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# 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

#### 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