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Register a new userTorsion classes in the cohomology of KHT Shimura varieties
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Authors
Pascal Boyer
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A particular case of Bergeron-Venkateshs conjecture predicts that torsion classes in the cohomology of Shimura varieties are rather rare. According to this and for Kottwitz-Harris-Taylor type of Shimura varieties, we first associate to each such torsion class an infinity of irreducible automorphic representations in characteristic zero, which are pairwise non isomorphic and weakly congruent. Then, using completed cohomology, we construct torsion classes in regular weight and then deduce explicit examples of such automorphic congruences.
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Let $Sh_K(G,mu)$ be a Shimura variety of KHT type, as introduced in Harris-Taylor book, associated to some similitude group $G/mathbb Q$ and a open compact subgroup $K$ of $G(mathbb A)$. For any irreducible algebraic $overline{mathbb Q}_l$-representation $xi$ of $G$, let $V_xi$ be the $mathbb Z_l$-local system on $Sh_K(G,mu)$. From my paper about p-stabilization, we know that if we allow the local component $K_l$ of $K$ to be small enough, then there must exists some non trivial cohomology classes with coefficient in $V_xi$. The aim of this paper is then to construct explicitly such torsion classes with the control of $K_l$. As an application we obtain the construction of some new automorphic congruences between tempered and non tempered automorphic representations of the same weight and same level at $l$.
Given a KHT Shimura variety provided with an action of its unramified Hecke algebra $mathbb T$, we proved in a previous work, see also the work of Caraiani-Scholze for other PEL Shimura varieties, that its localized cohomology groups at a generic maximal ideal $mathfrak m$ of $mathbb T$, appear to be free. In this work, we obtain the same result for $mathfrak m$ such that its associated galoisian $overline{mathbb F}_l$-representation $overline{rho_{mathfrak m}}$ is irreducible.
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 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.
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.
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