For reprints (or rather *electronic* reprints), just shoot me an email.

- Shapes of sextic fields and log-terms in Malle's conjecture (with Erik Holmes)

In preparation - The shapes of Galois quartic fields (with Piper H)

To appear in Transactions of the American Mathematical Society(arXiv) - Equidistribution of shapes of complex cubic fields of fixed quadratic resolvent

Submitted (see slides below for a talk on the subject)(arXiv) - Explicit computations of Hida families via overconvergent modular symbols (with Evan Dummit, Marton Hablicsek, Lalit Jain, Robert Pollack, and Daniel Ross)

- The shapes of pure cubic fields

- Iwasawa theory for symmetric powers of CM modular forms at non-ordinary primes, II (with Jonathan Pottharst)

- Counting elliptic curves with prescribed torsion (with Andrew Snowden)

- On symmetric power L-invariants of Iwahori level Hilbert modular forms (with Andrei Jorza)

- Gauss–Manin connections for p-adic families of nearly overconvergent modular forms (with Liang Xiao)

- Iwasawa theory for symmetric powers of CM modular forms at non-ordinary primes (with Antonio Lei)

- The exceptional zero conjecture for symmetric powers of CM modular forms: the ordinary case

- On Greenberg’s L-invariant of the symmetric sixth power of an ordinary cusp form

- L-invariants of low symmetric powers of modular forms and Hida deformations

PhD thesis, Princeton University, 2009.

Advisor: Andrew Wiles.

- Equidistribution of shapes of cubic fields of fixed quadratic resolvent

These are (expanded) slides from a 15-minute talk at CNTA XIV discussing my (upcoming) paper on the equidistribution of shapes of complex cubic fields with fixed trace-zero form on the corresponding geodesics in the modular curve. (The animations don't work in Preview on Mac.) *Critical integers of motivic L-functions and Hodge numbers*

This short note explains how to relate the critical integers of a motivic*L*-function (in the sense of Deligne) to the Hodge numbers of the motive. Specific examples are worked out (the Tate motive**Q**(1) and symmetric powers of modular forms).

Some data visualizations I've created. Click on an image for a description of it.

- Overconvergent modular symbols code (including Hida theory):

This code is available here on github. See the published article. - Iterate factorization trees:

This code is available here on github. It was first written at the AIM workshop on the Galois theory of orbits in arithmetic dynamics and computes the trees described in Section 3 of Rafe Jones and Nigel Boston's Settled polynomials over finite fields. - Shape of a number field:

This code is available here on github. It computes the "shape" of a number field. The*shape*of a number field*K*of degree*n*is the (*n*−1)-dimensional lattice in the Minkowski space given by the orthogonal complement of the vector 1. The output of this function is simply the (not necessarily reduced) Gram matrix of the lattice. - Artin representations in Sage:

This code is available here on github. The goal of this project is to be able to compute with Artin representations in Sage. Some highlights include Artin conductors,*L*-functions, and root numbers. The project also improves the Galois group code to allow non-Galois fields.