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Register a new userOn the $bar{mathbb F}_l$-cohomology of a simple unitary Shimura variety
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Authors
Pascal Boyer
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We study the torsion cohomology classes of Shimura varieties of type Kottwitz-Harris-Taylor and we show that up to an arbitrary place one can raise them to an automorphic representation. In application, to any mod $l$ system of Hecke eigenvalues appearing in the $bar{mathbb F}_l$-cohomology of a Shimuras variety of Kottwitz-Harris-Taylor type, we associate a $bar{mathbb F}_l$-Galois representation which Frobenius eigenvalues are given by Heckes. Compared to the highly more general construction of Scholze, we gain both the simplicity of the proof and the control at places ramified and at those dividing $l$.
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We first give a relative flexible process to construct torsion cohomology classes for Shimura varieties of Kottwitz-Harris-Taylor type with coefficient in a non too regular local system. We then prove that associated to each torsion cohomology class, there exists a infinity of irreducible automorphic representations in characteristic zero, which are pairwise non isomorphic and weakly congruent.
We exhibit cases of a level fixing phenomenon for galoisian automorphic representations of a CM field $F$, with dimension $d geq 2$. The proof rests on the freeness of the localized cohomology groups of KHT Shimura varieties and the strictness of its filtration induced by the spectral sequence associated to the filtration of stratification of the nearby cycles perverse sheaf at some fixed place $v$ of $F$. The main point is the observation that the action of the unipotent monodromy operator at $v$ is then given by those on the nearby cycles where its order of nilpotency modulo $l$ equals those in characteristic zero. Finally we infer some consequences concerning level raising and Iharas lemma.
The integral model of a GU(n-1,1) Shimura variety carries a universal abelian scheme over it, and the dual top exterior power of its Lie algebra carries a natural hermitian metric. We express the arithmetic volume of this metrized line bundle, defined as an iterated self-intersection in the Gillet-Soule arithmetic Chow ring, in terms of logarithmic derivatives of Dirichlet L-functions.
We prove that there is a natural plectic weight filtration on the cohomology of Hilbert modular varieties in the spirit of Nekovar and Scholl. This is achieved with the help of Morels work on weight t-structures and a detailed study of partial Frobenius. We prove in particular that the partial Frobenius extends to toroidal and minimal compactifications.
Let $F$ be a totally real field in which a fixed prime $p$ is inert, and let $E$ be a CM extension of $F$ in which $p$ splits. We fix two positive integers $r,s in mathbb N$. We investigate the Tate conjecture on the special fiber of $G(U(r,s) times U(s,r))$-Shimura variety. We construct cycles which we conjecture to generate the Tate classes and verify our conjecture in the case of $G(U(1,s) times U(s,1))$. We also discuss the general conjecture regarding special cycles on the special fibers of unitary Shimura varieties.
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