Tetrahedral elliptic curves and the local-global principle for isogenies
For a prime number \(\ell\) and an algebraic number field \(K\), the local-global principle for rational \(\ell\)-isogenies of an elliptic curve \(E/K\) establishes that if the reduction of \(E\) at almost every prime ideal of \(K\) of good reduction admits an \(\ell\)-isogeny over the residue field of the prime ideal, then \(E\) admits an \(\ell\)-isogeny over \(K\). Let \(\ell\) be odd and put \(\ell^*=(-1)^{(\ell-1)/2}\ell\). In [J. Théor. Nombres Bordx. 24, No. 2, 475--485 (2012; Zbl 1276.11095)] \textit{A. V. Sutherland} showed that, under the assumption that \(\sqrt{\ell^*}\not\in K\), if the local-global principal fails for \((\ell,E)\), then \(\ell\equiv 3 \mod 4\) and the image \(H_{E,\ell}\) of the projective Galois representation on the \(\ell\)-torsion group of \(E\) is a dihedral subgroup \(D_{2n}\) of \(\mathrm{PGL}_2(\mathbb F_\ell)\)~(\(n\) is a divisor of \((\ell-1)/2\)). Further he showed that for \(K=\mathbb Q\), only one counterexample occurs when \(\ell=7\). In this article, the authors deal with the case \(\sqrt{\ell^*}\in K\). They call a pair \((\ell,j_0)\) an exceptional pair if the principle fails for \(\ell\) and an elliptic curve \(E\) with the \(j\)-invariant \(j(E)=j_0\). They give a necessary and sufficient condition for \((\ell,j_0)\) to be exceptional. In particular, in the failure case, \(\ell\equiv 1\mod 4\) and there are two sorts of exceptional pairs:~\(H_{E,\ell}\cong D_{2n}\) or \(H_{E,\ell}\) is one of \(A_4,S_4\) and \(A_5\). The authors consider for quadratic fields the case where the principle does not hold . For \(K=\mathbb Q(\sqrt 5)~(\ell=5)\), they show that there are infinitely many exceptional pairs \((5,j(E))\) with \(H_{E,5}\cong D_4\) and those \(j(E)\) are explicitly given. For the non-dihedral pairs, they prove that the only non-dihedral exceptional prime \(\ell\) over any quadratic field is \(13\) over \(\mathbb Q(\sqrt{13})\), where the projective image is \(H_{E,13}\cong A_4\). For the proof, they compute defining equations and \(K\)-rational points of the modular curves \(X_s(5)\) and \(X_{S_4}(13)\). Here \(X_s(\ell)\) (resp.\(X_{S_4}(\ell)\)) is the modular curve corresponding to the normalizer of a split Cartan subgroup (resp \(S_4\)).
scientific article; zbMATH DE number 6348604