In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, Behrend's trace formula is a generalization of the
Grothendieck–Lefschetz trace formula
In algebraic geometry, the Grothendieck trace formula expresses the number of points of a variety over a finite field in terms of the trace of the Frobenius endomorphism on its cohomology groups. There are several generalizations: the Frobenius end ...
to a
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebraic ...
algebraic stack
In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's repr ...
over a finite field conjectured in 1993 and proven in 2003 by
Kai Behrend
Kai Behrend is a German mathematician. He is a professor at the University of British Columbia in Vancouver, British Columbia, Canada.
His work is in algebraic geometry and he has made important contributions in the theory of algebraic stacks, G ...
. Unlike the classical one, the formula counts points in the "
stacky way"; it takes into account the presence of nontrivial automorphisms.
The desire for the formula comes from the fact that it applies to the
moduli stack of principal bundles In algebraic geometry, given a smooth projective curve ''X'' over a finite field \mathbf_q and a smooth affine group scheme ''G'' over it, the moduli stack of principal bundles over ''X'', denoted by \operatorname_G(X), is an algebraic stack given b ...
on a curve over a finite field (in some instances indirectly, via the
Harder–Narasimhan stratification In algebraic geometry and complex geometry, the Harder–Narasimhan stratification is any of a stratification of the moduli stack of principal ''G''-bundles by locally closed substacks in terms of "loci of instabilities". In the original form due t ...
, as the moduli stack is not of finite type.) See the
moduli stack of principal bundles In algebraic geometry, given a smooth projective curve ''X'' over a finite field \mathbf_q and a smooth affine group scheme ''G'' over it, the moduli stack of principal bundles over ''X'', denoted by \operatorname_G(X), is an algebraic stack given b ...
and references therein for the precise formulation in this case.
Pierre Deligne
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Pr ...
found an example that shows the formula may be interpreted as a sort of the
Selberg trace formula
In mathematics, the Selberg trace formula, introduced by , is an expression for the character of the unitary representation of a Lie group on the space of square-integrable functions, where is a cofinite discrete group. The character is given b ...
.
A proof of the formula in the context of the
six operations
In mathematics, Grothendieck's six operations, named after Alexander Grothendieck, is a formalism in homological algebra, also known as the six-functor formalism. It originally sprang from the relations in étale cohomology that arise from a mor ...
formalism developed by Yves Laszlo and Martin Olsson is given by Shenghao Sun.
Formulation
By definition, if ''C'' is a category in which each object has finitely many automorphisms, the number of points in
is denoted by
:
with the sum running over representatives ''p'' of all isomorphism classes in ''C''. (The series may diverge in general.) The formula states: for a smooth algebraic stack ''X'' of finite type over a finite field
and the
"arithmetic" Frobenius , i.e., the inverse of the usual geometric Frobenius
in Grothendieck's formula,
:
Here, it is crucial that the
cohomology of a stack
In algebraic geometry, the cohomology of a stack is a generalization of étale cohomology. In a sense, it is a theory that is coarser than the Chow group of a stack.
The cohomology of a quotient stack (e.g., classifying stack) can be thought of a ...
is with respect to the
smooth topology In algebraic geometry, the smooth topology is a certain Grothendieck topology, which is finer than étale topology. Its main use is to define the cohomology of an algebraic stack with coefficients in, say, the étale sheaf \mathbb_l.
To understand ...
(not etale).
When ''X'' is a variety, the smooth cohomology is the same as etale one and, via the
Poincaré duality
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact a ...
, this is equivalent to Grothendieck's trace formula. (But the proof of Behrend's trace formula relies on Grothendieck's formula, so this does not subsume Grothendieck's.)
Simple example
Consider