#### DOCUMENTA MATHEMATICA,
Extra Volume: Kazuya Kato's Fiftieth Birthday (2003), 157-185

** David Burns and Cornelius Greither **
Equivariant Weierstrass Preparation
and Values of $L$-functions at Negative Integers

We apply an equivariant version of the $p$-adic Weierstrass Preparation
Theorem in the context of possible non-commutative generalizations of the
power series of Deligne and Ribet. We then consider CM abelian extensions
of totally real fields and by combining our earlier considerations with
the known validity of the Main Conjecture of Iwasawa theory we prove, modulo
the conjectural vanishing of certain $\mu$-invariants, a (corrected version
of a) conjecture of Snaith and the `rank zero component' of Kato's Generalized
Iwasawa Main Conjecture for Tate motives of strictly positive weight. We
next use the validity of this case of Kato's conjecture to prove a conjecture
of Chinburg, Kolster, Pappas and Snaith and also to compute explicitly
the Fitting ideals of certain natural étale cohomology groups in terms
of the values of Dirichlet $L$-functions at negative integers. This computation
improves upon results of Cornacchia and \O stv\ae r, of Kurihara and of
Snaith, and, modulo the validity of a certain aspect of the Quillen-Lichtenbaum
Conjecture, also verifies a finer and more general version of a well known
conjecture of Coates and Sinnott.

2000 Mathematics Subject Classification: 11R42 11R33 11R70

Keywords and Phrases: Iwasawa theory, values of $L$-functions, Euler characteristics, Fitting
ideals

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