Hello, and welcome! I am an Ussher assistant professor in number theory and cryptography at the School of Mathematics of Trinity College Dublin. Here is my CV.
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- Moduli-friendly Eisenstein series over the p-adics and the computation of modular Galois representations
arXiv preprint, April 2020.
We show how our p-adic method to compute Galois representations occurring in the torsion of Jacobians of algebraic curves can be adapted to modular curves. The main ingredient is the use of "moduli-friendly" Eisenstein series introduced by Makdisi, which allow us to evaluate modular forms at p-adic points of modular curves points and dispenses us of the need for equations of modular curves and for q-expansion computations. The resulting algorithm compares very favourably to the complex-analytic method.
The source code (still at an experimental stage) is available in a GitHub repository.
- A Prym variety with everywhere good reduction over Q(√61) (joint with Jeroen Sijsling and John Voight)
arXiv preprint, August 2019.
We compute an equation for a modular abelian surface A that has everywhere good reduction over the quadratic field K=Q(√61) and that does not admit a principal polarization over K.
- Explicit computation of a Galois representation attached to an eigenform over SL(3) from the H2 étale of a surface
arXiv preprint, October 2018 (updated June 2019).
We sketch a method to compute mod ℓ Galois representations contained in
the H2 étale of surfaces. We apply this method to the case of a
with values in GL(3,9) attached to an eigenform over a congruence
of SL(3). We obtain in particular a polynomial with Galois group
to the simple group PSU(3,9) and ramified at 2 and 3 only.
Data are available here.
- Hensel-lifting torsion points on Jacobians and Galois representations
Published in Math. Comp. 89.
We show how to compute any Galois representation appearing in the
torsion of the Jacobian of a given curve, given the characteristic
polynomial of the Frobenius at one prime.
The source code is available in a GitHub repository.
- Rigorous computation of the endomorphism ring of a Jacobian (joint with Edgar Costa, Jeroen Sijsling, and John Voight)
Published in Math. Comp. 88.
We describe several improvements to algorithms for the rigorous
computation of the endomorphism ring of the Jacobian of a curve defined
over a number field.
- Modular Galois representation data available for download
These data were computed and certified thanks to the algorithms described in the three articles below.
- Companion forms and explicit computation of PGL2 number fields with very little ramification
Published in Journal of Algebra 509.
This article shows how to generalise the algorithms to compute Galois
representations attached to modular forms to the case of forms of any
level. As an application, we compute representations attached to forms
which are supersingular or admit a companion form, and obtain previously
unknown number fields of Galois group PGL2(Fℓ) and record-breaking low root-discriminants. Finally, we establish a formula to predict the discriminant of such fields.
- Certification of modular Galois representations
Published in Math. Comp. 87.
This article presents methods to certify efficiently and rigorously that
the output of the algorithms described in the article below is correct.
- Computing modular Galois representations
Published in Rendiconti del Circolo Matematico di Palermo, Volume 62, No 3, December 2013.
This article describes how to explicitly compute modular Galois
representations associated with a modular newform, and studies the
related problem of computing the coefficients of this newform modulo a
- My PhD thesis, Computing modular Galois representations
Here is a short selection of slides of research talks which I have given:
And here are the slides of two talks (in French) that I gave in 2012 at informal mathematics seminars organised by and for PhD students:
This term (Michaelmas term 2020-21), I teach Introduction to number theory.
Last year (2019-2020), I taught the following modules:
Here are archives from some of the classes I taught at other universities in the past:
- In 2018-2019:
- In 2017-2018:
- From 2008 to 2010: