Cambridge Journal of Mathematics

Volume 8 (2020)

Number 3

Hermitian $K$-theory, Dedekind $\zeta$-functions, and quadratic forms over rings of integers in number fields

Pages: 505 – 607

DOI: https://dx.doi.org/10.4310/CJM.2020.v8.n3.a3

Authors

Jonas Irgens Kylling (Department of Mathematics, University of Oslo, Norway)

Röndigs Oliver (Mathematisches Institut, Universität Osnabrück, Germany)

Paul Arne Østvær (Department of Mathematics, University of Oslo, Norway)

Abstract

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.

Keywords

motivic homotopy theory, slice filtration, motivic cohomology, algebraic $K$-theory, Hermitian $K$-theory, higher Witt-theory, quadratic forms over rings of integers, special values of Dedekind $\zeta$-functions of number fields

2010 Mathematics Subject Classification

11R42, 14F42, 19E15, 19F27

Received 1 January 2019

Published 2 October 2020