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Weights of the boundary motive of some Shimura varieties

Abstract : Given a Shimura variety S associated to a reductive group G, we study the weight filtration in the cohomology of variations of Hodge structure µH(V ) and ℓ-adic sheaves µℓ(V) on S coming from algebraic representations V of G, with the aim of constructing motives for automorphic representations of G.In the first two chapters we review the theories that we use and we give some complements to them. In the first one we summarize the relationship between cohomology of Shimura varieties, automorphic representations and weights, whereas in the second one we recall relative Chow and Beilinson motives over PEL Shimura varieties and the applications of the theory of weight structures to this setting. In particular, we study in detail the action of the Hecke algebra at the level of motives. In the last two chapters we concentrate on the case of the group G =ResF|ℚGSp₄,F , for F a totally real number field, and to the associated Shimura varieties S (genus 2 Hilbert-Siegel varieties). In the third chapter, we study in detail the weight filtration on the degeneration of the sheaves µℓ(V) along the boundary of the Baily-Borel compactification of S. We are able to describe the weights in terms of an invariant of the representation V , called corank. From this, we deduce a complete characterization of the representations V such that the degeneration of µℓ(V) avoids the weights 0 and 1, and we find that they form a quite large class. In the fourth chapter, given such a representation V, we define motives for those automorphic representations of G which appear in the cohomology of µℓ(V). We then study the properties of such motives.
Keywords : Theory of weight
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Theses
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https://tel.archives-ouvertes.fr/tel-03104375
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Submitted on : Friday, January 8, 2021 - 7:38:06 PM
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Mattia Cavicchi. Weights of the boundary motive of some Shimura varieties. Number Theory [math.NT]. Université Sorbonne Paris Cité, 2019. English. ⟨NNT : 2019USPCD032⟩. ⟨tel-03104375⟩

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