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Following the natural instinct that when a group operates on a number fieldthen 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 ofany CM Galois extension of $\Q$ as a module for $\Gal(K/Q)$, in the spirit ofHerbrand-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 toconsider $L(0,\rho)$, for $\rho$ an Artin representation, in situations wherethis is known to be nonzero and algebraic, and it is important for us tounderstand if this is $p$-integral for a prime $\p$ of the ring of algebraicintegers $\bar{Z}$ 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!

A mod-p Artin-Tate conjecture, and generalized Herbrand-Ribet