arXiv subjects

Cécile Armana

Publications and source records attributed to Cécile Armana.

8 recordsLinked to original sources

arXiv2026

A computational approach to Drinfeld modules

This survey provides a practical and algorithmic perspective on Drinfeld modules over $\mathbb F_q[T]$. Starting with the construction of the Carlitz module, we present Drinfeld modules in any rank and some of their arithmetic properties. We emphasise the analogies with elliptic curves, and in the meantime, we also highlight key differences such as their rank structure and their associated Anderson motives. This document is designed for researchers in number theory, arithmetic geometry, algorithmic number theory, cryptography, or computer algebra, offering tools and insights to navigate the computational aspects of Drinfeld modules effectively. We include detailed SageMath implementations to illustrate explicit computations and facilitate experimentation. Applications to polynomial factorisation, isogeny computations, cryptographic constructions, and coding theory are also presented.

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arXiv2024

Ogg's conjectures over function fields

In the early 1970s, Andrew Ogg made several conjectures about the rational torsion points of elliptic curves over $\mathbb{Q}$ and the Jacobians of modular curves. These conjectures were proved shortly after by Barry Mazur as a consequence of his fundamental study of the arithmetic properties of modular curves and Hecke algebras. In this paper, we review the function field analogues of Ogg's conjectures, their current status, and the methods that have been applied to prove some of these conjectures. The methods are based on the ideas of Mazur and Ogg, but there are interesting differences and technical complications that arise in the function field setting, as well as intriguing possible new directions for generalizations.

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arXiv2024

Non-perfect pairings between Hecke algebra and modular forms over function fields

We study two analogs, for modular forms over $\mathbb{F}_{q}(T)$, of the pairing between Hecke algebra and cusp forms given by the first coefficient in the expansion. For Drinfeld modular forms, the $\mathbb{C}_{\infty}$-pairing is provided by the first coefficient of their $t$-expansion at infinity. For $\mathbb{Z}$-valued harmonic cochains, the $\mathbb{Z}$-pairing is given by their Fourier coefficient with respect to the trivial ideal. We prove that, contrarily to classical cusp forms, both pairings in weight $2$ are not perfect in a quite general setting, namely for the congruence subgroup $Γ_0(\mathfrak{n})$ with any prime ideal $\mathfrak{n}$ in $\mathbb{F}_{q}[T]$ of degree $\geq 5$. We show it by exhibiting a common element of the Hecke algebra in the kernels of both pairings and proving that it is non-zero using computations with modular symbols over $\mathbb{F}_{q}(T)$. Finally we present computational data on other kernel elements of these pairings.

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arXiv2024

Drinfeld singular moduli, hyperbolas, units

Let $q\geq2$ be a prime power and consider Drinfeld modules of rank 2 over $\mathbb{F}_q[T]$. We prove that there are no points with coordinates being Drinfeld singular moduli, on a family of hyperbolas $XY=γ$, where $γ$ is a polynomial of small degree. This is an effective André-Oort theorem for these curves. We also prove that there are at most finitely many Drinfeld singular moduli that are algebraic units, for every fixed $q\geq2$, and we give an effective bound on the discriminant of such singular moduli. We give in an appendix an inseparability criterion for values of some classical modular forms, generalising an argument used in the proof of our first result.

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arXiv2020

Sturm-type bounds for modular forms over functions fields

In this paper, we obtain two analogues of the Sturm bound for modular forms in the function field setting. In the case of mixed characteristic, we prove that any harmonic cochain is uniquely determined by an explicit finite number of its first Fourier coefficients where our bound is much smaller than the ones in the literature. A similar bound is derived for generators of the Hecke algebra on harmonic cochains. As an application, we present a computational criterion for checking whether two elliptic curves over the rational function field $\mathbb{F}_q(θ)$ with same conductor are isogenous. In the case of equal characteristic, we also prove that any Drinfeld modular form is uniquely determined by an explicit finite number of its first coefficients in the $t$-expansion.

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arXiv2016

Large Galois images for Jacobian varieties of genus 3 curves

Given a prime number l greater than or equal to 5, we construct an infinite family of three-dimensional abelian varieties over Q such that, for any A/Q in the family, the Galois representation ρ_{A, l}: Gal_Q -> GSp(6, l) attached to the l-torsion of A is surjective. Any such variety A will be the Jacobian of a genus 3 curve over Q whose respective reductions at two auxiliary primes we prescribe to provide us with generators of Sp(6, l).

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arXiv2014

Galois representations and Galois groups over Q

In this paper we generalize results of P. Le Duff to genus n hyperelliptic curves. More precisely, let C/Q be a hyperelliptic genus n curve and let J(C) be the associated Jacobian variety. Assume that there exists a prime p such that J(C) has semistable reduction with toric dimension 1 at p. We provide an algorithm to compute a list of primes l (if they exist) such that the Galois representation attached to the l-torsion of J(C) is surjective onto the group GSp(2n, l). In particular we realize GSp(6, l) as a Galois group over Q for all primes l in [11, 500000].

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arXiv2014

Une base explicite de symboles modulaires sur les corps de fonctions

Modular symbols for the congruence subgroup $Γ_0(\mathfrak{n})$ of $GL_{2}(\mathbf{F}_q[T])$ have been defined by Teitelbaum. They have a presentation given by a finite number of generators and relations, in a formalism similar to Manin's for classical modular symbols. We completely solve the relations and get an explicit basis of generators when $\mathfrak{n}$ is a prime ideal of odd degree. As an application, we give a non-vanishing statement for $L$-functions of certain automorphic cusp forms for $\mathbf{F}_q(T)$. The main statement also provides a key-step for a result towards the uniform boundedness conjecture for Drinfeld modules of rank $2$.

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