Quadratic Number Fields

Imagine you have a sphere in R^3 with radius equal to the square root of n. How many points with integer coordinates are there on this sphere? Gauss gave a formula in Disquisitiones Arithmeticae relating this number to the structure of quadratic fields. Generally, the rings of integers of quadratic fields do not have unique factorization into primes, and the failure of unique factorization is measured by the class number. Gauss related the number of integral points on the sphere to the class number.

Class numbers are connected to deep mathematics. Proving Gauss’ original conjectures and making them effective took place hand-in-hand with many of the great developments of number theory throughout the twentieth century, including Siegel’s work on zeros of Dirichlet L-functions and work by Gross and Zagier, and Goldfeld on elliptic curves, via ideas that have been important in some of the best theorems to date on the Birch and Swinnerton-Dyer Conjecture.

There’s still a lot we don’t know about class numbers. I have worked on understanding the divisibility properties of the class number. Predictions by Cohen and Lenstra describe many divisibility statistics. However we are a long way from proving their conjectures. I have studied divisibility questions for class numbers of imaginary quadratic fields, and some of their implications for the structure of elliptic curves, and some of my favorite ongoing work deals with class group structure for real quadratic number fields.

What are my techniques? Going back to the sphere question, suppose we do what is often done in combinatorics, and take the generating function of the numbers of points. The function you get turns out to have a ton of symmetries in terms of the matrix group SL_2(Z). Functions with these kinds of symmetries are modular forms and are what I use in my research.

Some of my ongoing work is related to the problem of explicitly constructing extensions of number fields with an abelian Galois group. Other than a few cases - the rational numbers and CM fields - it is not known how to construct such extensions. Famous conjectures of Stark relate the extensions to values of certain L-functions. My ongoing work expresses some of these values in terms of a type of modular form called “polyharmonic Maass forms". The study of such functions is a fairly new area, as only a few examples have appeared in the literature so far, which means that there are a lot of tantalizing questions about them that haven't yet been explored.

Integer Partitions

Say you want to know how many ways you can write n as a sum of positive integers, up to reordering the summands. The number of ways is the nth partition number p(n). 100 years ago, Ramanujan discovered three surprising divisibility properties, he discovered that p(5n+4) is always divisible by 5, p(7n+5) is always divisible by 7, and p(11n+6) is always divisible by 11. Trying to understand these and similar congruences led to important work on newforms by Atkin and Lehner and we now know that many similar congruences come about from the theory of Galois representations attached to modular forms.

A lot of cool math has been developed, just trying to understand the partition numbers, and understanding the partition numbers is what a lot of my research does. If we put these numbers into a generating function, the resulting function has lots of symmetries and is a modular form. Using the theory of modular forms, I have studied integer partitions both in terms of their asymptotic properties and arithmetic properties, including work towards understanding the Ramanujan-type congruences that can occur.

The partition function p(n) counts the nonincreasing sequences of positive integers whose sum is n. Despite having such an elementary definition, partition numbers are connected to deep mathematics. The Ramanujan congruences are three simple divisibility patterns for partition numbers, and understanding them led to some of the major developments in the theory of modular forms throughout the twentieth century. I am interested in a collection of statistical and arithmetic aspects of partitions.

One question I’m interested in is classifying the types of Ramanujan-type congruences that can occur, more specifically, finding a description for the triples (A,B,p) such that p(An+B)=0(mod p) for all n. For example, are there any of the form (Q*P, B, P), where Q, P are primes, and P is at least 13? This is not known, but my work with Ahlgren and Raum suggests that such triples, if they exist, are rare in a certain sense. Surprisingly, some undergraduate students that I supervised last fall found that such congruences do in fact exist when you look at colored partition numbers. This is a surprising finding, and in my future research I’m hopeful that understanding the difference between the colored and uncolored situations will help us answer the question for p(n).

Modular forms and L-functions

The study of L-functions such as the Riemann zeta function is one of the most classical fields of analytic number theory. The distribution of zeros of L-functions encode arithmetic information such as the distribution of primes and coefficients of modular forms. For example, the Generalized Riemann Hypothesis predicts that all the nontrivial zeros of L-functions have real part equal to ½, and knowing this for the Riemann zeta function would give us a very strong approximation for the prime counting function. With Liu, Thorner, and Zaharescu, I have been studying statistical approximations to the Generalized Riemann Hypothesis. As an application, we've explored the distributions of the fractional parts of the zeros of a class of L-functions. We would be able to make several of our applications more precise if we could quantify the error terms in some of our estimates, but this involves a certain Diophantine approximation question that does not seem straightforward, and it is something I hope to make progress on in the future.