Torsion points of elliptic curves with bad reduction at some primes (Q2839759)

Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\): it is well known that in \(E(\mathbb{Q})\) there are no \(p\)-torsion points for \(p\geq 11\). The paper aims at finding conditions for the absence of 5 and 7-torsion points as well. The main result is the following:NEWLINENEWLINELet \(p=5\) or 7 and let \(E\) be an elliptic curve having bad reduction only at primes \(\ell\neq p\) such that \(\ell\not\equiv \pm 1\pmod{p}\), then \(E(\mathbb{Q})\) has no \(p\)-torsion.NEWLINENEWLINEThe author provides two different proofs of this fact. In the first, assuming the presence of a \(p\)-torsion points (which implies semistable reduction at all primes \(\neq p\)), one proves the decomposition \(\mathcal{E}[p]\simeq \mathbb{Z}/p\times \mu_p\) as group schemes over \(\mathbb{Z}[\frac{1}{N}]\) (where \(\mathcal{E}\) is the Neron model of \(E\) and \(N\) the product of bad reduction primes). Then using the sequence of isogenies \(E {\buildrel \varphi_1\over\longrightarrow} E_1 {\buildrel \varphi_2\over\longrightarrow} E_2 {\buildrel \varphi_3\over\longrightarrow} E_3 \dots \) with \(Ker(\varphi_i)=\mu_p\) for any \(i\) and Shafarevich theorem on the finiteness of the isomorphism classes, the author proves that some of the \(E_i\)'s have complex multiplication and this leads to a contradiction for the semistable reduction.NEWLINENEWLINE The second proof again assumes the presence of a \(p\)-torsion point and reduces to the case in which \(\mathbb{Q}(E[p])\) is a (ramified) extension of degree \(p\) of \(\mathbb{Q}(\zeta_p)\): using the Tate parametrization at the ramified prime \(\mathfrak{q}\) (which is of bad reduction), one proves that \(\mathfrak{q}|\ell \equiv \pm 1\pmod{p}\). This second proof can be generalized to a number field \(K\) (with some extra hypotheses on \(K(\zeta_p)\,\)) and the author concludes with some computations of bad reduction primes for curves admitting a 5 or 7-torsion point.