\(K_ 3\) and \(p\)- adic Riemann-Hurwitz formulas
type: Item
instance of: scholarly article
title: \(K_ 3\) and \(p\)- adic Riemann-Hurwitz formulas
zbMATH Open document ID: 0801.11049
DOI: 10.1007/BF00961536
published in: \(K\)-Theory
publication date: 1994-04-17T00:00:00Z
review text: In the theory of \(\mathbb{Z}_ p\)-extensions of a number field one has the notion of a so-called \(\lambda\)-invariant which plays a role analogous to the genus of an algebraic curve. So, for suitable extensions one may relate the corresponding \(\lambda\)-invariants by a formula of the Riemann-Hurwitz type. More precisely, let \(F\) be a number field, \(F_ \infty\) its cyclotomic \(\mathbb{Z}_ p\)-extension, and let \({\mathcal X}_ \infty= \text{Gal} (M_ \infty/ F_ \infty)\), where \(M_ \infty\) is the maximal abelian \(p\)- ramified pro-\(p\)-extension of \(F_ \infty\). \(t{\mathcal X}_ \infty\) denotes the \(\Lambda\)-torsion submodule \({\mathcal X}_ \infty\) for the Iwasawa algebra \(\Lambda\). Furthermore, let \(X_ \infty= X_ \infty(F)= \text{Gal} (L_ \infty/ F_ \infty)\), where \(L_ \infty\) is the maximal abelian unramified pro-\(p\)-extension of \(F_ \infty\), and \(X_ \infty'= X_ \infty' (F_ \infty)= \text{Gal} (L_ \infty' /F_ \infty)\), with \(L_ \infty'\subset L_ \infty\) the subextension which totally decomposes all places of \(F_ \infty\). \(\lambda_{S_ p} (F_ \infty)\), \(\lambda(F_ \infty)\) and \(\lambda' (F_ \infty)\) denote the \(\lambda\)-invariants corresponding to \(t{\mathcal X}_ \infty\), \(X_ \infty\) and \(X_ \infty'\), respectively. One can also associate an invariant \(\mu\) to the extension \(F_ \infty/F\). One defines, for an extension \(M/L\), the capitulation \(\text{Cap} (M/L)= \text{Ker} (K_ 2(L)\to K_ 2(M))\). One can now state the main result of the paper: Assume (a) \(F\) contains the group \(\mu_{2p}\) of \(2p\)th-roots of unity and (b) the \(\mu\)- invariant of \(F_ \infty/F\) vanishes, then, for a \(p\)-extension \(E_ \infty/ F_ \infty\) such that \(\text{Cap} (E_ \infty /F)=0\), one has the formulas: \[ \lambda_{S_ p} (E_ \infty)= \lambda_{S_ p} (F_ \infty)\cdot [E_ \infty: F_ \infty]+ \sum (e_ w -1), \] where the sum runs over all places \(w\) of \(E_ \infty\) not dividing \(p\), and \(e_ w\) is the ramification index of \(…
Mathematics Subject Classification ID: 11S70
Mathematics Subject Classification ID: 19D99
zbMATH DE Number: 529891
zbMATH Keywords: K-groups
zbMATH Keywords: \(p\)-adic Riemann-Hurwitz formulas
zbMATH Keywords: formula of Riemann-Hurwitz type
zbMATH Keywords: \(\lambda\)-invariant
zbMATH Keywords: capitulation
author: Thong Nguyen Quang Do
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 529891