Control theorems for Selmer groups of nearly ordinary deformations (Q2890364)

Let \({\mathcal F}\) be a free module of rank two over a complete local noetherian ring \(R\) of characteristic zero with finite residue field of characteristic \(p\) on which the absolute Galois group \(G_F\) of a number field \(F\) acts continuously. Fix a Zariski-dense subset \(S\) of \(\text{Hom}_{\text{cont}}(R,\bar{{\mathbb Q}_p})\). If \(({\mathcal F}, R, S)\) satisfies certain properties, it was suggested by \textit{R. Greenberg} [``\(L\)-functions and arithmetic'', Lond. Math. Soc. Lect. Note Ser. 153, 211--233 (1991; Zbl 0727.11043); ``Iwasawa theory and \(p\)-adic deformations of motives'', Proc. Symp. Pure Math. 55, Pt. 2, 193--223 (1994; Zbl 0819.11046)] that there exists a non-zero divisor \(p\)-adic \(L\)-function \(L_p ({\mathcal F})\in R\) interpolating special values of the complex \(L\)-function \(L(M_{\kappa},0)\), \(\kappa\in S\), in the sense that, for all \(\kappa\in S\), \(\kappa(L_p({\mathcal F})\) is equal to \(L(M_{\kappa},0)\) divided by a suitable complex period of \(M_{\kappa}\) and multiplied by Euler factors for all \({\mathfrak p}\mid p\).NEWLINENEWLINENEWLINELet \({\mathcal A}\) be the discrete \(G_F\)-representation \({\mathcal F} \otimes_R D_p(R)\) where \(D_p\) is the Pontrjagin dual. Similarly is defined \({\mathcal A}_{\kappa}\) for \(\kappa \in S\). Greenberg defined Selmer groups \(\text{Sel}_{\mathcal A}\) and \(\roman{Sel}_{{\mathcal A}_{\kappa}}\) as subgroups of a suitable Galois cohomology group with coefficients in \({\mathcal A}\) and \({\mathcal A}_{\kappa}\) respectively.NEWLINENEWLINENEWLINEThe combination of the Iwasawa Main Conjecture proposed by Greenberg and the conjecture that \(D_p(\text{Sel}_{\mathcal A})\) (resp. \(D_p(\text{Sel}_{{\mathcal A}_{\kappa}})\)) is linked with \(L_p( {\mathcal F})\) (resp. \(L(M_{\kappa},0)\)), with the interpolation property defining \(L_p({\mathcal F})\), suggests that both \(D_p( \text{Sel}_{\mathcal A})\) and \(D_p(\text{Sel}_{\mathcal A}) \otimes_R\kappa(R)\) should be linked with \(\kappa(L_p( {\mathcal F}))\) and thus the natural map NEWLINE\[NEWLINE D_p(\roman{Sel}_{\mathcal A})\otimes_R \kappa(R)\to D_p(\roman{Sel}_{{\mathcal A}_{\kappa}}) NEWLINE\]NEWLINE should be very close to an isomorphism. When this holds, we say that the Selmer groups satisfy a \textit{control theorem at \(\kappa\)}.NEWLINENEWLINEIn this paper the authors consider the Pontryagin dual \(D_p (\text{Sel}_{\mathcal A})\) of \(\roman{Sel}_{\mathcal A}\) as well as the second cohomology group \(\tilde{H}^2_f(F,{\mathcal F})\) of the Selmer complex \(R\Gamma_f(F,{\mathcal F})\) as defined and studied by \textit{J. Nekovář} [Selmer complexes. Astérisque 310. Paris: Société Mathématique de France (2007; Zbl 1211.11120)] and prove control theorems for these objects. The precise statements are given in Proposition 3.5 and Corollary 3.6.