Selmer groups and generalized class field towers (Q2890244)

Let \(K\) be a number field and let \(A/K\) be an abelian variety: a celebrated ``control theorem'' of \textit{B. Mazur} [Invent. Math. 18, 183--266 (1972; Zbl 0245.14015)] asserts that for every prime number \(p\) such that \(A\) has good ordinary reduction at all \(v\mid p\), and for every \(\mathbb{Z}_p\)-extension \(K_\infty/K\) the natural maps induced by restriction NEWLINE\[NEWLINE \mathrm{Sel}_p(A/K_n)\rightarrow \mathrm{Sel}_p(A/K_\infty)^{\mathrm{Gal}(K_\infty/K_n)} NEWLINE\]NEWLINE have finite kernels and cokernels, whose orders are bounded as \(K_n\) varies through the intermediate layers of \(K_\infty/K\). The purpose of the paper under review is to address a similar control theorem in a different setting, namely when \(K_\infty/K\) is a pro-\(p\) extension unramified outside a finite set \textit{which does not contain any prime above} \(p\). Since it is well known (see, for instance, Theorem 13.2 in [\textit{L. C. Washington}, Introduction to cyclotomic fields. 2nd ed. New York, NY: Springer (1997; Zbl 0966.11047)]) that only primes above \(p\) can ramify in a \(\mathbb{Z}_p\)-extension, this case is somehow ``orthogonal'' to the one treated by Mazur; more precisely, one conjectures that in the setting considered by the author the Galois group \(\mathrm{Gal}(K_\infty/K)\) is never \(p\)-adic analytic, in contrast to \(\mathbb{Z}_p\).NEWLINENEWLINEThe main results of the paper are the following Theorems A and B for which we need a bit of notation. First of all, fix two finite sets \(R,S\) of primes of \(K\) such that \(R\) does not contain any prime above \(p\) and let \(K_\infty\) be the maximal pro-\(p\) extension of \(K\) which is unramified outside of \(R\) and such that all primes in \(S\) split completely: accordingly, put \(\Sigma=\mathrm{Gal}(K_\infty/K)\). The author denotes by \(\mathcal{E}\) the set of all finite extensions \(K'/K\) contained in \(K_\infty/K\) and considers, for \(K'\in\mathcal{E}\), the map induced by restriction NEWLINE\[NEWLINEs_{K'}:\mathrm{Sel}_p(A/K')\rightarrow \mathrm{Sel}_p(A/K_\infty)^{\mathrm{Gal}(K_\infty/K')}\;.NEWLINE\]NEWLINE He introduces a condition (C) which is a bit technical to be repeated here (see page \(883\)) and writes \(r_p(\cdot)\) for the \(p\)-rank of a pro-finite abelian group.NEWLINENEWLINETheorem A. For any \(K'\in\mathcal{E}\) the kernel and the cokernel of \(s_{K'}\) are finite. Moreover, NEWLINE\begin{itemize}NEWLINE\item[(i)] if \(A(K)[p]=\{0\}\), then \(\mathrm{ker}\;s_{K'}=0\) for any \(K'\in\mathcal{E}\). Assuming condition {(C)}, \(\mathrm{coker}\;s_{K'}=0\) for every \(K'\in\mathcal{E}\); NEWLINE\item[(ii)] if \(A(K)[p]\neq \{0\}\) and assuming that \(\Sigma\) is not \(p\)-adic analytic, NEWLINE\begin{itemize}NEWLINE\item[(a)] if condition {(C)} holds, then \(r_p(\mathrm{ker}\;s_{K'})\) is unbounded as \(K'\) varies in \(\mathcal{E}\); NEWLINE\item[(b)] if the Mordell-Weil rank of \(A(K')\) is bounded as \(K'\) varies, if the Tate-Šafarevič group \(\text{Ш}(A/K')[p^\infty]\) is finite for every \(K'\in\mathcal{E}\) and if \(K\) is totally imaginary in case \(p=2\), then \(r_p(\mathrm{coker}\;s_{K'})\) is unbounded as \(K'\) varies in \(\mathcal{E}\). NEWLINE\end{itemize} NEWLINE\end{itemize}NEWLINENote the striking dichotomy in the behaviour of the \(p\)-rank of the kernels (and cokernels) of \(s_{K'}\) according as \(A(K)[p]\) is trivial or not. As a consequence, the author proves (Corollary A) that if the above hypothesis are satisfied and \(A/K\), \(A'/K\) are two isogenous abelian varieties over \(K\) such that \(A(K)[p]=\{0\}\) but \(A'(K)[p]\neq \{0\}\) while \(\mathrm{Sel}_p(A'/K)=\{0\}\) then condition (C) implies that \(\text{Ш}(A/K_\infty)\) is \(0\) while \(\text{Ш}(A'/K_\infty)\) contains an infinite elementary abelian \(p\)-group. The second main result is the followingNEWLINENEWLINETheorem B. If \(\Sigma\) is not \(p\)-adic analytic and \(A\) is an abelian variety defined over \(K\) with \(A(K)[p]\neq\{0\}\) and condition {(C)} is satisfied, then \(r_p\big(\text{Ш}(A/K')\big)\) is unbounded as \(K'\) varies through \(\mathcal{E}\).NEWLINENEWLINEThe paper consists of five sections. The first two are introductory, where notations are fixed and some preliminaries about the size of the cohomology of \(\Sigma\) are discussed. Section 2 also contains one crucial result which appears frequently in the proofs, namely (Theorem 2.5) that \(A(K_\infty)[p^\infty]\) is finite. Section 3 computes explicity some cohomology group both of \(\Sigma\) and of \(G_T\), the Galois group of the maximal extension of \(K\) unramified outside all primes where \(A\) has bad reduction, the infinite primes and the \(p\)-adic primes. Although this extension is not directly related to \(K_\infty\), which is unramified at primes above \(p\), its cohomology groups play a fundamental role in the author's strategy. Indeed, in order to analyze the kernel and the cokernel of \(s_{K'}\) he follows \textit{R. Greenberg} [Compos. Math. 136, No. 3, 255--297 (2003; Zbl 1158.11319)]) and studies the commutative diagram NEWLINE\[NEWLINE\begin{tikzcd} NEWLINE0\rar & \mathrm{Sel}_p(A/K_\infty)^{\Sigma_{K^\ast}}\rar & H^1(G_T(K_\infty),A[p^\infty])^{\Sigma_{K^\ast}}\rar["{\psi_{K_\infty}}"] & \mathrm{img}(\psi_{K_\infty})\rar & 0\\NEWLINE0 \rar & \mathrm{Sel}_p(A/K^\ast)\uar["s_{K^\ast}"]\rar & H^1(G_T(K^\ast),A[p^\infty])\uar["h_{K^\ast}" '] \rar["\lambda_{K^\ast}" '] & \mathrm{img}(\lambda_{K^\ast})\rar \uar["g^\ast_{K^\ast}" '] & 0 \rlap{\,.} NEWLINE\end{tikzcd} NEWLINE\]NEWLINE Section 4 is the core of the paper, containing the proofs of Theorems A. and B. and of Corollary A. The main ingredients are \textit{A. Mattuck}'s theorem describing explicitly the Mordell-Weil group of an abelian variety over a local field [Ann. Math. (2) 62, 92--119 (1955; Zbl 0066.02802)], the result that \(A(K_\infty)[p^\infty]\) is finite and some estimates on the cohomology of \(p\)-adic analytic groups due to \textit{A. Lubotzky} and \textit{A. Mann} [J. Algebra 105, 506--515 (1987; Zbl 0626.20022)]. The section ends with a nice example of an elliptic curve satisfying the hypothesis of Theorem B. The final Section 5. contains Theorem 5.1 concerning the situation where \(R\) is allowed to contain primes above \(p\): a nice combination of results of \textit{S. L. Zerbes} [Proc. Lond. Math. Soc. (3) 98, No. 3, 775--796 (2009; Zbl 1167.11037)] and \textit{L. V. Kuz'min} [Math. USSR, Izv. 9(1975), 693--726 (1976; Zbl 0342.12007)] shows that, even assuming that \(A\) has good ordinary reduction above \(p\), the boundness of \(\mathrm{ker}\;s_{K'}\) and of \(\mathrm{coker}\;s_{K'}\) should not be expected.