  Following the natural instinct that when a group operates on a number field
then every term in the class number formula should factorize `compatibly'
according to the representation theory (both complex and modular) of the group,
we are led to some natural questions about the $p$-part of the classgroup of
any CM Galois extension of $\Q$ as a module for $\Gal(K/Q)$, in the spirit of
Herbrand-Ribet's theorem on the $p$-component of the class number of
$Q(\zeta_p)$. In trying to formulate these questions, we are naturally led to
consider $L(0,\rho)$, for $\rho$ an Artin representation, in situations where
this is known to be nonzero and algebraic, and it is important for us to
understand if this is $p$-integral for a prime $\p$ of the ring of algebraic
integers $\bar{Z}$ in $C$, that we call {\it mod-$p$ Artin-Tate conjecture}.
The most minor term in the class number formula, the number of roots of unity,
plays an important role for us --- it being the only term in the denominator,
is responsible for all poles!
