In this article, originally motivated by examples coming from eigenvarieties, the author aims to provide new foundations for the algebraic side of the Iwasawa theory of Selmer groups of Galois representations over \(p\)-adic analytic spaces. More specifically, his approach allows to put the finite-slope non ordinary case onto an equal footing with ordinary cases in which \(p\) is inverted (in the classical situation, when working over a complete Noetherian local ring, inverting \(p\) amounts to passing to the generic fiber of the associated formal scheme). By developing a parallel theory for \((\varphi,\Gamma)\)-modules varying in analytic families of the same type, the author is able to mimic R. Greenberg’s “ordinary” Iwasawa theory [Adv. Stud. Pure Math. 17, 97–137 (1989; Zbl 0739.11045)] for Galois representations which look ordinary only on the level of their associated \((\varphi,\Gamma)\)-modules. Without being technically too precise one can give the following outline of the main results.
First, the author develops a theory of group cohomology of a profinite group \(G\) with coefficients in families of representations over a \(p\)-adic analytic space \(X\) over \(Q_p\). A family of \(G\)-representations over \(X\) is a locally finitely generated, flat \(\mathcal O_X\)-module \(M\), equipped with a continuous map \(G\to\operatorname{Aut}_{\mathcal O_X}^{\mathrm{cont}}(M)\). The first main theorem asserts that, if \(G\) has finite cohomology on all discrete \(G\)-modules of finite \(p\)-power order, vanishing in degrees \(\geq e\), then the continuous cohomology with values in \(\Gamma(Y, M)\), where \(Y\subseteq X\) ranges over affinoid subdomains, gives rise to a perfect complex of coherent \(\mathcal O_X\)-modules, vanishing in degrees \(\geq e\); if \(f: X'\to X\) is a morphism, there is a canonical isomorphism \(Lf^\ast R\Gamma_{\mathrm{cont}} (G, M)\to R\Gamma_{\mathrm{cont}}(G, f^\ast M)\). This allows to translate into the author’s context the theory of Selmer complexes.
Second, taking \(G\) to be the absolute Galois group \(G_K\) of a local \(p\)-adic field \(K\), one can formulate a notion of families of \((\varphi, \Gamma_K)\)-modules over \(X\), define their Galois cohomology, and give their basic functorial properties. The second main theorem asserts the existence of a functorial isomorphism \(R\Gamma_{\mathrm{cont}} (G_K,M)\to R\Gamma(G_K,D(M))\), where \(D(M)\) is the family of \((\varphi,\Gamma_K)\)-modules associated to \(M\). Note that the cohomology groups \(\mathrm H^{\mathrm i}(G_K,D)\) are finitely generated [K. S. Kedlaya et al., J. Am. Math. Soc. 27, No. 4, 1043–1115 (2014; Zbl 1314.11028)] and the above notion of ordinariness is very broad: for modular forms, it includes all cases “of finite slope” (i.e. having nonzero \(U_p\)-eigenvalue up to \(p\)-stabilization and twisting) and, on the automorphic side, having local Weil-Deligne representation at \(p\) that is nonsupercuspidal and of nonscalar Frobenius.
Third, the author studies the \(p\)-adic Hodge theory of \((\varphi,\Gamma_K)\)-modules, extending to them well-known notions and results for Galois representations. He thus defines ordinary \((\varphi,\Gamma_K)\)-modules and formulates the (strict) ordinary local condition in their Galois cohomology, which he compares to the Bloch-Kato local conditions. He concludes with a semicontinuity result on the rank of Selmer groups in an ordinary (in his sense) family. Various recent works (e.g. Kedlaya et al., loc. cit.) show the abundance of ordinary families.