Research – Ben Breen

My research interests lie in number theory and arithmetic geometry. I work on explicit methods for computing Hilbert modular forms and developing Cohen-Lenstra type heuristics for units and narrow class groups of number fields.

Articles

An explicit trace formula for Hilbert modular forms. In preparation.
Unit signatures, class groups and narrow class groups of $S_n$ -fields of even degree. In preparation.
On unit signatures and narrow class groups of odd abelian number fields: Galois structure and heuristics (with Ila Varma and John Voight), Submitted to Compositio
Wild ramification in a family of low-degree extensions arising from iteration. (with Rafe Jones, Tommy Occhipinti, and Michelle Yuen), JP J. Algebra Number Theory 37 (2015), 69-104.
Fourier transforms on $\text{SL}_2(\mathbb{Z}/p^n\mathbb{Z})$ and related numerical experiments. (with Daryl R. Deford, Jason D. Linehan, Daniel N. Rockmore), arXiv:1710.02687

Projects

Hilbert Modular Forms

The graded ring of complex modular forms for $\Gamma = \text{SL}_2(\mathbb{Z})$ is generated by two Eisenstein series

$\bigoplus_{k \; \in \;2\mathbb{Z}} M_k(\Gamma) = \mathbb{C} [E_4, E_6].$

Hilbert modular forms are an extension of classical modular forms to the setting of totally real number fields. The graded rings of complex Hilbert modular forms are also finitely generated, but few cases have been worked out. For example, the ring of parallel weight forms for $F = \mathbb{Q}(\sqrt{5})$ is

$\bigoplus_{k \; \in \;2\mathbb{Z}} M_k(\Gamma_F) = \mathbb{C} [X_2, X_6,X_{10},X_{20}] / (R_{40}),$

where $R_{40}$ is a relation between the generators in weight $40$ . I am part of a collaboration at Dartmouth developing an implementation in Magma to compute rings of Hilbert Modular forms. This is available on my Github.

Code

All of my code is all available on my Github. Here are links to some of the Magma/Sage packages