# Research Overview

My research is primarily in algebraic number theory and related fields. A common theme in my work is to render theoretical topics more concrete by explicitly working out detailed examples. With coauthor John Jones, I have built extensive tables of number fields and p-adic fields.** **In higher degrees, where complete tables are out of reach, I have found and studied various extreme number fields.** **In parallel to my number field studies, I have studied polynomials** **which are extreme or special in some way. Some early papers study general issues in Galois theory which have arisen in these number-theoretic studies. More recent topics include modular forms,** **motives, and** **Hurwitz covers, with connections with number fields being a recurring theme. I also have an interest in game theory.

## Preprints

Preprints are posted on the Arxiv as well as here.

## Publications

The versions posted here are typically slightly different from the published versions.

## Presentations

Presentations often contain results not yet in preprints or publications.

## Number fields

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 A_{6} and thirteen S_{6} 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.

## p-adic fields

A recurring theme of my work is close attention to ramification. In [13], John Jones and I constructed a database of p-adic fields of small degree. [10] treated the complicated case of nonic 3-adic fields, finding that there are 795 of them. [20] treated the even more complicated case of octic 2-adic fields, finding that there are 1823 of them. The database has expanded further since these publications, and Jones has incorporated its data into the LMFDB.

The paper [17] studies p-adic fields in arbitrary degree, centering on generating functions for the total mass of such fields. A theme is that just tame ramification is described by ordinary partitions, wild ramification can be approached in a similar combinatorial spirit. The theoretical paper [22] introduces a tame-wild principle. This principle says that from a certain viewpoint, wild ramification is often constrained by tame ramification.

## Extreme number fields

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 M_{12}; no other sporadic group is known to similarly be involved in a wild one-prime field.

## Special Polynomials

[6] obtains explicit formulas for discriminants of polynomials related to the Painlevé differential equations. These formulas show that the discriminants factor into products of small primes only. [14] likewise obtains explicit formulas for discriminants of a certain generalization of cyclotomic polynomials related to fractals in the limit of large degree. [25] catalogs certain polynomials with prescribed bad primes. These polynomials are useful in constructing non-solvable number fields with ramified at those primes only.

## Galois theory

[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 S_{6} 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.

## Modular Forms

My thesis concerns Shimura curves which behave very similarly to the classical modular curves X_{0}(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 PGL_{2}(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.

## Motives

[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 SU_{3}(3).2=G_{2}(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.

## Hurwitz covers

[27], joint with Akshay Venkatesh, studies the monodromy of the covering maps from Hurwitz moduli spaces to configuration spaces. Always assuming enough branch points of each type, it gives necessary and sufficient conditions for this monodromy to be full, meaning the alternating or symmetric group on the degree. In particular, from any nonabelian finite simple group G one can build full covers of arbitrarily large degree which have bad reduction at exactly the primes dividing the order of G.

The Bhargava mass heuristic, combined with the numerics of [17], suggests that for any finite set of primes P, there should be only finitely many full fields ramified within P. The full monodromy theorem, on the other hand, points strongly in the other direction whenever P contains the set of primes dividing the order of a nonabelian finite simple group. It says that there are infinitely many full fields ramified within P, if specialization of families behaves anywhere near generically. [29] specializes many coverings of surfaces, considering fibers above points to obtain Hurwitz number fields. It gives evidence that specialization indeed rapidly becomes completely generic as degrees get larger.

[30] specializes Hurwitz covers to special curves in configuration spaces to obtain Hurwitz-Belyi maps. Just as Hurwitz number fields are extreme outliers among all number fields, Hurwitz-Belyi maps are extreme outliers among all Belyi maps. The interesting phenomenon that one can sometimes give infinitely many of these maps at once by varying certain parameters is illustrated in [31].

## Game Theory

I have developed a side interest in game theory through supervising senior seminars and co-teaching an honors game theory course. [16] gives a clear picture of the typical behavior of the unique Nash equilibrium in random zero-sum matrix games. Changing some signs in the argument, it simultaneously gives a clear picture of the enormously many Nash equilibria in random coordination matrix games. [12] is a sequel paper further clarifying the case of coordination games.