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Algorithms for computing norms and characteristic polynomials on general Drinfeld modules

Xavier Caruso, Antoine Leudière

TL;DR

Two families of algorithms to compute characteristic polynomials of endomorphisms and norms of isogenies of Drinfeld modules of any rank, defined over any base curve are provided, demonstrating that these algorithms are the most asymptotically performant.

Abstract

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.

Algorithms for computing norms and characteristic polynomials on general Drinfeld modules

TL;DR

Two families of algorithms to compute characteristic polynomials of endomorphisms and norms of isogenies of Drinfeld modules of any rank, defined over any base curve are provided, demonstrating that these algorithms are the most asymptotically performant.

Abstract

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 , 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.
Paper Structure (36 sections, 32 theorems, 95 equations, 1 figure, 7 algorithms)

This paper contains 36 sections, 32 theorems, 95 equations, 1 figure, 7 algorithms.

Table of Contents

  1. Context.
  2. Algorithmic results.
  3. Comparison with previous results.
  4. Anderson motives.
  5. The central simple algebra method.
  6. Acknowledgements.
  7. Background
  8. Drinfeld modules
  9. Drinfeld modules and isogenies
  10. Torsion points, Tate module, and Anderson motives
  11. Norms and characteristic polynomials
  12. Restriction of Drinfeld modules
  13. Algorithmics
  14. Complexity model
  15. Polynomial matrices
  16. ...and 21 more sections

Key Result

Theorem 1

Let $\phi$ be a Drinfeld ${\mathbb{F}_q}[T]$-module of rank $r$ over a field $K$, and let $u$ be an endomorphismWe refer to § ssec:backgrounddrinfeld for the definition of an endomorphism of a Drinfeld module and of its degree. of $\phi$ of degree $n$. The characteristic polynomial of $u$ can be com bit operations.

Figures (1)

  • Figure 1: The best algorithm for computing the characteristic polynomial of the Frobenius endomorphism, depending on the size of $r$, $d$ and $m$. Assumptions:$2 \leqslant \omega \leqslant 3$ and $\omega \leqslant \Omega \leqslant \omega + 1$.

Theorems & Definitions (82)

  • Theorem 1: see Theorems \ref{['theo:motive-charpoly-comp']} and \ref{['theo:charpoly-finitefield']}
  • Theorem 2
  • Theorem 3: see Theorems \ref{['theo:iso-norm']} and \ref{['theo:iso-norm-finite']}
  • Definition 1.1: Drinfeld modules
  • Example 1.2
  • Definition 1.3: Morphisms
  • Definition 1.4: $A$-module
  • Definition 1.5: Tate module
  • Remark 1.6
  • Definition 1.7: Anderson motive
  • ...and 72 more