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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.