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Behrend's trace formula
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<p>In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, Behrend's trace formula is a generalization of the <a href="/facts/Grothendieck%E2%80%93Lefschetz_trace_formula/8uRWu5u8">Grothendieck–Lefschetz trace formula</a> to a <a href="/facts/Smooth_stack/9mVm3h0g">smooth</a> <a href="/facts/Algebraic_stack/flrtwvwU">algebraic stack</a> over a finite field conjectured in 1993 and proven in 2003 by <a href="/facts/Kai_Behrend/CumDNxt8">Kai Behrend</a>. Unlike the classical one, the formula counts points in the "<a href="/facts/Stack_(mathematics)/MkASN1rA">stacky way</a>"; it takes into account the presence of nontrivial automorphisms. </p><p>The desire for the formula comes from the fact that it applies to the <a href="/facts/Moduli_stack_of_principal_bundles/tjRzzo5K">moduli stack of principal bundles</a> on a curve over a finite field (in some instances indirectly, via the <a href="/facts/Harder%E2%80%93Narasimhan_stratification/KqtuXeUI">Harder–Narasimhan stratification</a>, as the moduli stack is not of finite type.) See the <a href="/facts/Moduli_stack_of_principal_bundles/tjRzzo5K">moduli stack of principal bundles</a> and references therein for the precise formulation in this case. </p><p><a href="/facts/Pierre_Deligne/wYC8fhfq">Pierre Deligne</a> found an example that shows the formula may be interpreted as a sort of the <a href="/facts/Selberg_trace_formula/Pkrz73Y9">Selberg trace formula</a>. </p><p>A proof of the formula in the context of the <a href="/facts/Six_operations/lAz8n9tc">six operations</a> formalism developed by Yves Laszlo and Martin Olsson is given by Shenghao Sun. </p>