Tamagawa numbers for motives with (non-commutative) coefficients

Abstract

Let $M$ be a motive which is defined over a number field and admits an action of a finite dimensional semisimple $\mathbb{Q}$-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 $\mathfrak{A}$ in $A$ for which there exists a 'projective $\mathfrak{A}$-structure' on $M$. The existence of such a structure is guaranteed if $\mathfrak{A}$ is a maximal order, and also occurs in many natural examples where $\mathfrak{A}$ 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.