Tamagawa numbers for motives with (non-commutative) coefficients

By David Burns, M. Flach. Documenta Mathematica, 2001

210 citations recorded by the source corpus.

Let M be a motive which is defined over a number field and admits an action of a finite dimensional semisimple 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 ofM . 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. 2000 Mathematics Subject Classification: Primary 11G40; Secondary 11R65 19A31 19B28

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/161f9f2b86b37fb3ce757fac4cd0d4f45b75adf9

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In Discrete Mathematics and Combinatorics

1. Werner Bley
2. David Burns
3. M. Flach

In Geometry and Topology

1. Werner Bley
2. David Burns
3. M. Flach

In Mathematical Physics

1. Werner Bley
2. David Burns
3. M. Flach

Citations in the corpus

• On the equivariant Tamagawa number conjecture for Abelian extensions of a quadratic imaginary field (2006, methodology): “The aim of this paper is to provide new evidence for the validity of the Equivariant Tamagawa Number Conjectures (for short ETNC) as formulated by Burns and Flach in [4].”
• Equivariant Weierstrass Preparation and Values of L-functions at Negative Integers (2003, unknown): “This conjecture refined the ‘Tamagawa number conjectures’ previously formulated by Bloch and Kato in [3] and by Fontaine and Perrin-Riou in [20], and led naturally to the subsequent formulation by Flach and the first named author in [9] of a Tamagawa number conjecture for motives over number fields ”
• The main conjecture of Iwasawa theory for totally real fields (2010, unknown): “This was generalised to include noncommutative groups G by Burns and Flach [4], Huber and Kings [21], Coates et al. [8], and Fukaya and Kato [16].”
• On the Equivariant Tamagawa Number Conjecture for Tate Motives, Part II. (2006, methodology): “Beilinson) regulator map itself as used in [9].”