On the \(\mathfrak M_H(G)\)-conjecture (Q2900375)

In this paper the authors study what they call the \(\mathfrak M_H(G)\)-conjectures which naturally arise in Iwasawa theory. Their validity is essential even for the formulation of main conjectures in (noncommutative) equivariant Iwasawa theory. They also propose a version of this conjecture for the dual Selmer group of an ordinary Hida family over an admissible \(p\)-adic Lie extension of the rational numbers.NEWLINENEWLINEMore precisely, fix an odd prime \(p\) and let \(F\) be a number field. Let \(F_{\infty}/F\) be an \textit{admissible} Galois extension of \(F\) that is:NEWLINENEWLINE(i) \(G := \text{Gal}(F_{\infty}/F)\) is a \(p\)-adic Lie group;NEWLINENEWLINE(ii) \(F_{\infty}/F\) is unramified outside a finite set of primes of \(F\);NEWLINENEWLINE(iii) \(F_{\infty}\) contains the cyclotomic \(\mathbb Z_p\)-extension \(F^{\text{cyc}}\) of \(F\).NEWLINENEWLINEPut \(H := \text{Gal}(F_{\infty} / F^{cyc})\) and \(\Gamma := G/H \simeq \mathbb Z_p\). Write \(\Lambda(G)\) for the Iwasawa algebra over \(G\) (and similarly for \(H\)). If \(W\) is a finitely generated (left) \(\Lambda(G)\)-module, we write \(W(p)\) for its \(p\)-primary part. Then \(\mathfrak M_H(G)\) denotes the set of all such \(W\) with the property that \(W/W(p)\) is finitely generated over \(\Lambda(H)\).NEWLINENEWLINELet \(A\) be an abelian variety defined over \(F\) and with good ordinary reduction at all places above \(p\). Then the original \(\mathfrak M_H(G)\)-conjecture of \textit{J.~Coates} et al., [Publ. Math., Inst. Hautes Étud. Sci. 101, 163--208 (2005; Zbl 1108.11081)] asserts that \(X(A/F_{\infty})\), the dual of the Selmer group of \(A\) over \(F_{\infty}\), belongs to \(\mathfrak M_H(G)\). When \(H\) is finite, this recovers a conjecture of \textit{B.~Mazur} [Invent. Math. 18, 183--266 (1972; Zbl 0245.14015)].NEWLINENEWLINELet \(K\) be a finite extension of \(\mathbb Q_p\) and let NEWLINE\[NEWLINE\rho: \text{Gal}(\overline F / F) \rightarrow \Aut_K(V)NEWLINE\]NEWLINE be a continuous Galois representation, where \(V\) is a two dimensional \(K\)-vector space. Under certain conditions on \(\rho\) the authors define the Selmer group \(\text{Sel}(T/F_{\infty})\) and its dual \(X(T/F_{\infty})\), where \(T\) is an \(\mathcal O_K\)-lattice in \(V\) which is stable under the Galois action. The \(\mathfrak M_H(G)\)-conjecture then asserts that \(X(T/F_{\infty})\) belongs to \(\mathfrak M_H(G)\) (one has to use Iwasawa algebras with coefficients in \(\mathcal O_K\) rather then \(\mathbb Z_p\), and one has to replace \(p\)-primary parts by \(\pi\)-primary parts, where \(\pi \in \mathcal O_K\) is a uniformizer). The authors then give some general evidence for this conjecture.NEWLINENEWLINESeveral equivalent formulations are given in the case that also \(H \simeq \mathbb Z_p\). In particular, the case \(G = \mathbb Z_p \times \mathbb Z_p\) is studied in some detail.NEWLINENEWLINEIn the final section, the authors propose an analogue of this conjecture for the dual of the Selmer group of an ordinary Hida family over an admissible \(p\)-adic Lie extension \(F_{\infty}\) over \(\mathbb Q\). Note that their definition of the Selmer group differs somewhat from other cases considered in the literature. The interest of this conjecture certainly comes from the aim to formulate a main conjecture for such Hida families. The authors hope to come back to this question in a subsequent paper.NEWLINENEWLINEFor the entire collection see [Zbl 1237.11001].