The Lefschetz trace formula for algebraic stacks (Q690210)

The author obtains the following: Let \(G\) be a nonsingular linear algebraic group over \(\mathbb{F}_ q\), acting on a smooth equidimensional algebraic space \(X\) of finite type over \(\mathbb{F}_ q\). Let \({\mathfrak X}\) be the associated algebraic stack and \(\Phi_ q\) be the arithmetic Frobenius acting on \(H^*(\overline {\mathfrak X}_{sm},\mathbb{C})\). Then \(\text{tr } \Phi_ q | H^*(\overline {\mathfrak X}_{sm},\mathbb{C})\) is absolute convergent and satisfies the trace formula: \[ q^{\dim} {^{\mathfrak X}\text{tr}} [\Phi_ q | H^*(\overline {\mathfrak X}_{sm}, \mathbb{C})]= \sum_{\xi \in [{\mathfrak X} (\mathbb{F}_ q)]} {1 \over \# \Aut \xi}. \] The above formula can be applied to the situation of Deligne-Mumford stacks to obtain similar trace formulae and equations for zeta functions. Since many of the moduli spaces can be constructed by Deligne-Mumford stacks, the above formula suggests further applications in studying the cohomology of these moduli spaces.