Most of my research relates to number fields in some way. A classic problem is to list all number fields of a given degree with absolute discriminant less than some bound. Many of my papers with John Jones address some variant of this problem. [4] found that there are 398 sextic fields ramified only within {2,3}. The sequel paper 7 found that there are only 10 septic fields ramified within {2,3}. The long paper [15] focused on the important special case of Galois number fields; a sample result is that there are five A₆ and thirteen S₆ such fields with root discriminant less than 44.76. The project of extensive tabulation of number fields reached a mature phase in [23]. This paper describes a website which lets the user obtain many complete lists. More recent research in this vein is [32] which shifts the focus from number fields to Artin L-functions, and [38] which focuses on a complete list in a difficult octic case.

The paper [5] responds to the classic problem in the case of degree three. It adds a conjectural secondary term to an asymptotic formula of Davenport and Heilbronn. With this secondary term, the asymptotic formula becomes a much closer fit to the data collected by Belabas. The conjecture has since been proved by Thorne & Taniguchi and by Bhargava, Shankar, & Tsimerman.

Some of my papers are focused on number fields of large degree which have light ramification for their Galois group. [B] includes a number field of degree 15875, Galois group the full symmetric group, and ramification set {2,5}; the existence of this and other similar number fields is surprising from the viewpoint of Bhargava's mass heuristic. [21] centers on a polynomial for a degree twenty-five non-solvable number field ramified only at 5; this polynomials responds concretely and affirmatively to a conjecture of Gross. [24] includes a number field ramified only at eleven with Galois group involving the Mathieu group M₁₂; no other sporadic group is known to similarly be involved in a wild one-prime field.

[1] and [2] were both written as satellite papers to [3], each pursuing a general Galois-theoretic topic associated to sextic fields. The latter describes how the non-trivial outer automorphism of S₆ plays a helpful role in cataloging separable extensions of degree at most six of a given ground field. Of interest here is that one has to leave the traditional setting of fields, as the twin of a given sextic field can be a product of fields. [8] generalizes a trick of Serre for fully computing Frobenius classes in alternating groups. The problem of completely computing Frobenius classes in general groups was later solved by Dokchitser and Dokchitser.

My thesis concerns Shimura curves which behave very similarly to the classical modular curves X₀(N). An intersection formula from this thesis was proved in greater generality with Keating in [18]. Newforms with rational coefficients and no CM are studied in [33]. The classical Maeda conjecture says that there are no such forms at level N=1 beyond the familiar ones occurring in weights 12, 16, 18, 20, 22, and 26. This paper make an analogous conjecture for all levels N. For example, all such forms with weight k>26 are quadratic twists of 32 forms of weight 2 and 6, covering all even weights up through k=50.

[34] works backwards from the rational modular forms of [33] to get an interesting collection of PGL₂(p) fields, illustrating three parallel theories of companion forms in the process. When the ground field Q is replaced by a general totally real field, the theory of companion forms becomes much more complicated. [28], joint with Dembele and Diamond, formulates a general conjecture in this context. This conjecture was subsequently proved by Calegari, Emerton, Gee, and Mavrides.

[A] classifies rigid local systems, which are group-theoretic objects underlying families of motives. The classification reveals some unexpected features in Katz's "fascinating bestiary". Among these features is that the two classical cases—the hypergeometric and Pochhammer—are extreme outliers, in that they have the maximal number of parameters for their degree. Next are twelve submaximal series, which have been studied further by Oshima.

By reduction modulo a prime and then specialization, rigid local systems yield number fields with Lie-type groups and highly restricted ramification. [9] presents examples of this construction, using mostly hypergeometric systems as starting points, and keeping ramification within {2,3}. [26] presents more examples of this construction, not using a broader notion of rigidity. In the end 376 fields are produced with Galois group SU₃(3).2=G₂(2) and ramification within {2,3}.

With Rodriguez Villegas and Watkins, I am currently pursuing the hypergeometric case in detail. The research announcements [35] and [36], and also all recent talks with hypergeometric in their title are part of this project.

With Broadhurst, I am studying motives whose Frobenius traces involve Kloosterman sums and whose period matrices involve integrals of Bessel functions of interest in physics. The note [37] presents some numerically calculated special values of L-functions, and the last section of [39] gives intricate conjectural quadratic relations between entries of the period matrix.