论文标题
Ekedahl派生的$ \ ell $ -Adic设置中的nomizu-van est定理
A Nomizu-van Est theorem in Ekedahl's derived $\ell$-adic setting
论文作者
论文摘要
Nomizu和van est的定理计算紧凑的Nilmanifold的共同体，或等于$ \ Mathbb {Q} $的单位线性代数群的算术亚组的组共同体。我们证明了与$ \ mathbb {q} _ {\ ell} $相似的单位线性代数群的紧凑型开放子群的共同体的结果类似，并在连续$ \ ell $ addic表示的复合体中具有系数。我们与Ekedahl定义的三角类别合作，该类别扮演着``连续$ \ ell $ adic表示的派生类别''的角色。这是由Pink的公式计算在Shimura品种中的最小压实中的$ \ ell $ adiC本地系统的派生的直接图像，以及其在Shimura品种上的自动形态变形支链上的应用。关键的技术结果是通过具有有限的多项式Cochains的显式复合物（具有有限类型的多项式复合物），其具有扭转系数的系数，其系数在单位表示中进行了扭转系数的计算。
A theorem of Nomizu and van Est computes the cohomology of a compact nilmanifold, or equivalently the group cohomology of an arithmetic subgroup of a unipotent linear algebraic group over $\mathbb{Q}$. We prove a similar result for the cohomology of a compact open subgroup of a unipotent linear algebraic group over $\mathbb{Q}_{\ell}$ with coefficients in a complex of continuous $\ell$-adic representations. We work with the triangulated categories defined by Ekedahl which play the role of ``derived categories of continuous $\ell$-adic representations''. This is motivated by Pink's formula computing the derived direct image of an $\ell$-adic local system on a Shimura variety in its minimal compactification, and its application to automorphic perverse sheaves on Shimura varieties. The key technical result is the computation of the cohomology with coefficients in a unipotent representation with torsion coefficients by an explicit complex of polynomial cochains which is of finite type.
