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Computing Crystalline Cohomology and p-Divisible Groups for Curves over Finite Fields

Jeremy Booher

TL;DR

The paper presents a practical, polynomial-time scheme to compute the $p$-divisible group $ ext{Jac}(X)[p^ abla]$ of a curve over a finite field by obtaining the Dieudonné module from $H_{ ext{crys}}^1(X)$ via Frobenius $F$ and Verschiebung $V$, realized through a lift to $f Z_q$ and Tuitman’s $p$-adic point counting. Central to the method is identifying crystalline cohomology with the de Rham cohomology of a lift and embedding it as a lattice inside rigid cohomology, enabling explicit computation of $F$ and $V$ with controlled $p$-adic precision. The work introduces an idealized algorithm for computing $H_{ ext{dR}}^1( ilde X/f Z_q)$ using enhanced differentials of the second kind, and it provides a detailed precision analysis and implementation notes, including a fix for precision issues arising from lattice mismatches in existing Magma implementations. The results are supported by a refined complexity analysis showing polynomial-time behavior under reasonable lift and ramification assumptions, and the authors provide practical guidance and examples demonstrating feasibility on concrete curves.

Abstract

Let $X$ be a smooth projective curve over a finite field of characteristic $p$. We describe and implement a practical algorithm for computing the $p$-divisible group $Jac(X)[p^\infty]$ via computing its Dieudonné module, or equivalently computing the Frobenius and Verschiebung operators on the first crystalline cohomology of $X$. We build on Tuitman's $p$-adic point counting algorithm, which computes the rigid cohomology of $X$ and requires a ``nice'' lift of $X$ to be provided.

Computing Crystalline Cohomology and p-Divisible Groups for Curves over Finite Fields

TL;DR

The paper presents a practical, polynomial-time scheme to compute the -divisible group of a curve over a finite field by obtaining the Dieudonné module from via Frobenius and Verschiebung , realized through a lift to and Tuitman’s -adic point counting. Central to the method is identifying crystalline cohomology with the de Rham cohomology of a lift and embedding it as a lattice inside rigid cohomology, enabling explicit computation of and with controlled -adic precision. The work introduces an idealized algorithm for computing using enhanced differentials of the second kind, and it provides a detailed precision analysis and implementation notes, including a fix for precision issues arising from lattice mismatches in existing Magma implementations. The results are supported by a refined complexity analysis showing polynomial-time behavior under reasonable lift and ramification assumptions, and the authors provide practical guidance and examples demonstrating feasibility on concrete curves.

Abstract

Let be a smooth projective curve over a finite field of characteristic . We describe and implement a practical algorithm for computing the -divisible group via computing its Dieudonné module, or equivalently computing the Frobenius and Verschiebung operators on the first crystalline cohomology of . We build on Tuitman's -adic point counting algorithm, which computes the rigid cohomology of and requires a ``nice'' lift of to be provided.
Paper Structure (24 sections, 10 theorems, 30 equations, 1 figure, 1 table, 2 algorithms)

This paper contains 24 sections, 10 theorems, 30 equations, 1 figure, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. The Situation
  4. Theoretical Setup
  5. Practical Setup
  6. Cohomology Theories
  7. Rigid Cohomology
  8. Algebraic de Rham Cohomology
  9. Crystalline Cohomology
  10. Computing the de Rham Cohomology of the Lift
  11. More on Enhanced Differentials of the Second Kind
  12. Computing with Enhanced Differentials of the Second Kind
  13. Implementation and Assumptions
  14. Analyzing p-adic Precision
  15. Some Lattices
  16. ...and 9 more sections

Key Result

Theorem 4.2

Let $S$ be the spectrum of a field or a DVR. Then $\mathop{\mathrm{\mathrm{H}}}\nolimits_{\textrm{dR}}^1(C/S)$ is isomorphic to the first homology of the total complex of global sections of the pole-order resolution in Figure fig:acyclic resolution.

Figures (1)

  • Figure 1: The Pole-Order Resolution $\mathcal{D}(n)$ of the de Rham Complex.

Theorems & Definitions (52)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Example 2.3: c.f. BT20
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 3.1
  • Definition 3.2
  • Remark 3.4
  • ...and 42 more