On the values of Artin $L$-series at $s=1$ and annihilation of class groups

Hugo Castillo ¹   ; Andrew Jones ²

¹ Department of Mathematics King's College London Strand WC2R 2LS, United Kingdom
² 1 Sheridan Grove Milton Keynes MK4 4GP, United Kingdom

Acta Arithmetica, Tome 160 (2013) no. 1, pp. 67-93

Résumé

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 $p$.

Détail

DOI : 10.4064/aa160-1-5

Keywords: finite galois cm extension totally real field validity appropriate special equivariant tamagawa number conjecture leads natural construction each odd prime explicit elements non commutative fitting invariants mathbb certain tame ray class group hence analogous fitting invariants p primary part ideal class group these elements involve values artin l series characters group gal results become unconditional under certain natural hypotheses extension prime