Degree
Doctor of Philosophy (PhD)
Department
Mathematics
Document Type
Dissertation
Abstract
A great deal of progress in number theory throughout history has been motivated by trying to solve equations. One of the most famous challenges is to show there are no positive integer solutions to $x^{n}+y^{n} = z^{n}$ for $n > 2$, posed by Fermat around 1637. Special cases, such as the $n = 3$ and $n = 4$ cases, can be established using various algebraic manipulations. However, a general solution was elusive until the late 1990s when the combined work of Wiles \cite{Wiles} and Taylor--Wiles \cite{TaylorWiles} give a full proof.
One of the key insights used in proving Fermat's conjecture involves reducing the problem to establishing a correspondence between two seemingly unrelated objects, certain elliptic curves with coefficients in $\Q$, and appropriate holomorphic functions called modular forms. This correspondence between elliptic curves over $\Q$ and modular forms typically called the modularity of elliptic curves over $\Q$, is one example of the far-reaching links between algebraic geometry and number theory predicted in the Langlands program.
The aims of this thesis are twofold. We study both modularity results and Sato--Tate distributions related to certain families of hypergeometric varieties. In the first part, we develop an explicit method to establish the modularity of a large class of hypergeometric varieties, including various families of elliptic curves. This method gives an explicit construction of the target modular form from hypergeometric information, which elucidates the relationship between hypergeometric character sums and modular forms in many cases.
In the second part, a new inductive method is introduced to establish Sato--Tate distributions for certain families of elliptic curves, which are examples of hypergeometric varieties. The recursions inherent in several Hecke trace formulas for particular spaces of cusp forms are linked with the recursions satisfied by the Catalan numbers. This recursive perspective allows us to establish several new examples of hypergeometric moments and leads to a further understanding of the known results in this area.
Date
3-26-2025
Recommended Citation
Grove, Brian, "Explicit Modularity and Moments for Certain Hypergeometric Character Sums" (2025). LSU Doctoral Dissertations. 6699.
https://repository.lsu.edu/gradschool_dissertations/6699
Committee Chair
Long, Ling
