Let \(E/\mathbb Q\) be an elliptic curve defined over \(\mathbb Q.\) For every prime \(p\), there is a natural homomorphism \[ \rho_{E,p} : G_{\mathbb Q}:=\mathrm{Gal}(\bar{\mathbb Q} / \mathbb Q) \to \mathrm{Aut}_{\mathbb Z_{p}}(T_{p}(E)) \] giving the action of \(G_{\mathbb Q}\) on the \(p\)-adic Tate module \(T_{p}(E)\).
A famous theorem of J.-P. Serre [Invent. Math. 15, 259–331 (1972; Zbl 0235.14012)] asserts that if \(E\) has no CM, then the image \(\rho_{E,p}(G_{\mathbb Q})\) has finite index in \( \mathrm{Aut}_{\mathbb Z_{p}}(T_{p}(E))\) and has index 1 for all but finitely many \(p\). The existence of a \(\mathbb Q\)-rational \(p\)-isogeny of \(E\) as well as the property of being a CM curve are constraints on the image of the map \(\rho_{E,p}\) and give rise to the following definition:
The curve \(E/\mathbb Q\) will be called \(p\)-exceptional if \(E\) has an isogeny of degree \(p\) defined over \(\mathbb Q\) and the image of \(\rho_{E,p}\) does not contain a Sylow pro-\(p\) subgroup of \(\mathrm{Aut}_{\mathbb Z_{p}}(T_{p}(E)).\)
R. Greenberg [Am. J. Math. 134, No. 5, 1167–1196 (2012; Zbl 1308.11060)] proved that for \(p>7\) there are no non-CM \(p\)-exceptional curves and, for \(p=5\) that the index of \(\rho_{E,p}\) in \(\mathrm{Aut}_{\mathbb Z_{p}}(T_{p}(E))\) cannot be divisible by \(5^{2}\) and is divisible by \(5\) if and only if \(E\) has a cyclic \(\mathbb Q\)-isogeny of degree \(5^{2}\) or two independent \(\mathbb Q\)-isogenies of degree \(5\).
In this paper the case of \(p=7\) is considered. The main result is that the only 7-exceptional elliptic curves, are the elliptic curves with complex multiplication by \(\mathbb Q(\sqrt{-7})\). This case has “additional interesting complications”. The problem is reduced to find the \(\mathbb Q\)-rational points of a curve of genus 12. This is itself an interesting problem. For the proof is used the method of Chabauty. The case of elliptic curves \(E\) defined over an arbitrary field \(K\) of characteristic different from \(7\), with a \(K\)-rational isogeny of degree 7 is also considered.