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A computational approach to Drinfeld modules

Cécile Armana, Elena Berardini, Xavier Caruso, Antoine Leudière, Jade Nardi, Fabien Pazuki

TL;DR

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.

Abstract

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.

A computational approach to Drinfeld modules

TL;DR

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.

Abstract

This survey provides a practical and algorithmic perspective on Drinfeld modules over . 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.
Paper Structure (66 sections, 37 theorems, 151 equations, 2 figures, 1 table)

This paper contains 66 sections, 37 theorems, 151 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Purpose and organisation of the survey
  3. Setting and notation
  4. Euler-Poincaré characteristic.
  5. Discovering the Carlitz module
  6. Looking for an exponential function
  7. Analytic construction of the Carlitz module
  8. Finding a formula for the function e
  9. Setting a new structure of e-module on C_∞
  10. Normalising the lattice Lambda
  11. Application to the cyclotomic theory
  12. The Carlitz zeta function
  13. Drinfeld modules and their Anderson motives
  14. Ore polynomials
  15. Kernels of Ore polynomials
  16. ...and 51 more sections

Key Result

Theorem 2.2

For all $a \in A$, the extension $K(\phi[a])/K$ is Galois and its Galois group is canonically isomorphic to $(A/aA)^\times$.

Figures (2)

  • Figure 1: An analogy to keep in mind when starting to work with Drinfeld modules.
  • Figure 2: Isogenies of small degree and height of invariants for elliptic curves and Drinfeld modules of rank $2$.

Theorems & Definitions (104)

  • Remark 1.1
  • Remark 2.1
  • Theorem 2.2: Carlitz, see Rosen_2002
  • proof : Sketch of the proof
  • Definition 3.1: Ore polynomials
  • Proposition 3.2
  • Proposition 3.3
  • Remark 3.4
  • Remark 3.5
  • Definition 3.6
  • ...and 94 more