A mod-p Artin-Tate conjecture, and generalized Herbrand-Ribet
| Autor: | Prasad, Dipendra |
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| Rok vydání: | 2017 |
| Předmět: | |
| Zdroj: | Pacific J. Math. 303 (2019) 299-316 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.2140/pjm.2019.303.299 |
| Popis: | 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! Comment: Although several changes, mostly minor revision |
| Databáze: | arXiv |
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