In the 1970s and 1980s, Jean-Pierre Serre formulated a conjecture connecting two-dimensional Galois representations and modular forms. The conjecture came to be known as Serre's modularity conjecture. It was recently proved by Khare and Wintenberger in 2008. Serre's conjecture has various important consequences in number theory. Most notably, it played a key role in the proof of Fermat's last theorem. A natural question is, what is the analogue of Serre's conjecture for higher dimensional Galois representations? In 2002, Ash, Doud and Pollack formulated a precise statement for a higher dimensional analogue of Serre's conjecture. They also provided numerous computational examples as evidence for this generalized conjecture. We consider the three-dimensional version of the Ash-Doud-Pollack conjecture. We find specific examples of three-dimensional Galois representations and computationally verify the generalized conjecture in all these examples.
College and Department
Physical and Mathematical Sciences; Mathematics
BYU ScholarsArchive Citation
Dang, Vinh Xuan, "Three-Dimensional Galois Representations and a Conjecture of Ash, Doud, and Pollack" (2011). All Theses and Dissertations. 2697.
Algebraic Number Theory, Representation Theory, Serre's Conjecture