On the values of Artin $L$-series at $s=1$ and annihilation of class groups
Volume 160 / 2013
Acta Arithmetica 160 (2013), 67-93
MSC: 11G40.
DOI: 10.4064/aa160-1-5
Abstract
Let $L$ be a finite Galois CM-extension of a totally real field $K$. We show that the validity of an appropriate special case of the Equivariant Tamagawa Number Conjecture leads to a natural construction for each odd prime $p$ of explicit elements in the (non-commutative) Fitting invariants over $\mathbb {Z}_p[G]$ of a certain tame ray class group, and hence also in the analogous Fitting invariants of the $p$-primary part of the ideal class group of $L$. These elements involve the values at $s=1$ of the Artin $L$-series of characters of the group ${\rm Gal}(L/K)$. We also show that our results become unconditional under certain natural hypotheses on the extension $L/K$ and prime $p$.
