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

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

2010-04-17

arxiv.org/abs/1004.2961v2

Mathematics

Number Theory

17 pages

Scientific paper

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.

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