We study the problem of counting the number of roots of an irreducible polynomial $f(X) in mathbb{Z}[X]$ modulo rational primes. We consider the family of polynomials $f_n(X) = X^n-X-1$, which have Galois groups isomorphic to $S_n$. The approach we take is to attach Galois representations to the counting problem and then to relate these to automorphic forms. In particular, we attempt to attach the representations to holomorphic forms on $GL_2$. We show this only works when $n leq 5$, and we present the solutions to the problem in the $n = 4$ and $5$ cases, following methods due to Serre, Crespo, and Buhler for explicitly constructing Galois representations. The solution to the $n = 5$ case is novel, requiring Hilbert modular forms. In solving the problem, we produce the first example of an icosahedral Hilbert form that is not the base change of a classical form.
Subject (authority = RUETD)
Topic
Mathematics
RelatedItem (type = host)
TitleInfo
Title
Rutgers University Electronic Theses and Dissertations
Rutgers University. Graduate School - New Brunswick
AssociatedObject
Type
License
Name
Author Agreement License
Detail
I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.