The Lichtenbaum Conjecture at the Prime 2

Abstract

<p>The Analytical Class Number Formula, a classical result of Dirichlet, asserts that two important transcendental invariants associated to a number field F, namely the first non-zero Taylor coefficient of the Dedekind zeta-function, ζF, at 0 and the regulator of the group of unites of F differ by a rational number. Moreover, this rational number is the quotient of two algebraic invariants of F, namely the ideal class number and the order of the group of the roots of unity in F. The Lichtenbaum Conjecture attempts to exhibit the same type of relation between the first non-zero Taylor coefficient of ζF at 1 - m for m ≥ 2 and the Boral regulator in K-theory. They differ by the quotient of the orders of the torsion parts of consecutive higher K-groups (the even K-groups appear as generalizing the ideal class group, while the odd ones appear as generalizing the group of units).</p> <p>The study of this conjecture is done at each prime p using p-adic Chern characters from K-theory to étale cohomology and an interplay between étale cohomology duality results and Iwasawa theory results. Using a different regulator the Lichtenbaum Conjecture has been proved at all odd primes for all abelian number number fields by Kolster, Nguyen Quang Do and Fleckinger. We develop similar methods and succeed to obtain a description of the 2-powers appearing in the formula for the case when m is odd. We also note that a motivic context is possible for the formulation of the Lichtenbaum Conjecture.</p>