Hermitian \(K\)-theory, Dedekind \(\zeta \)-functions, and quadratic forms over rings of integers in number fields. (English) Zbl 1456.11221
Summary: We employ the slice spectral sequence, the motivic Steenrod algebra, and Voevodsky’s solutions of the Milnor and Bloch-Kato conjectures to calculate the hermitian \(K\)-groups of rings of integers in number fields. Moreover, we relate the orders of these groups to special values of Dedekind \(\zeta \)-functions for totally real abelian number fields. Our methods apply more readily to the examples of algebraic \(K\)-theory and higher Witt-theory, and give a complete set of invariants for quadratic forms over rings of integers in number fields.
MSC:
| 11R70 | \(K\)-theory of global fields |
| 11R42 | Zeta functions and \(L\)-functions of number fields |
| 14F42 | Motivic cohomology; motivic homotopy theory |
| 19E15 | Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) |
| 19F27 | Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects) |
