\(K_ 3\) and \(p\)- adic Riemann-Hurwitz formulas
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 \(E_ \infty /F_ \infty\) in \(w\); \[ \lambda' (E_ \infty)-1= (\lambda' (F_ \infty)-1)\cdot [E_ \infty: F_ \infty]+ \sum (n_ w -1), \] where the sum runs over all places \(w\) of \(E_ \infty\) dividing \(p\) and where \(n_ w\) is the degree of \(E_ \infty/ F_ \infty\) in \(w\). Similar results were known for CM-fields \(F\). As a matter of fact, one has a relation between \(\lambda_{S_ p}\) and \(\lambda'\) and the second formula can be deduced from the first one. An essential ingredient of the proof of the first formula is the vanishing of a Herbrand quotient involving \(K_ 3 (E_ \infty)\). Here the condition on the capitulation comes into play via a Hochschild-Serre spectral sequence. After an introduction giving some background information on the problem and heuristics for introducing \(K\)-groups, the necessary tools and results on \(K\)-groups for number fields and capitulations are resumed in a special section. The second section is concerned with the statement and the proof of the main results. In the third section two applications are discussed.
scientific article; zbMATH DE number 529891