Math Conjectures

Equivariant Tamagawa Number Conjecture

OPENLandmarkConjectureProposed 2001 · Full conjecture

Canonical statement

Let MM be a pure motive over a number field with coefficients in a finite-dimensional semisimple Q\mathbb Q-algebra AA, and let A⊂A\mathcal A\subset A be an order defining a projective integral structure on MM. The full conjecture first asserts the analytic continuation needed to define the leading term at s=0s=0 of the equivariant LL-function of MM. Burns and Flach use that leading term, together with period and regulator maps and global and local étale cohomology, to define a canonical class
TΩ(M,A)∈K0(A,R). T\Omega(M,\mathcal A)\in K_0(\mathcal A,\mathbb R).
The equivariant Tamagawa number conjecture asserts TΩ(M,A)=0T\Omega(M,\mathcal A)=0.
View source LaTeX
Let \(M\) be a pure motive over a number field with coefficients in a finite-dimensional semisimple \(\mathbb Q\)-algebra \(A\), and let \(\mathcal A\subset A\) be an order defining a projective integral structure on \(M\). The full conjecture first asserts the analytic continuation needed to define the leading term at \(s=0\) of the equivariant \(L\)-function of \(M\). Burns and Flach use that leading term, together with period and regulator maps and global and local étale cohomology, to define a canonical class
\[
  T\Omega(M,\mathcal A)\in K_0(\mathcal A,\mathbb R).
\]
The equivariant Tamagawa number conjecture asserts \(T\Omega(M,\mathcal A)=0\).
NT-LFUNC-039:[BlochKato1990Tamagawa][BurnsFlach2001ETNC][Flach2004ETNCSurvey]
Number theoryAlgebraic geometry

Notes

The Bloch–Kato Tamagawa-number philosophy relates leading terms of motivic LL-functions to periods, regulators, and arithmetic cohomology [BlochKato1990Tamagawa]. Burns and Flach made this integral and equivariant: once the required analytic continuation supplies the leading term at s=0s=0, the analytic and cohomological data determine a canonical class TΩ(M,A)T\Omega(M,\mathcal A) in a relative algebraic KK-group, and the ETNC asserts that this class vanishes [BurnsFlach2001ETNC].

This single vanishing statement specializes to or refines analytic class-number formulas, Stark conjectures, and Birch–Swinnerton-Dyer-type leading-term predictions. Important Tate, abelian, CM, and modular cases are known, sometimes only after localization at a prime or under auxiliary hypotheses; Flach's survey explains the common framework and the scope of these results [Flach2004ETNCSurvey].

The full conjecture includes the analytic existence needed to define the leading term; it is not known for arbitrary motives, integral structures, and semisimple coefficient algebras, with noncommutative coefficients posing particular difficulties. The entry records the Burns–Flach ETNC itself rather than merging it with every Beilinson–Bloch–Kato conjecture that it refines.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.