Quadratic Number Fields
Updated: Mar 11, 2021
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.