Local-Global Compatibility and the Action of Monodromy on nearby Cycles

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Local-Global Compatibility and the Action of Monodromy on nearby Cycles

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Title: Local-Global Compatibility and the Action of Monodromy on nearby Cycles
Author: Caraiani, Ana
Citation: Caraiani, Ana. 2012. Local-Global Compatibility and the Action of Monodromy on nearby Cycles. Doctoral dissertation, Harvard University.
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Abstract: In this thesis, we study the compatibility between local and global Langlands correspondences for \(GL_n\). This generalizes the compatibility between local and global class field theory and is related to deep conjectures in algebraic geometry and harmonic analysis, such as the Ramanujan-Petersson conjecture and the weight monodromy conjecture. Let L be a CM field. We consider the case when \(\Pi\) is a cuspidal automorphic representation of \(GL_n(\mathbb{A}_L^\infty)\), which is conjugate self-dual and regular algebraic. Under these assumptions, there is an l-adic Galois representation \(R_l(\Pi)\) associated to \(\Pi\), which is known to be compatible with the local Langlands correspondence in most cases (for example, when n is odd) and up to semisimplification in general. In this thesis, we complete the proof of the compatibility when \(l \neq p\) by identifying the monodromy operator N on both the local and the global sides. On the local side, the identification amounts to proving the Ramanujan-Petersson conjecture for \(\Pi\) as above. On the global side it amounts to proving the weight-monodromy conjecture for part of the cohomology of a certain Shimura variety.
Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA
Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:10086046
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