The Artinian \(\Lambda\)-module and the pairing on the cyclotomic \(\mathbb{Z}_ l\)-extensions. (English) Zbl 0756.11032
Let \(p\) be a prime number, \(k\) an algebraic number field containing \(\zeta_{2p}\), \(k_ \infty\) the cyclotomic \(\mathbb{Z}_ p\)-extension of \(k\). In Iwasawa theory, one usually studies certain abelian Galois groups \(\text{Gal}(M/k_ \infty)\) as left Noetherian modules over \(\Lambda=\mathbb{Z}_ p[[T]]\); their Pontryagin or Kummer duals are naturally right Artinian \(\Lambda\)-modules. In this paper, the author constructs a \(\Lambda\)-pairing \(X\times Y\to\mu_{p^ \infty}\) which is nondegenerate up to \(\Lambda\)-divisible parts and finite factors, between two Artinian \(\Lambda\)-modules \(X\), \(Y\) defined in the following setting:
For any finite prime \(v\), fix a pro-\(p\)-extension \(\Omega^ v/k_ v\) such that \(\text{Gal}(\Omega^ v/k_ v)\) is free of finite rank. Let \(K/k\) be a finite \(p\)-extension; if \(L/K\) is a \(p\)-extension such that \(K_ v\cdot L\subset\Omega^ v\) for any \(v\), \(L/K\) is called an \(\Omega\)-extension. A local abelian \(p\)-extension \(F/K_ v\) is called \(\Omega^ v\)-orthogonal if the \(p\)-adic completion of \(N_{F/K_ v}(F^*)\) contains \(\operatorname{Hom}(\text{Gal}(\Omega^ v\cdot F/F,\mathbb{Z}_ p(1))\). If \(M/K\) is an abelian \(p\)-extension and every \(K_ v\cdot M/K_ v\) is \(\Omega^ v\)-orthogonal, then \(M/K\) is called an \(\Omega^ \perp\)-extension. Obviously, one can define \(\Omega^{ab}(k_ \infty)\), the maximal abelian \(\Omega\)-extension of \(k_ \infty\), and \(\Omega^ \perp(k_ \infty)\), the maximal \(\Omega^ \perp\)-extension of \(k_ \infty\). Then \(X\) and \(Y\) are respectively the Kummer duals of \(\text{Gal}(\Omega^{ab}(k_ \infty)/k_ \infty)\) and \(\text{Gal}(\Omega^ \perp(k_ \infty)/k_ \infty)\). The construction of the pairing is very technical and lengthy. As a typical example, one can take \(\Omega^{ab}(k_ \infty)=\) the maximal abelian pro-\(p\)- extension of \(k_ \infty\) in which every \(v\) totally splits, and \(\Omega^ \perp(k_ \infty)=\) the maximal abelian \(p\)-ramified pro-\(p\)- extension \(F\) of \(k_ \infty\) such that any \(\zeta_{p^ n}\in k(\zeta_{p^ n})\) is a local norm in \(F/k(\zeta_{p^ n})\).
Remark: More straightforward constructions of pairings related to the above example can be found in [K. Wingberg, Compos. Math. 55, 333- 381 (1985; Zbl 0608.12012), or the reviewer, Sémin. Théor. Nombres, Paris 1986-87, Prog. Math. 75, 271-297 (1988; Zbl 0687.12005)].
For any finite prime \(v\), fix a pro-\(p\)-extension \(\Omega^ v/k_ v\) such that \(\text{Gal}(\Omega^ v/k_ v)\) is free of finite rank. Let \(K/k\) be a finite \(p\)-extension; if \(L/K\) is a \(p\)-extension such that \(K_ v\cdot L\subset\Omega^ v\) for any \(v\), \(L/K\) is called an \(\Omega\)-extension. A local abelian \(p\)-extension \(F/K_ v\) is called \(\Omega^ v\)-orthogonal if the \(p\)-adic completion of \(N_{F/K_ v}(F^*)\) contains \(\operatorname{Hom}(\text{Gal}(\Omega^ v\cdot F/F,\mathbb{Z}_ p(1))\). If \(M/K\) is an abelian \(p\)-extension and every \(K_ v\cdot M/K_ v\) is \(\Omega^ v\)-orthogonal, then \(M/K\) is called an \(\Omega^ \perp\)-extension. Obviously, one can define \(\Omega^{ab}(k_ \infty)\), the maximal abelian \(\Omega\)-extension of \(k_ \infty\), and \(\Omega^ \perp(k_ \infty)\), the maximal \(\Omega^ \perp\)-extension of \(k_ \infty\). Then \(X\) and \(Y\) are respectively the Kummer duals of \(\text{Gal}(\Omega^{ab}(k_ \infty)/k_ \infty)\) and \(\text{Gal}(\Omega^ \perp(k_ \infty)/k_ \infty)\). The construction of the pairing is very technical and lengthy. As a typical example, one can take \(\Omega^{ab}(k_ \infty)=\) the maximal abelian pro-\(p\)- extension of \(k_ \infty\) in which every \(v\) totally splits, and \(\Omega^ \perp(k_ \infty)=\) the maximal abelian \(p\)-ramified pro-\(p\)- extension \(F\) of \(k_ \infty\) such that any \(\zeta_{p^ n}\in k(\zeta_{p^ n})\) is a local norm in \(F/k(\zeta_{p^ n})\).
Remark: More straightforward constructions of pairings related to the above example can be found in [K. Wingberg, Compos. Math. 55, 333- 381 (1985; Zbl 0608.12012), or the reviewer, Sémin. Théor. Nombres, Paris 1986-87, Prog. Math. 75, 271-297 (1988; Zbl 0687.12005)].
Reviewer: T.Nguyen Quang Do (Besançon)
