On \(\ell\)-adic sheaves on Shimura varieties and their higher direct images in the Baily-Borel compactification. (English) Zbl 0748.14008

Let \(S\) be the canonical model of a Shimura variety associated to a reductive group \(G\). Let \(j: S\hookrightarrow S^*\) denote the open embedding into the canonical model of the Baily-Borel compactification. Let \(i:S_ 1\hookrightarrow S^*\) be the locally closed embedding of a boundary stratum; here \(S_1\) is the canonical model of a Shimura variety associated to another, explicitly given, reductive group \(G_1\). (This has been described in the author’s dissertation [“Arithmetical compactification of mixed Shimura varieties”, Bonn. Math. Schr. 209 (1989; see the preceding review Zbl 0748.14007)].
An algebraic representation of \(G\) on a \(\mathbb{Q}_\ell\)-vector space \(V\) determines a smooth \(\ell\)-adic sheaf \({\mathcal V}\) on \(S\). The main result of the present article is a description of the \(\ell\)-adic sheaves \(i^*R^nj_*{\mathcal V}\) on \(S_1\). A simple group cohomological formula yields an algebraic representation of \(G_1\) and hence an \(\ell \)-adic sheaf on \(S_1\): this is canonically isomorphic to \(i^*R^nj_*{\mathcal V}\). The analogous statement in the analytic category is well-known and much easier to prove.


14G35 Modular and Shimura varieties
11G18 Arithmetic aspects of modular and Shimura varieties
14F30 \(p\)-adic cohomology, crystalline cohomology
11F75 Cohomology of arithmetic groups


Zbl 0748.14007
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