Research Overview
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My research in number theory focuses on the connection between the arithmetic of quadratic forms and automorphic forms. A prototypical result along these lines is Jacobi’s formula for the exact number of integer solutions $(x, y, z, w)$ of the equation
$$x^2 + y^2 + z^2 + w^2 = m$$
for any integer $m \geq 1$. He was able to prove the exact formula that the number of solutions is given by
$$r_4(m) = 8 \sum_{0 < d \mid m, 4\nmid d} d$$
which is 8 times the sum of the positive divisors $d$ of $m$ where the number 4 doesn’t divide $d$. This formula is not easy to prove, and Jacobi used the rather surprising method of elliptic functions (which at first glance belongs much more to complex analysis than to number theory) to derive this result. Elliptic functions, and their close cousins called “theta functions”, are objects that unify analysis and arithmetic as functions with a large number of arithmetically interesting symmetries.
The study of similar kinds of functions over the last 100 years has revealed the incredibly important nature of such functions (i.e. automorphic forms) for using analytic methods and ideas to study arithmetic problems.
My research program is to use similar ideas to study questions about the arithmetic of arbitrary quadratic forms over the integers and rational numbers. Below are several topics that I find interesting which motivate my research program:
- Mass formulas and Class numbers of Quadratic Forms
- Comparing Theta functions with Hecke eigenforms
- Local-global principles for quadratic forms (in small numbers of variables)
- Explicit representation and finiteness theorems for quadratic forms
- Uniqueness results and linear relations between theta series for quadratic forms
- Arithmetic parametrizations of class groups and related objects by quadratic forms
- “Theta functions” for other arithmetic embedding problems related to automorphic forms