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# Behrend's trace formula

In algebraic geometry, Behrend's trace formula is a generalization of the Grothendieck–Lefschetz trace formula to a smooth algebraic stack over a finite field, conjectured in 1993 [1] and proven in 2003 [2] by Kai Behrend. 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 on a curve over a finite field (in some instances indirectly, via the Harder–Narasimhan stratification, as the moduli stack is not of finite type.[3][4]) See the moduli stack of principal bundles and references therein for the precise formulation in this case.

Deligne found an example[5] that shows the formula may be interpreted as a sort of the Selberg trace formula.

A proof of the formula in the context of the six operations formalism developed by Laszlo and Olsson[6] is given by Shenghao Sun.[7]

## Formulation

By definition, if C is a category in which each object has finitely many automorphisms, the number of points inis 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 fieldand the"arithmetic" Frobenius, i.e., the inverse of the usual geometric Frobeniusin Grothendieck's formula,[8][9]

Here, it is crucial that the cohomology of a stack is with respect to the smooth topology (not etale).

When X is a variety, the smooth cohomology is the same as etale one and, via the Poincaré duality, 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, theclassifying stackof the multiplicative group scheme (that is,). By definition,is the category ofprincipal-bundles over, which has only one isomorphism class (since all such bundles are trivial byLang's theorem). Its group of automorphisms is, which means that the number of-isomorphisms is.
On the other hand, we may compute the l-adic cohomology ofdirectly. We remark that in the topological setting, we have(wherenow denotes the usual classifying space of a topological group), whose rational cohomology ring is a polynomial ring in one generator (Borel's theorem), but we shall not use this directly. If we wish to stay in the world of algebraic geometry, we may instead "approximate"by projective spaces of larger and larger dimension. Thus we consider the mapinduced by the-bundle corresponding toThis map induces an isomorphism in cohomology in degrees up to 2N. Thus the even (resp. odd) Betti numbers ofare 1 (resp. 0), and the l-adic Galois representation on the *(2n)*th cohomology group is the nth power of the cyclotomic character. The second part is a consequence of the fact that the cohomology ofis generated by algebraic cycle classes. This shows that

Note that

Multiplying by, one obtains the predicted equality.

## References

[1]
Citation Linkwww.math.ubc.caBehrend, K. The Lefschetz Trace Formula for the Moduli Stack of Principal Bundles. PhD dissertation.
Sep 26, 2019, 10:52 PM
[2]
Citation Linkwww.math.ubc.caBehrend, K. Derived l-adic categories for algebraic stacks. Memoirs of the American Mathematical Society Vol. 163, 2003
Sep 26, 2019, 10:52 PM
[3]
Citation Linkarxiv.orgK. Behrend, A. Dhillon, Connected components of moduli stacks of torsors via Tamagawa numbers
Sep 26, 2019, 10:52 PM
[4]
Sep 26, 2019, 10:52 PM
[5]
Sep 26, 2019, 10:52 PM
[6]
Citation Link//arxiv.org/abs/math/0512097v2*Laszlo, Yves; Olsson, Martin (2006). "The six operations for sheaves on Artin stacks I: Finite Coefficients". arXiv:math/0512097v2.
Sep 26, 2019, 10:52 PM
[7]
Sep 26, 2019, 10:52 PM
[8]
Citation Linkopenlibrary.orgTo define Frobenius on a stack X, let . Then we have , which is the Frobenius on X, also denoted by .
Sep 26, 2019, 10:52 PM
[9]
Sep 26, 2019, 10:52 PM
[10]
Sep 26, 2019, 10:52 PM
[11]
Sep 26, 2019, 10:52 PM
[12]
Citation Linkwww.math.ubc.caThe Lefschetz Trace Formula for the Moduli Stack of Principal Bundles.
Sep 26, 2019, 10:52 PM
[13]
Sep 26, 2019, 10:52 PM
[14]
Citation Linkarxiv.orgConnected components of moduli stacks of torsors via Tamagawa numbers
Sep 26, 2019, 10:52 PM
[15]