A geometric Jacquet-Langlands correspondence for U(2) Shimura varieties
A geometric Jacquet-Langlands correspondence for U(2) Shimura varieties
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U(2) Shimura 品种的几何 Jacquet-Langlands 对应关系
DOI：
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发表时间：
2004
影响因子：
1
通讯作者：
David Helm
中科院分区：
文献类型：
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作者：
David Helm
设G是环G上的酉群，G与CM-域F相关联，CM-域F具有全真实的部分F+，在F+的所有阿基米德位置上都具有签名（1，1）.在对F+的某些假设下，我们证明了G的某些自守表示与除无穷远处同构于G的群G′的表示之间的Jacquet-Langlands对应可以在G和G′的Shimura簇的上同调中实现.我们通过研究在素数p处附着在G上的Shimura簇X的坏约化得到了这些Jacquet-Langlands对应，并且X具有“Γ0（p）”水平结构，并构造了X的“Deligne-Rapoport”模型.这个模型的特殊纤维的不可约分量具有全局结构，对于远离无穷远同构于G的酉群G′，可以用Shimura簇X′来明确描述。然后，Rapoport-Zink的权谱序列给出了X的tale上同调的权滤子的某些部分根据X′的上同调的表达式。这就用G′的代数模形式的空间来标识这个权过滤的一部分。这一结果意味着G和G′的Jacquet-Langlands对应的某些情况是在算术空间之间的一个标准映射，而不是简单地作为同构表示类之间的一个抽象双射。
Let G be a unitary group over ℚ, associated to a CM-field F with totally real part F+, with signature (1, 1) at all the archimedean places of F+. Under certain hypotheses on F+, we show that Jacquet-Langlands correspondences between certain automorphic representations of G and representations of a group G′ isomorphic to G except at infinity can be realized in the cohomology of Shimura varieties attached to G and G′. We obtain these Jacquet-Langlands correspondences by studying the bad reduction of a Shimura variety X attached to G at a prime p for which X has “Γ0(p)” level structure, and construct a “Deligne-Rapoport” model for X. The irreducible components of the special fiber of this model have a global structure that can be explicitly described in terms of Shimura varieties X′ for unitary groups G′ isomorphic to G away from infinity. The weight spectral sequence of Rapoport-Zink then yields an expression for certain pieces of the weight filtration on the étale cohomology of X in terms of the cohomology of X′. This identifies a piece of this weight filtration with a space of algebraic modular forms for G′. This result implies certain cases of the Jacquet-Langlands correspondence for G and G′ in terms of a canonical map between spaces of arithmetic interest, rather than simply as an abstract bijection between isomorphism classes of representations.