curves over number fields, identifying new exceptional cases linked to subgroup structures and modular curves.

Contribution

It provides a comprehensive classification of such failures over quadratic fields, including new sources from exceptional subgroups and modular curve models.

Findings

Only one failure over $\\mathbb{Q}$ at $l=7$ with a unique $j$-invariant.

New failures arise from exceptional subgroups of $\mathrm{PGL}_2(\mathbb{F}_l)$.

Constructed models of modular curves $X_{\text{s}}(5)$ and $X_{S_4}(13)$ reveal new elliptic curve families.

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.