arXiv subjects

Antoine Leudière

Publications and source records attributed to Antoine Leudière.

6 recordsLinked to original sources

arXiv2026

Computing submodules of points of general Drinfeld modules over finite fields

We present an algorithm for computing the structure of any submodule of the module of points of a Drinfeld $A$-module over a finite field, where $A$ is a function ring over $\mathbb F_q$. When the function ring is $A = \mathbb F_q[T]$, we additionally compute a Frobenius decomposition of said submodule. Our algorithms apply in particular to kernels of isogenies and torsion submodules. They are presented within the frameworks of Frobenius normal forms, presentations of modules, and Fitting ideals. They rely largely on efficient and classical linear algebra methods, combined with fast arithmetic of Ore polynomials. We analyze the complexity of our algorithms, explore optimizations, and provide an implementation in SageMath. Finally, we compute a simple invariant attached to a Drinfeld $\mathbb F_q[T]$-module that encodes all the polynomials in $\mathbb F_q[T]$ whose associated torsion is rational.

math.NT↗

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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arXiv2025

The exceptional set in Cassel's theorem on small cyclotomic integers

In a 1965 paper, R. Robinson made five conjectures about the classification of cyclotomic algebraic integers for which the maximum absolute value in any complex embedding (the house) is small, modulo the equivalence relation generated by Galois conjugation and multiplication by roots of unity. In response to one of these conjectures, Cassels showed in 1969 that when the house is at most $\sqrt{5}$, one obtains three parametric families plus an effectively computable finite set of equivalence classes of exceptions. Building on the work of Jones, Calegari-Morrison-Snyder, and Robinson-Wurtz, we determine this exceptional set. By specializing to the case where the house is strictly less than 2, we resolve the final outstanding conjecture from Robinson's 1965 paper.

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arXiv2024

Algorithms for computing norms and characteristic polynomials on general Drinfeld modules

We provide two families of algorithms to compute characteristic polynomials of endomorphisms and norms of isogenies of Drinfeld modules. Our algorithms work for Drinfeld modules of any rank, defined over any base curve. When the base curve is $\mathbb P^1_{\mathbb F_q}$, we do a thorough study of the complexity, demonstrating that our algorithms are, in many cases, the most asymptotically performant. The first family of algorithms relies on the correspondence between Drinfeld modules and Anderson motives, reducing the computation to linear algebra over a polynomial ring. The second family, available only for the Frobenius endomorphism, is based on a formula expressing the characteristic polynomial of the Frobenius as a reduced norm in a central simple algebra.

cs.SC↗

arXiv2024

Computing a Group Action from the Class Field Theory of Imaginary Hyperelliptic Function Fields

We explore algorithmic aspects of a simply transitive commutative group action coming from the class field theory of imaginary hyperelliptic function fields. Namely, the Jacobian of an imaginary hyperelliptic curve defined over $\mathbb F_q$ acts on a subset of isomorphism classes of Drinfeld modules. We describe an algorithm to compute the group action efficiently. This is a function field analog of the Couveignes-Rostovtsev-Stolbunov group action. We report on an explicit computation done with our proof-of-concept C++/NTL implementation; it took a fraction of a second on a standard computer. We prove that the problem of inverting the group action reduces to the problem of finding isogenies of fixed $τ$-degree between Drinfeld $\mathbb F_q[X]$-modules, which is solvable in polynomial time thanks to an algorithm by Wesolowski. We give asymptotic complexity bounds for all algorithms presented in this paper.

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arXiv2023

Drinfeld modules in SageMath

We present the first implementation of Drinfeld modules fully integrated in the SageMath ecosystem. First features will be released with SageMath 10.0.

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