Dissertations, Theses, and Capstone Projects
Date of Degree
9-2025
Document Type
Doctoral Dissertation
Degree Name
Doctor of Philosophy
Program
Mathematics
Advisor
Krzysztof Klosin
Committee Members
Brooke Feigon
Kenneth Kramer
Subject Categories
Number Theory
Keywords
Galois representations, modular forms, automorphic forms, Selmer groups, congruences
Abstract
We prove some residually reducible p-adic Galois representations of number fields arise from modular forms. We study the universal deformation ring arising from deformations satisfying the Fontaine--Laffaille condition at primes over p. Under certain conditions, we establish the reduced universal deformation ring is a discrete valuation ring. The method uses pseudocharacters and certain bounds on Selmer groups, where the ideal of reducibility as defined by Bellaïche and Chenevier is shown to be maximal and principal. A self-dual assumption is not needed in our argument. The main result on deformations applies to n-dimensional representations. In applications, congruences between Hermitian modular forms due to Klosin are used with our results to prove modularity of some irreducible, residually reducible 4-dimensional p-adic representations of the imaginary quadratic field Q(i) and of imaginary quadratic fields with odd discriminant. Using other congruences, we provide explicit examples with 2-dimensional representations of Q. We also obtain an R=T result where the universal deformation ring is not necessarily a discrete valuation ring.
Recommended Citation
Akers, Geoffrey, "Galois Deformation Rings and Modularity in the Residually Reducible Case" (2025). CUNY Academic Works.
https://academicworks.cuny.edu/gc_etds/6372
