Nicolas Mascot

Introduction

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.

I am located at office 1.9 in the Hamilton building. You can email me by clicking on this link.

June 2023 workshop

I am the organiser of a workshop on Explicit methods in automorphic forms and arithmetic geometry. See this page for details.

Research

My ORCID is 0000-0001-5296-7422, and my MR Author ID is 1040021.

Explicit computation of Galois representations occurring in families of curves
arXiv preprint, 2024.
We extend our method to compute division polynomials of Jacobians of curves over Q to curves over Q(t), in view of computing mod ell Galois representations occurring in the étale cohomology of surfaces over Q. Although the division polynomials which we obtain are unfortunately too complicated to achieve this last goal, we still obtain explicit families of Galois representations over P¹_Q, and we study their degeneration at places of bad reduction of the corresponding curve.
The code and data maybe be accessed here.

A method to prove that a modular Galois representation has large image
arXiv preprint, May 2022.
This short note presents an easy-to-test sufficient condition for a mod ℓ Galois representation attached to a modular form to have large image.

Computing the trace of an algebraic point on an elliptic curve (joint with Denis Simon)
Published in Expositiones Mathematicae, 2023.
We present a simple and efficient algorithm to compute the sum of the algebraic conjugates of a point on an elliptic curve.

Moduli-friendly Eisenstein series over the p-adics and the computation of modular Galois representations
Published in Research in Number Theory, 2022.
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, as well as in the "nicolas-KKM" git branch of PARI/GP.

A Prym variety with everywhere good reduction over Q(√61) (joint with Jeroen Sijsling and John Voight)
Published in Arithmetic Geometry, Number Theory, and Computation in the Springer Simons Symposia series, 2022.
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
Published in Foundations of Computational Mathematics, 2022.
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 representation with values in GL(3,9) attached to an eigenform over a congruence subgroup of SL(3). We obtain in particular a polynomial with Galois group isomorphic 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, as well as in the "nicolas-KKM" git branch of PARI/GP.

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(F_ℓ) 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 small prime.

My PhD thesis, Computing modular Galois representations

Talks

Here is a short selection of research talks which I have given:

Slides and video of a presentation of the functionalities of my GP package to compute with plane algebraic curves

A presentation of the functionalities of my PARI/GP package to compute Galois representations occurring in the torsion of Jacobians of curves

Computing modular Galois representations p-adically using Makdisi's moduli-friendly forms

A p-adic method to compute Galois representations afforded in the ℓ-torsion of Jacobians of curves, with applications to higher étale cohomology spaces of higher-dimensional varieties

Explicit computation of lightly-ramified number fields with Galois group PGL₂(F_ℓ) through representations attached to companion or supersingular modular forms

A method based on complex geometry to compute Galois representations attached to modular forms, and a method to certify the correctness of the result

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:

The Jacobian variety of a compact Riemann surface: study of the Riemann relations between the periods of a compact Riemann surface, and statement and proof of the Abel-Jacobi theorem

Khinchin's constant, an amusing phenomenon about continued fractions statistics in ergodic number theory

Teaching

This term (Hilary term 2024-25), I am teaching MAU34104 Group representations. I am also the Mathematics course coordinator and the Mathematics Capstone project coordinator.

Before that, I taught the following modules:

Michaelmas term 2024-25: MAU34109 Algebraic number theory
Michaelmas term 2023-24: MAU34101 Galois theory
Hilary term 2022-23: MAU34104 Group representations
Michaelmas term 2022-23: MAU34109 Algebraic number theory
Michaelmas term 2021-22: MAU23101 Introduction to number theory
Michaelmas term 2021-22: MAU34101 Galois theory
Hilary term 2020-21: MAU34104 Group representations
Hilary term 2020-21: MAU22102 Rings, fields, and modules
Michaelmas term 2020-21: MAU23101 Introduction to number theory
Hilary term 2019-20: MAU22102 Rings, fields, and modules
Michaelmas term 2019-20: MAU34101 Galois theory

Here are archives from some of the classes I taught at other universities in the past:

In 2018-2019:

Math 261 - Number theory

In 2017-2018:

From 2008 to 2010:

"Colles" questions (oral examinations) that I gave from 2008 to 2010 (in French)

And here is a translation of a French 1993 ENS exam on Dirichlet's theorem on primes in arithmetic progressions that I particularly enjoyed as a student.