Ranks of Selmer groups in an analytic family (Q2841369)

First introduce some notations. Let \(K\) be a number field, \(\Sigma\) a finite set of finite places of \(K\) which contains the set \(\Sigma_p\) of places above a (fixed) odd prime number \(p\). Let \(G_{K,\Sigma}\) be the Galois group of the maximal algebraic extension of \(K\) unramified outside \(\Sigma\). For any place \(v\) of \(K\), \(K_v\) is the completion of \(K\) at \(v\) and \(G_v=G_{K_v}\) is the absolute Galois group of \(K_v\). There is a natural morphism \(G_v\rightarrow G_{K,\Sigma}\) (up to conjugacy) which allows us to view a representation of \(G_{K,\Sigma}\) as a representation of \(G_v\).NEWLINENEWLINELet \(S\) be an affinoid algebra over \(\mathbb{Q}_p\) and \(d\geq 1\) be an integer. By a family of representations \(V\) of \(G_{K,\Sigma}\) of dimension \(d\) over \(S\), we mean a locally free \(S\)-module of rank \(d\) with a continuous \(S\)-linear action of \(G_{K,\Sigma}\). For any point \(x\) in the rigid analytic space \(\mathrm{Sp} S\) attached to \(S\), we get by specialization a \(p\)-adic representation \(G_{K,\Sigma}\rightarrow L(x)\) denoted by \(V_x\), where \(L(x)\) is the residue field at \(x\). The family \(V\) is said \textit{trianguline} if, viewed as a family of representations of \(G_v\), the attached \((\phi,\Gamma)\)-module \(D^{\dagger}(V)\) over the relative Robba ring \(\mathcal{R}_S\) is trianguline in the sense of Colmez.NEWLINENEWLINEThe paper under review studies the variation of the dimension of \(H^1_f(G_{K,\Sigma},V_x)\) when \(x\) runs along points of \(\mathrm{Sp} S\) for suitable families \(V\) of representations of \(G_{K,\Sigma}\). Here, the Selmer group \(H^1_f(G_{K,\Sigma},V_x)\) is defined as NEWLINE\[NEWLINE\mathrm{ker}\big(H^1(G_{K,\Sigma},V_x)\rightarrow \prod_{v\in\Sigma_p}H^1(G_v,V_x\otimes B_{\text{crys}})\big).NEWLINE\]NEWLINE In general, the variation is chaotic, see the example given in the introduction of the paper. However, the author proves that for families \(V\) which are trianguline at every \(p\in\Sigma_p\), non-critical and non-exceptional, and with a fixed number of non-negative Hodge-Tate weights, the dimension of the Selmer group varies lower-semi-continuously.NEWLINENEWLINEIn application to families of representations carried by the eigenvarieties, one needs to check that such families are trianguline. Theorem 2 of the paper provides a result of this type stating that, if \(V\) is a refined family (in the sense of Chenevier-Bellaïche) of rank 2, and \(x\in \mathrm{Sp} S\) is a classical point such that \(V_x\) is non-critical, then there exists an affinoid neighborhood of \(x\) on which \(V\) becomes trianguline. This property, also called global triangulation, has been largely generalized by the work of \textit{R. Liu} [Comment. Math. Helv. 90, No. 4, 831--904 (2015; Zbl 1376.11029)], and \textit{K. S. Kedlaya} et al. [J. Am. Math. Soc. 27, No. 4, 1043--1115 (2014; Zbl 1314.11028)].