The Brauer-Kuroda formula for higher S-class numbers in dihedral extensions of number fields

TL;DR

This paper provides an algebraic proof of a conjectural formula relating higher S-class numbers and regulators in dihedral extensions of number fields, extending classical results without relying on the Lichtenbaum conjecture.

Contribution

It offers a new algebraic proof of a conjectural formula connecting motivic cohomology, regulators, and S-class numbers in dihedral Galois extensions.

Findings

Algebraic proof of the conjectural formula

Interpretation of the regulator as a higher unit index

Extension of classical class number formulas to higher S-class numbers

Abstract

Let p be an odd prime and let L/k be a Galois extension of number fields whose Galois group is isomorphic to the dihedral group of order 2p. Let S be a finite set of primes of L which is stable under the action of Gal(L/k). The Lichtenbaum conjecture on special values of the Dedekind zeta function at negative integers, together with Brauer formalism for Artin's L-functions, gives a (conjectural) formula relating orders of motivic cohomology groups of rings of S-integers and higher regulators of S-units of the subextensions of L/k. In analogy with the classical case of special values at 0, we give an algebraic proof of this formula, i.e. without using the Lichtenbaum conjecture nor Brauer formalism. Our method also gives an interpretation of the regulator term as a higher unit index.