# David Loeffler: Research

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 U
_{3}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.)