Contents Online
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