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

Antoine Leudière, Pierre-Jean Spaenlehauer

TL;DR

The problem of inverting the group action reduces to the problem of finding isogenies of fixed $\tau$-degree between Drinfeld $\mathbb F_q[X]$-modules, which is solvable in polynomial time thanks to an algorithm by Wesolowski.

Abstract

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.

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

TL;DR

The problem of inverting the group action reduces to the problem of finding isogenies of fixed -degree between Drinfeld -modules, which is solvable in polynomial time thanks to an algorithm by Wesolowski.

Abstract

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 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 -degree between Drinfeld -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.
Paper Structure (17 sections, 21 theorems, 23 equations, 5 algorithms)

This paper contains 17 sections, 21 theorems, 23 equations, 5 algorithms.

Table of Contents

  1. Context
  2. Main results
  3. Related works and applications
  4. Organization of the paper
  5. Acknowledgements
  6. Drinfeld modules
  7. Ore polynomials
  8. General Drinfeld modules
  9. Finite Drinfeld modules
  10. Class field theory of imaginary hyperelliptic function fields
  11. Complex multiplication for rank-two finite Drinfeld modules
  12. Rank-one Drinfeld modules on imaginary hyperelliptic curves
  13. A group action from class field theory
  14. Algorithms
  15. Computation of the group action
  16. ...and 2 more sections

Key Result

Proposition 2.2

Assume $[L:\mathbb{F}_q]$ is odd, let $\phi\in\mathop{\mathrm{Dr}}\nolimits_2(\mathbb{F}_q[X], L)$ be an ordinary rank-2 Drinfeld module, and assume that $\xi$ defines an imaginary hyperelliptic curve $\mathcal{H}$. Then $\mathop{\mathrm{End}}\nolimits_{{\overline{L}}}(\phi) = \mathop{\mathrm{End}}\

Theorems & Definitions (45)

  • Definition 1.1: gos98
  • Definition 1.2: gos98, gek91
  • Definition 2.1: gekeler1983
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Lemma 2.5
  • proof
  • ...and 35 more