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In http://www.math.ubc.ca/~behrend/ladic.pdf, the author uses his generalization of Lefschetz trace formula to smooth algebraic stacks to prove an interesting identity (Proposition 6.4.11.):

$\sum_{k} \frac{1}{p^{k+1}} tr T_p |_{S_{k+2}}=1- \frac{1}{p^3-p} - \sum_{E / \mathbb F_p}\frac{1}{\#E(\mathbb F_p) \# Aut E(\mathbb F_p)}$.

Here $T_p$ is the $p$-th Hecke operator on the space of cusp forms $S_{k+2}$ of weight $k + 2$. The sum on the right hand side extends over all isomorphism classes of elliptic curves over $\mathbb F_p$.

In the proof, he applies the trace formula to the algebraic stack of curves of genus one. Can we generalize this identity to other Shimura varieties ? For example, can we apply Behrend trace formula to some moduli stacks describing torsors of abelian varieties?

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