On the structure of Selmer groups of lambda-adic deformations over \(p\)-adic Lie extensions
Let \(E\) be an elliptic curve defined over a number field \(K\). Let \(R\) be a finite flat extension of \(\Lambda=\mathbb{Z}_pT\) and \(\rho:G_{\mathbb{Q}} \rightarrow \mathrm{GL}_2(\mathbb{R})\) a \(\Lambda\)-adic deformation of the classical representation \(\rho_0\) defined by the action of \(G_{\mathbb{Q}}\) on the \(p\)-adic Tate module \(T_pE\) of \(E\). Let \(K_\infty/K\) be a \(p\)-adic Lie extension unramified outside a finite set of primes \(S\) and with Galois group \(G\) not containing \(p\)-torsion (but containing a closed subgroup \(H\) such that \(G/H\simeq \mathbb{Z}_p\,\)). Let \(\mathcal{T}\) be the \(RG\)-module underlying the representation \(\rho\) and \(\mathcal{A}=\mathcal{T}\otimes_R \widehat{R}\) be the associated discrete module.NEWLINENEWLINEThe paper mainly deals with the structure of the (Pontrjagin duals of the) Selmer groups \(Sel(\mathcal{A}/K_\infty)\). The main tool is the homotopy theory presented in [\textit{U. Jannsen}, Adv. Stud. Pure Math. 17, 171--207 (1989; Zbl 0732.11061)] (see also [\textit{Y. Ochi} and \textit{O. Venjakob}, J. Algebr. Geom. 11, No.~3, 547--580 (2002; Zbl 1041.11041)] for a similar application) which the authors generalize to the setting of \(RG\)-modules, obtaining a big (``powerful'') diagram and spectral sequences analogue to the classical ones. Then, with some additional assumptions on the dimension of \(G\) and the surjectivity of the maps defining the Selmer groups, the authors provide a computation of \(RG\)-ranks of cohomology groups and prove that \(Sel(\mathcal{A}/K_\infty)^\vee\) has no nontrivial pseudo-null submodules (using computations with the \(Ext\) functors and the reflexivity of certain modules appearing in the powerful diagram).NEWLINENEWLINEExamples of extensions satisfying the hypotheses are the false Tate extension and the extension generated by the \(p^\infty\)-torsion of the elliptic curve \(E\). In the final sections the authors discuss the surjectivity of the maps defining the Selmer groups under the hypothesis \(H^2(K_S/K_\infty,\mathcal{A})=0\) (where \(K_S\) is the maximal algebraic extension of \(K\) unramified outside \(S\)) and consider the deformation \(R\) provided by Hida theory to study (with some control theorems) the behaviour of the Iwasawa invariants of the various specializations of such deformation.
scientific article; zbMATH DE number 6097014