My field of research is number theory. More specifically, I'm interested in the following (closely related) areas:

• Special values of L-functions
• Cohomology of Galois representations and Iwasawa theory
• P-adic modular and automorphic forms
• Computational methods for automorphic forms

Almost all of my research is carried out in collaboration with Sarah Livia Zerbes (and frequently with other collaborators as well). See also my publications page.

Euler systems notes

For an introduction to my research area, see the following lecture notes:

• Notes of Sarah's and my lectures on Euler systems and the Bloch–Kato conjecture at the 2021 Alpbach Summer School
• Notes and videos of our course on Euler Systems at the 2018 Arizona Winter School.
• Notes of my Euler Systems mini-course at the Iwasawa centenary conference, Tokyo, 2017

Sage and Magma scripts

Here is some Magma code (mostly by my collaborator Lassina Dembele) for calculating Hilbert modular forms of non-paritious weight over real quadratic fields.

Here is a Sage script for computing the Atkin-Lehner W_N operators on modular forms spaces.

Study groups

Notes for a talk on Bernstein–Zelevinsky derivatives for the 2021 London study group

In spring 2014, Alex Bartel and I organised a study group on Galois cohomology; see here.

In summer 2011 I organised a study group on p-adic rigid analytic geometry; see here.

Notes and other junk

Some of the computer programs I have written can be found here; but newer versions of many of these are incorporated in Sage.

• Computing with algebraic automorphic forms (notes of my lectures at the 2011 Heidelberg summer school)
• Slides for my talk Calculating Automorphic Forms for Unitary Groups Using SAGE at the Sage Days 6 conference, 11/11/2007.
• Notes for my talk Approaches to computing overconvergent p-adic modular forms and the accompanying implementation presentation at the Heilbronn Institute workshop "Computing with automorphic forms", summer 2008
• Slides for my talk Calculating p-adic modular forms at Sage Days 16, 25/6/2009.
• A note on the norm of the U_p operator
• A note on the norms of overconvergent eigenfunctions
• Attempts to factorise the U₃ operator
• Study group talks:
  • De Rham cohomology (November 2005)
  • The highest weight theorem (October 2006)
  • Supercuspidal representations of GL_n(F) (February 2007)
  • A spectral sequence primer (May 2007)
• Cambridge things:
  • Elliptic Curves notes for the Lent 2005 Part III course
  • My Part III essay: Congruences between modular forms.
  • Spectral Measures, my entry for the 2003 Yeats Mathematical Essay Prize at Trinity College
• General junk:
  • A note on topological groups (in which I show that if G is compact and Hausdorff its conjugacy class space is Hausdorff in the quotient topology).
  • An introduction to Burnside's algorithm for calculating character tables of finite groups.
  • A short note on roots of cubic polynomials (published in volume 59 of Eureka (March 2008), the journal of the Cambridge undergrad maths society), in which I investigate when all of the roots of a complex cubic have the same absolute value. (Although this was motivated by my research on automorphic forms for unitary groups, the actual content is elementary.)