Isogenies of degree \(p\) of elliptic curves over local fields and Kummer theory.
The author is interested in the determination of the Selmer group of a \(p\)-isogeny of elliptic curves over number fields. For that purpose, he considers the following local situation. Let \(p\) be a prime number, \(K\) a finite extension of \(\mathbb Q_p\) and \(\overline{K}\) an algebraic closure of \(K\). Let \(E/K\) and \(E'/K\) be two elliptic curves and \(\nu: E\to E'\) an isogeny defined over \(K\) of degree \(p\). Let \(G_K\) be the Galois group of \(\overline{K}\) over \(K\). We have the exact sequence of \(G_K\)-modules \[ 1\to\text{Ker}(\nu)\to E(\overline{K})@>\nu>> E'(\overline{K})\to 1. \] Taking Galois cohomology yields the injection \[ 1\to E'(K)/\nu(E(K))@>\delta>> H^1(G_k,\text{Ker}(\nu)), \] \(\delta\) being the connecting homomorphism. The author assumes that \(\text{Ker}(\nu)\) is contained in \(E(K)\) and the group \(\mu_p\) of the \(p\)th roots of unity is contained in \(K\). Fixing an isomorphism \(\kappa\) between \(\text{Ker}(\nu)\) and \(\mu_p\), we obtain by Kummer theory a homomorphism, still denoted \(\delta\), \[ \delta:E'(K)/\nu(E(K))\to K^*/K^{*p}, \] whose image does not depend on the choice of the isomorphism \(\kappa\). The author describes the image of \(\delta\) in case \(E\) has good reduction over \(K\), or multiplicative reduction over \(K\) and \(p\neq2\), in terms of filtrations of \(K^*/K^{*p}\) and \(E'(K)/\nu(E(K))\). Let us mention the result obtained by the author in case \(E\) has good ordinary reduction over \(K\). Let \(v\) be the normalised valuation over \(K\) and \(U^0\) the unit group of the valuation ring of \(K\). For all \(i\geq1\), let \(U^i\) be the subgroup of \(U^0\) of the elements \(x\) such that \(v(x-1)\geq i\). We note \[ C^i= U^i/(K^{*p}\cap U^i) \quad\text{for all }i\geq 0, \] and \(e_0\) the ramification index of the extension \(K/\mathbb Q_p(\mu_p)\). One has the filtration \[ \{1\}=C^{pe_0+1}\subseteq\cdots C^1\subseteq C^0\subseteq K^*/K^{*p}. \] Let \(\pi\) be the reduction map. The author proves that the image of \(\delta\) is \(C^1\) if \(\pi(\text{ker}(\nu))=\{0\}\), and \(C^{e_0p}\) if \(\pi(\ker(\nu))\neq\{0\}\).
scientific article; zbMATH DE number 1930963
scientific article