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Tamagawa Numbers for Motives with (Non-Commutative) Coefficients

Burns, D. and Flach, M. (2001) Tamagawa Numbers for Motives with (Non-Commutative) Coefficients. Documenta Mathematica, 6 . pp. 501-570. ISSN 1431-0635.

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Let $M$ be a motive which is defined over a number field and admits an action of a finite dimensional semisimple $\bq$-algebra $A$. We formulate and study a conjecture for the leading coefficient of the Taylor expansion at $0$ of the $A$-equivariant $L$-function of $M$. This conjecture simultaneously generalizes and refines the Tamagawa number conjecture of Bloch, Kato, Fontaine, Perrin-Riou et al. and also the central conjectures of classical Galois module theory as developed by Fröhlich, Chinburg, M. Taylor et al. The precise formulation of our conjecture depends upon the choice of an order $\A$ in $A$ for which there exists a `projective $\A$-structure' on $M$. The existence of such a structure is guaranteed if $\A$ is a maximal order, and also occurs in many natural examples where $\A$ is non-maximal. In each such case the conjecture with respect to a non-maximal order refines the conjecture with respect to a maximal order. We develop a theory of determinant functors for all orders in $A$ by making use of the category of virtual objects introduced by Deligne.

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Additional Information:Received: September 6, 2001. Revised: January 28, 2002. Communicated by Don Blasius
Record Number:CaltechAUTHORS:BURdm01
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:758
Deposited By: Archive Administrator
Deposited On:27 Sep 2005
Last Modified:02 Oct 2019 22:36

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