Ramification in the division fields of an elliptic curve
The author studies the ramification properties of the fields generated by torsion points of elliptic curves. In the first section, he starts with the following situation. Let \(p\) be a prime number, \(K\) a finite extension of the \(p\)-adic field \(\mathbb Q_p\) and \(\overline{K}\) an algebraic closure of \(K\). Let \(E\) be an elliptic curve defined over \(K\) and \(\ell\) a prime number distinct from \(p\). For any \(n\geq 1\), let us denote by \(E[\ell^n]\) the subgroup of the \(\ell^n\)-torsion points of \(E(\overline{K})\) and by \(K(E[\ell^n])\) the \(\ell^n\)-division field of \(E\) over \(K\). Let us recall that by the Néron-Ogg-Shafarevich criterion, \(E\) has good reduction over \(K\) if and only if for all \(n\geq 1\) the extension \(K(E[\ell^n])/K\) is unramified. Below the symbols \(I_\nu\), \(I_\nu^*\) \((\nu\geq 0)\) denote the Kodaira type of \(E\) over \(K\). The author proves the following results: 1) Suppose \(\ell\geq3\). The extension \(K(E[\ell])/K\) is unramified if and only if the Kodaira type of \(E\) is \(I_\nu\) with \(\nu\geq0\) and \(\nu\equiv0\bmod\ell\). 2) Suppose \(\ell=2\). The extension \(K(E[2])/K\) is unramified if and only if the Kodaira type of \(E\) is \(I_\nu\) or \(I_\nu^*\) with \(\nu\geq0\) and \(\nu\) even. 3) Assume that \(K(E[\ell])/K\) is unramified and \(E\) has bad reduction over \(K\). 3.1) If the Kodaira type of \(E\) is \(I_\nu\) \((\nu\geq1)\), then \(K(E[\ell^n])/K\) is unramified if and only if \(1\leq n\leq v_\ell(\nu)\) (\(v_\ell(\nu)\) being the \(\ell\)-adic valuation of \(\nu\)). 3.2) If the Kodaira type of \(E\) is \(I_\nu^*\) with \(\nu\geq0\), then \(K(E[\ell^n])/K\) is unramified if and only if \(n=1\). He also describes the pairs \((E,p)\) for which the extension \(K(E[\ell])/K\) is wildly ramified. In the second section, the author obtains results in case \(\ell=2\) and \(K=\mathbb Q_p\), \(p\) being any prime number. In the last section, as an application, he is interested in a characterization of the quadratic fields whose class numbers are divisible by 3, in terms of the existence of an elliptic curve defined over \(\mathbb Q\) having precise reduction properties; there is in fact an imprecision in the statement of the corresponding theorem: a counterexample is given by the field \(k=\mathbb Q(\sqrt{177})\), whose ring of integers is principal, and the elliptic curve \(E\) numbered 885B1 in Cremona's tables. Anyway, the results proved by the author allow probably to get a right analoguous statement with a little modification. Moreover, he gives some conditions on \(p\) which assure the nonexistence of elliptic curves over \(\mathbb Q\) having good reduction outside \(p\).
scientific article; zbMATH DE number 2063943
scientific article