×

zbMATH — the first resource for mathematics

On the cohomology with compact supports of Shimura varieties for \(GSp(4)_ \mathbb{Q}\). (Sur la cohomologie à supports compacts des variétés de Shimura pour \(GSp(4)_ \mathbb{Q}\).) (French) Zbl 0877.11037
Let \(G\) be the algebraic group \(GSp(4)\), and let \(K= K_N \subset G(\mathbb A_f)\) for an integer \(N \geq 3\), where \(K_N = \text{Ker} (G (\widehat{\mathbb Z}) \to G(\widehat{\mathbb Z} / N \widehat{\mathbb Z}))\). Then \(K\) is a compact open subgroup of \(G(\mathbb A_f)\) and determines a Shimura variety \(S_K\) over \(\mathbb Q\) with \(S_K (\mathbb C) = G(\mathbb Q) \backslash [(G (\mathbb R)/K'_\mathbb R) \times (G(\mathbb A_f) /K)]\), where \(K'_\mathbb R = \mathbb R^\times_+ \cdot K_\mathbb R\) and \(K_\mathbb R = \{ g \in G(\mathbb R) \mid g^t g = 1 \}\). \(S_K\) is a smooth quasi-projective variety of dimension 3 over \(\mathbb Q\), and, given a \(\mathbb Q\)-algebra \(R\), \(S_K (R)\) is the set of isomorphism classes of principally polarized abelian surfaces over \(R\) with the level \(N\) structure. For each prime \(\ell\), the \(\ell\)-adic cohomology spaces \(H^i_c (S_K \otimes_\mathbb Q \overline{\mathbb Q}, \mathbb Q_\ell)\) are zero for \(i \not\in [0,6]\) and are of finite dimension over \(\mathbb Q_\ell\). They are also equipped with an action of \(\text{Gal} (\overline{\mathbb Q}/ \mathbb Q)\) for each \(i\). Let \(\mathcal C_c (G(\mathbb A_f) // K)\) denote the \(\mathbb C\)-algebra of convolutions of functions \(f: G(\mathbb A_f) \to \mathbb C\) that are of compact support and left and right \(K\)-invariant. If \(\mathcal C_c (G(\mathbb A_f) // K)_\mathbb Q \subset \mathcal C_c (G(\mathbb A_f) // K)\) is the \(\mathbb Q\)-structure of this \(\mathbb C\)-algebra, then \(\mathcal C_c (G(\mathbb A_f) // K)_\mathbb Q\) acts on \(H^i_c (S_K \otimes_\mathbb Q \overline{\mathbb Q}, \mathbb Q_\ell)\), and this action commutes with the action of \(\text{Gal} (\overline{\mathbb Q}/ \mathbb Q)\).
In this paper, the author calculates the virtual \((\text{Gal} (\overline{\mathbb Q}/ \mathbb Q) \times \mathcal C_c (G(\mathbb A_f) // K)_\mathbb Q)\)-module \(W_\ell = \sum^6_{i=0} (-1)^i H^i_c (S_K \otimes_\mathbb Q \overline{\mathbb Q}, \mathbb Q_\ell)\) by using the method of the comparison of the Grothendieck-Lefschetz formula and the Arthur-Selberg trace formula, which has been developed by Ihara, Langlands and Kottwitz.

MSC:
11G18 Arithmetic aspects of modular and Shimura varieties
14G35 Modular and Shimura varieties
11F70 Representation-theoretic methods; automorphic representations over local and global fields
14G20 Local ground fields in algebraic geometry
PDF BibTeX XML Cite
Full Text: DOI