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

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.

arXiv comment64 pages

References in corpus (5)

• The six operations in equivariant motivic homotopy theory
• A comparison of motivic and classical homotopy theories
• The Generalized Slices of Hermitian K-Theory
• The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in hermitian K-theory
• Quotients of MGL, their slices and their geometric parts

Cited by in corpus (1)

• Topological models for stable motivic invariants of regular number rings