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.
- Publication:
-
arXiv e-prints
- Pub Date:
- November 2018
- DOI:
- 10.48550/arXiv.1811.03940
- arXiv:
- arXiv:1811.03940
- Bibcode:
- 2018arXiv181103940I
- Keywords:
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- Mathematics - K-Theory and Homology;
- Mathematics - Algebraic Topology;
- 11R42;
- 14F42;
- 19E15;
- 19F27
- E-Print:
- 64 pages