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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
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