Dissertations, Theses, and Capstone Projects
Date of Degree
9-2024
Document Type
Dissertation
Degree Name
Ph.D.
Program
Mathematics
Advisor
Krzysztof Klosin
Committee Members
Kenneth Kramer
Brooke Feigon
Subject Categories
Number Theory
Keywords
Galois representations, automorphic forms
Abstract
In this paper, we study Fontaine-Laffaille, self-dual deformations of a mod p non-semisimple Galois representation of dimension n with its Jordan–Hölder factors being three mutually non-isomorphic absolutely irreducible representations. We show that under some conditions regarding the orders of certain Selmer groups, the universal deformation ring is a discrete valuation ring. Given enough information on the Hecke algebra, we also prove an R = T theorem in the general context. We then apply our results to abelian surfaces with cyclic rational isogenies and certain 6-dimensional representations arising from automorphic forms congruent to Ikeda lifts. Assuming the Bloch-Kato conjecture, our result identifies special L-value conditions for the existence of a unique abelian surface isogeny class and an R = T theorem for certain 6-dimensional Galois representations.
Recommended Citation
Huang, Xiaoyu, "On Deformation Rings of Residual Galois Representations with Three Jordan–Hölder Factors" (2024). CUNY Academic Works.
https://academicworks.cuny.edu/gc_etds/6074