Torsion points and reduction of elliptic curves (Q2833609)

Let \(E\) be an elliptic curve defined over a number field \(K\) and let \(p\) be a prime number. The paper deals with the relation between the existence of a \(K\)-rational \(p\)-torsion point of \(E\) and the reduction of \(E\) at certain primes of \(K\). Let \(p\geqslant 5\), define NEWLINE\[NEWLINE \mathfrak{S}_{K,p}:=\{ \mathfrak{q},\,\mathfrak{q}\;\text{a\;prime\;of}\;K,\;\mathfrak{q}\cap\mathbb{Z}=(\ell)\;\text{with}\;\ell\neq p\;\text{and}\;\ell^{f_{\mathfrak{q}}}\not\equiv 1 \pmod{p}\,\} NEWLINE\]NEWLINE (where \(f_{\mathfrak q}\) is the inertia degree of \(\mathfrak{q}\)) and assume that \(E\) has good reduction outside \(\mathfrak{S}_{K,p}\). The author uses a result of \textit{R. Schoof} [Math. Ann. 325, No. 3, 413--448 (2003; Zbl 1058.11038)] (which holds under some hypotheses on the behaviour of primes of \(K\) dividing \(p\)) on the prolongation of group schemes like \(E[p]\) and Shafarevich's Theorem, to show that if \(E\) has a \(K\)-rational \(p\)-torsion point, then \(E\) has CM by an imaginary quadratic subfield of \(K\) and has good reduction everywhere. This theorem applies in particular to cyclotomic fields, showing that for a regular prime \(p\geqslant 11\), an elliptic curve \(E/\mathbb{Q}\) with good reduction everywhere over \(\mathbb{Q}(\zeta_p)\) cannot have \(\mathbb{Q}(\zeta_p)\)-rational \(p\)-torsion points.