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Register a new userGalois irreducibility implies cohomology freeness for KHT Shimura varieties
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Authors
Pascal Boyer
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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.
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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 $l$.
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.
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.
We determine the irreducible components of Igusa varieties for Shimura varieties of Hodge type and use that to determine the irreducible components of central leaves. In particular, we show that the discrete Hecke-orbit conjecture is false in general. Our method combines recent work of DAddezio on monodromy of compatible local systems with a generalisation of a method of Hida, using the Honda-Tate theory for Shimura varieties of Hodge type developed by Kisin-Madapusi Pera-Shin. We also determine the irreducible components of Newton strata in Shimura varieties of Hodge type by combining our results with recent work of Zhou-Zhu.
In this article, we generalize the work of H.Hida and V.Pilloni to construct $p$-adic families of $mu$-ordinary modular forms on Shimura varieties of Hodge type $Sh(G,X)$ associated to a Shimura datum $(G,X)$ where $G$ is a connected reductive group over $mathbb{Q}$ and is unramified at $p$, such that the adjoint quotient $G^mathrm{ad}$ has no simple factors isomorphic to $mathrm{PGL}_2$.
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