125 research outputs found
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 -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 -component of the class number of
. In trying to formulate these questions, we are naturally led to
consider , for 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 -integral for a prime \p of the ring of algebraic
integers in , that we call {\it mod- 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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