Almost rational torsion points on semistable elliptic curves (Q2726327)

An \textit{almost rational torsion point} on an abelian variety \(A\) over a perfect field \(K\) is a geometric torsion point \(P\) of \(A\) with the property that whenever \(\sigma\) and \(\tau\) are elements of the absolute Galois group of \(K\) such that \(P^\sigma + P^\tau = 2P\), we must have \(P = P^\sigma = P^\tau\). The definition is due to Ribet, who used the concept to show that the Manin-Mumford conjecture follows from results of Serre. Ribet's argument requires the analysis of almost rational torsion points on abelian varieties of dimension greater than~\(1\). NEWLINENEWLINENEWLINEThe present paper considers the case of almost rational torsion points on semistable elliptic curves over \(\mathbb{Q}\). The main theorem is that a non-rational torsion point \(P\) on such an elliptic curve \(E\) is almost rational if and only if it can be written as a sum \(Q + R + S\), where \(Q\) generates a \(\mu_3\) subgroup of \(E[3]\), where \(R\) is an element of \(E(\mathbb{Q})[9]\), and where \(S\) is an element of \(E(\mathbb{Q}(\zeta_3))[16]\). Using this theorem the author easily produces an elliptic curve whose set of almost rational torsion points does not form a group. NEWLINENEWLINENEWLINEThe proof of the main theorem depends on the study of the Galois module~\(M\) generated by the Galois conjugates of a non-trivial almost rational torsion point~\(P\). The author uses the modularity of \(E\) to show that if \(\ell\) is the largest prime dividing the order of~\(P\), then \(E[\ell]\) is a reducible Galois module, so that \(\ell\) is at most~\(7\) (by Mazur's theorem). The proof is completed by a case-by-case analysis of the possible ramification of the module \(M\) at the primes~\(p \leq 7\).