Papers and Preprints
Papers and Preprints
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- The Geometry of the unipotent component of the moduli space of Weil-Deligne representations
Abstract: 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.
2. Singularities of Steinberg deformation rings
2. Singularities of Steinberg deformation rings
With Jack Shotton.
Let l and p be distinct primes, let F be a local field with residue field of characteristic p, and let X be the irreducible component of the moduli space of Langlands parameters for GL3 over $\Z_l$ corresponding to parameters of Steinberg type. We show that X is Cohen-Macaulay and compute explicit equations for it. We also compute the Weil divisor class group of the special fibre of X, motivated by work of Manning for GL2. Our methods involve the calculation of the cohomology of certain vector bundles on the flag variety, and build on work of Snowden, Vilonen-Xue, and Ngo.
Thesis
Thesis
GOUDA__The_Geometry_of_Unipotent_Deformations_and_Applications (Complete).pdf The Geometry of Unipotent deformations and Applications
The Geometry of Unipotent deformations and Applications
Abstract: 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.
The Online Durham e-Thesis can be found at: http://etheses.dur.ac.uk/15143/
Talks
Talks
The Springer correspondence and an introduction to geometric representation theory
May 2023
The Unipotent component of the moduli space of Weil-Deligne Representations and the Cohen-Macaulay property
Dec 2022
The geometry of moduli spaces of inertially trivial Weil-Deligne representations
Sept 2021
Algebraic number theory and the maximal unramified extension of a number field
Feb 2020
Complex multiplication and class field theory
Oct 2019
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