Counting points on Igusa varieties. (English) Zbl 1218.11061
The author presents a counting-point formula for the cohomology of Igusa varieties. He constructs a Shimura variety \(X\) defined over the reflex field \(E\) along with an integral model, starting from an integral Shimura PEL datum of type (A) or (C). Let \(G\) be the associated algebraic group over \({\mathbb Q}\), \(p\) a prime number such that \(G\) is unramified at \(p\), \(J_b\) the \({\mathbb Q}_p\)-group arising as the automorphism group of an isocrystal of type \(b\). The Igusa variety \(\text{Ig}_b\) is defined as a projective system. The group \(G({\mathbb A}^{\infty,p})\times J_b({\mathbb Q}_p)\) acts on \(H_c(\text{Ig}_b, \mathcal{ L}_\xi)\), while \(G({\mathbb A}^\infty)\times \text{Gal}(\bar{E}/E)\) acts on \(H(X,\mathcal {L}_\xi)\), where \(\mathcal {L}_\xi\) is an \(l\)-adic sheaf constructed from an algebraic representation of \(G\). If \(\varphi\in C_c^\infty(G({\mathbb A}^{\infty,p})\times J_b({\mathbb Q}_p))\) is acceptable, then
\[
\text{tr}(\varphi|H_c(\text{Ig}_b, \mathcal {L}_\xi))=\sum_{(\gamma_0;\gamma,\delta)\in KT_b^{\text{eff}}}\text{vol}(I_\infty(\mathbb R)^1)^{-1}|A(I_0)| \text{tr}\xi(\gamma_0)\cdot O_{(\gamma,\delta)}^{G({\mathbb A}^{\infty,p})\times J_b({\mathbb Q}_p)}(\varphi).
\]
Reviewer: Florin Nicolae (Berlin)
MSC:
| 11G18 | Arithmetic aspects of modular and Shimura varieties |
| 11G25 | Varieties over finite and local fields |
| 14G15 | Finite ground fields in algebraic geometry |
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