Fine Selmer group of Hida deformations over non-commutative \(p\)-adic Lie extensions (Q436041)

Let \(E\) be an elliptic curve of conductor \(N\), defined over an abelian number field \(K\). For an odd prime \(p\) let \(K_{cyc}\) be the cyclotomic \(\mathbb{Z}_p\)-extension of \(K\). Fix a finite set of primes \(S\) containing the infinite places and the primes dividing \(Np\) and consider a \(p\)-adic Lie extension \(\mathcal{L}\) of \(K\) contained in \(K_S\) (the maximal algebraic extension of \(K\) unramified outside \(S\)). By Hida theory one can associate to a \(\Lambda\)-adic newform \(\mathcal{F}\) a quotient \(\mathbb{H}_{\mathcal{F}}^{\text{ord}}\) of the the Hecke algebra \(\mathbb{H}_{Np\infty}^{\text{ord}}\) and a ``large'' irreducible representation \(\rho:G_{\mathbb{Q}} \rightarrow \Aut_{\mathbb{H}_{\mathcal{F}}^{\text{ord}}}(\mathcal{T}_{\mathcal{F}})\) (where \(\mathcal{T}_{\mathcal{F}}\) is a torsion free \(\mathbb{H}_{\mathcal{F}}^{\text{ord}}\)-module of generic rank 2 and we let \(\mathcal{A}\) be its discrete dual). Assuming that {\parindent=12mm \begin{itemize}\item[(Nor)] \(\mathbb{H}_{\mathcal{F}}^{\text{ord}}\simeq \mathcal{O}[[\Gamma']]\) (where \(\mathcal{O}\) is a finite extension of \(\mathbb{Z}_p\) and \(\Gamma'\simeq 1+p\mathbb{Z}_p\) is the group of diamond operators for the tower of modular curves \(\{Y_1(p^t)\}_{t\geq 1}\,\)) and NEWLINE\item [(Irr)] the residual representation of \(\mathcal{T}_{\mathcal{F}}\) is an absolutely irreducible \(G_{\mathbb{Q}}\)-module, NEWLINENEWLINE\end{itemize}} one associates to any arithmetic point \(\xi \in\mathfrak{X}_{\text{arith}}(\mathbb{H}_{\mathcal{F}}^{\text{ord}}) \subset \text{Hom}_{\mathbb{Z}_p}(\mathbb{H}_{\mathcal{F}}^{\text{ord}},\overline{\mathbb{Q}_p})\) a cuspidal ordinary form \(f_\xi\) which arises as the specialization of \(\mathcal{F}\) at \(\xi\).NEWLINENEWLINE The paper deals with the structure of (Pontryagin duals of) fine Selmer groups \(\mathcal{R}(\mathcal{A}/\mathcal{L})\) (resp. \(\mathcal{R}(\mathcal{A}_{f_\xi}/\mathcal{L})\,\)) defined over \(\mathcal{L}\) by the Galois representation associated to \(\mathcal{F}\) (resp. to the \(f_\xi\)'s). It is conjectured that \(\mathcal{R}(\mathcal{A}/K_{\text{cyc}})^\vee\) is a finitely generated \(\mathbb{H}_{\mathcal{F}}^{\text{ord}}\)-module and, if this is true and \(\dim \text{Gal}(\mathcal{L}/K)\geq 2\), one also expects \(\mathcal{R}(\mathcal{A}/\mathcal{L})^\vee\) to be a finitely generated torsion \(\mathbb{H}_{\mathcal{F}}^{\text{ord}}[[\text{Gal}(\mathcal{L}/K_{cyc})]]\)-module.NEWLINENEWLINEIn a previous paper, \textit{S. Jha} and \textit{R. Sujatha} [J. Algebra 338, No.~1, 180--196 (2011; Zbl 1245.11071)] proved the equivalence of the first conjecture and of analogous statements for one (or for every) specialization \(f_\xi\,\), i.e., \(\mathcal{R}(\mathcal{A}_{f_\xi}/K_{\text{cyc}})^\vee\) is finitely generated over the (local) ring \(\mathcal{O}_{f_\xi}\) generated by the Fourier coefficients of \(f_\xi\,\).NEWLINENEWLINEIn the present paper the author proves a similar result for the second conjecture, using a control theorem for the maps \(\mathcal{R}(\mathcal{A}_{f_\xi}/\mathcal{L}) \rightarrow \mathcal{R}(\mathcal{A}/\mathcal{L})[p_\xi]\) (where \(p_\xi\) is the ideal such that \(\mathcal{T}_{\mathcal{F}}/p_\xi \simeq \mathcal{T}_{f_\xi}\,\)) to link the fine Selmer groups of the \(\Lambda\)-adic newform to the ones of its specializations and, under some additional hypotheses, obtain equalities between the \(\mathcal{O}_{f_\xi}[[\text{Gal}(\mathcal{L}/K)]]\)-ranks of \(\mathcal{R}(\mathcal{A}_{f_\xi}/\mathcal{L})^\vee\) and \(\mathcal{R}(\mathcal{A}/\mathcal{L})^\vee/p_\xi\,\). In the final sections the author provides the statements of analogous results for the Selmer groups and numerical examples where \(\mathcal{L}\) is the false Tate extension.