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

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 $\text{Q}$ as a module for $\text{Gal}(K\mathrm{/}Q)$, 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 $\text{p}$ 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!