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The Geometry of Unipotent Deformations and Applications

FUNCK, DANIEL,PETER (2023) The Geometry of Unipotent Deformations and Applications. Doctoral thesis, Durham University.

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This thesis studies primarily the local properties the unipotent connected component of the moduli space of Langlands parameters, the local rings of
which give us Galois deformation rings, a crucial ingredient in the Taylor-Wiles-Kisin
patching method that is used to prove global Langlands correspondences. We study
first the simpler ‘considerate’ case to give a criterion for smoothness of the connected
components when G = GLn. We also study the local rings of various unions of connected components to show that the Galois deformation rings are Cohen-Macaulay.
We study further the Steinberg component in the case of ‘extreme inconsiderateness’ to show that the Steinberg component has at most rational singularities, so
in particular is normal and Cohen-Macaulay. Finally, we give an application of the
smoothness result, to give a freeness result of the module of certain Hida families
of automorphic forms over its Hecke algebra, which in turn will give a multiplicity
result for the Galois representations of these Hida families.

Item Type:Thesis (Doctoral)
Award:Doctor of Philosophy
Keywords:Number Theory; Algebraic Geometry; Deformation rings; Langlands Parameters
Faculty and Department:Faculty of Science > Mathematical Sciences, Department of
Thesis Date:2023
Copyright:Copyright of this thesis is held by the author
Deposited On:20 Sep 2023 14:37

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