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Nicolas Mascot

Publications and source records attributed to Nicolas Mascot.

13 recordsLinked to original sources
arXiv2026

Poincar\'e \`a la Makdisi

We show how to extend Makdisi's algorithms to compute explicitly with the Poincar\'e torsor on the Jacobian of an algebraic curve.

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arXiv2026

Equationless quadratic Chabauty for non-split Cartan modular curves

The aim of this article is to describe an equationless method for determining the rational points on the non-split Cartan curve $X_{\rm{ns}}^+(N)$ of prime level $N \geqslant 13$. Instead of using a projective model for the modular curve, our method uses the moduli interpretation of the curve, namely we work directly with elliptic curves and Cartan level structures. To accomplish this, we use the geometric version of the quadratic Chabauty method. We show that this can be combined with algorithms for divisor arithmetic developed by Makdisi and Mascot so as to apply to modular curves. As an illustration, we rederive the set of rational points on the curve $X_{\rm{ns}}^+(13)$.

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arXiv2023

Explicit computation of Galois representations occurring in families of curves

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 P^1_Q, and we study their degeneration at places of bad reduction of the corresponding curve.

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arXiv2022

A method to prove that a modular Galois representation has large image

Let $ρ$ be a mod $\ell$ Galois representation attached to a newform $f$. Explicit methods are sometimes able to determine the image of $ρ$, or even the number field cut out by $ρ$, provided that $\ell$ and the level $N$ of $f$ are small enough; however these methods are not amenable to the case where $\ell$ or $N$ are large. The purpose of this short note is to establish a sufficient condition for the image of $ρ$ to be large and which remains easy to test for moderately large $\ell$ and $N$.

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arXiv2021

Moduli-friendly Eisenstein series over the p-adics and the computation of modular Galois representations

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 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.

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arXiv2019

Hensel-lifting torsion points on Jacobians and Galois representations

Let $ρ$ be a mod $\ell$ Galois representation. We show how to compute $ρ$, given the characteristic polynomial of the image of the Frobenius at one prime $p$ and a curve $C$ whose Jacobian contains $ρ$ in its $\ell$-torsion. The main ingredient is a method to $p$-adically lift torsion points on a Jacobian in the framework of Makdisi's algorithms.

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arXiv2016

Companion forms and explicit computation of PGL2 number fields with very little ramification

In previous works, we described algorithms to compute the number field cut out by the mod ell representation attached to a modular form of level N=1. In this article, we explain how these algorithms can be generalised to forms of higher level N. As an application, we compute the Galois representations attached to a few forms which are supersingular or admit a companion mod ell with ell=13 (and soon ell=41), and we obtain previously unknown number fields of degree ell+1 whose Galois closure has Galois group PGL(2,ell) and a root discriminant that is so small that it beats records for such number fields. Finally, we give a formula to predict the discriminant of the fields obtained by this method, and we use it to find other interesting examples, which are unfortunately out of our computational reach.

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arXiv2016

Certification of modular Galois representations

We show how the output of the algorithm to compute modular Galois representations described in our previous article can be certified. We have used this process to compute certified tables of such Galois representations obtained thanks to an improved version of this algorithm, including representations modulo primes up to 31 and representations attached to a newform with non-rational (but of course algebraic) coefficients, which had never been done before. These computations take place in the Jacobian of modular curves of genus up to 26. The resulting data are available on the author's webpage, http://www2.warwick.ac.uk/fac/sci/maths/people/staff/mascot/galreps.

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arXiv2013

Computing modular Galois representations

We compute modular Galois representations associated with a newform $f$, and study the related problem of computing the coefficients of $f$ modulo a small prime $\ell$. To this end, we design a practical variant of the complex approximations method presented in the book edited by B. Edixhoven and J.-M. Couveignes. Its efficiency stems from several new ingredients. For instance, we use fast exponentiation in the modular jacobian instead of analytic continuation, which greatly reduces the need to compute abelian integrals, since most of the computation handles divisors. Also, we introduce an efficient way to compute arithmetically well-behaved functions on jacobians, a method to expand cuspforms in quasi-linear time, and a trick making the computation of the image of a Frobenius element by a modular Galois representation more effective. We illustrate our method on the newforms $Δ$ and $E_4 Δ$, and manage to compute for the first time the associated faithful representations modulo $\ell$ and the values modulo $\ell$ of Ramanujan's $τ$ function at huge primes for $\ell \in {11,13,17,19,29}$. In particular, we get rid of the sign ambiguity stemming from the use of a non-faithful representation as in J. Bosman's work. As a consequence, we can compute the values of $τ(p) \bmod 2^11.3^6.5^3.7.11.13.17.19.23.29.691 \approx 2.8.10^19$ for huge primes $p$. These representations lie in the jacobian of modular curves of genus up to 22.

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