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

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    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 pp-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 pp-component of the class number of Q(ζp)Q(\zeta_p). In trying to formulate these questions, we are naturally led to consider L(0,ρ)L(0,\rho), for ρ\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 pp-integral for a prime \p of the ring of algebraic integers Zˉ\bar{Z} in CC, that we call {\it mod-pp 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!Comment: Although several changes, mostly minor revisio
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