Computing the cohomology of constructible étale sheaves on curves

1. Problem Statement

The paper addresses the computational challenge of determining the étale cohomology complex $R\Gamma(X, \mathcal{F})$ for a constructible sheaf $\mathcal{F}$ of $\mathbb{Z}/n\mathbb{Z}$-modules on an irreducible curve $X$ over a field $k_0$.

• Context: While the computability of these cohomology groups was established theoretically (by Poonen, Testa, van Luijk, Madore, and Orgogozo), existing algorithms are generally inefficient (primitive recursive) and not suitable for practical implementation.
• Specific Goal: The author aims to provide an explicit, functorial expression for the cohomology complex that involves only finite groups. Crucially, the method must compute the Galois action of $\text{Gal}(\bar{k}_0/k_0)$ on this complex, which is essential for applications like point counting on higher-dimensional varieties.
• Constraints: The integer $n$ must be invertible in the base field $k_0$ (i.e., the characteristic of $k_0$ does not divide $n$).

2. Methodology

The approach combines algebraic geometry, Galois theory, and algorithmic complexity analysis. The core strategy involves reducing the computation of the cohomology of a constructible sheaf on a curve to the cohomology of a finite Galois group acting on a module, plus local contributions from singularities and points at infinity.

A. The Trivializing Cover

The method relies on constructing a specific Galois cover $V \to U$ (where $U$ is a smooth open affine part of the curve) that "trivializes" the sheaf and kills the first cohomology group $H^1$.

• Construction: For a scheme $X$, the author defines a canonical cover $X\langle n \rangle \to X$ corresponding to the subgroup $S \subset \pi_1(X)$ which is the closure of $\pi_1(X)^n [\pi_1(X), \pi_1(X)]$.
• Explicit Realization: For curves, this cover is constructed by taking $n$-th roots of rational functions $f_i$ associated with a basis of $H^1(U, \mu_n)$. Specifically, if $H^1(U, \mu_n)$ is generated by pairs $(D_i, f_i)$ where $nD_i = \text{div}(f_i)$, the cover is defined by the field extension $K(U)(\sqrt[n]{f_1}, \dots, \sqrt[n]{f_r})$.
• Galois Action: The paper details how to construct a cover defined over the base field $k_0$ (or a finite extension) that respects the Galois action, ensuring the resulting complex is a module over the Galois group.

B. Explicit Cohomology Formula (The Main Theorem)

The paper proves that for a curve $X$ with at worst multicross singularities, the complex $R\Gamma(X, \mathcal{F})$ is represented by the cone of a specific morphism between two complexes.

1. Global Part: Computed via the Hochschild-Serre spectral sequence, reducing to the cohomology of the Galois group $G = \text{Aut}(V/U)$ acting on the generic fiber $M$. This involves crossed homomorphisms $\text{Hom}_{cr}(G, M)$.
2. Local Part: Contributions from the closed complement $Z = X \setminus U$ and the normalization $\tilde{X}$.
   • For points at infinity or singularities, the cohomology is expressed in terms of inertia groups $I_z$ and their wild inertia subgroups $P_z$.
   • The local terms involve $H^1(I_z/P_z, M^{P_z})$ and crossed homomorphisms relative to these inertia groups.
3. The Complex: The resulting complex is a finite chain complex of $\mathbb{Z}/n\mathbb{Z}$-modules (or $\Lambda[G_0]$-modules) with explicitly defined differentials involving restriction maps and adjunction units.

C. Algorithmic Implementation

The author translates the theoretical construction into a concrete algorithm (Algorithm 4: RGamma):

1. Input: A curve $X$, a constructible sheaf $\mathcal{F}$ (described by gluing data: a cover $V$, a module $M$, and local fibers).
2. Step 1: Compute the basis of $H^1(V, \mu_n)$ using existing algorithms (e.g., Huang-Ierardi or Couveignes).
3. Step 2: Construct the cover $V\langle n \rangle$ and compute its automorphism group $G$.
4. Step 3: Compute inertia subgroups $I_z$ for points at infinity/singularities.
5. Step 4: Construct the global complex using crossed homomorphisms and the local cone construction.
6. Step 5: Compute the Galois action by tracking the action on the generators of the cover and the module.

3. Key Contributions

• Explicit Functorial Expression: The paper provides the first explicit, finite-group-based formula for the entire cohomology complex $R\Gamma(X, \mathcal{F})$ (not just individual cohomology groups) for constructible sheaves on singular curves.
• Galois Action Computation: It details a method to compute the action of the absolute Galois group on the cohomology complex, a critical requirement for arithmetic applications.
• Algorithmic Complexity:
  • The proposed algorithm is polynomial time in the degree of the curve model, the genus, the number of points at infinity, and the size of the sheaf, provided the base field is finite.
  • It improves upon Jin's algorithm (which computes $H^1$ for locally constant sheaves) by offering a lower complexity bound for locally constant sheaves on smooth projective curves over finite fields.
  • The complexity is dominated by the computation of $H^1(X, \mu_n)$, for which efficient algorithms already exist.
• Handling Singularities: The method naturally handles curves with multicross singularities by utilizing the normalization and strict henselization, avoiding the need for desingularization which can be computationally expensive.

4. Results and Examples

The paper validates the theory through two detailed examples:

1. Sheaves on Subsets of $\mathbb{P}^1$:
   • The author computes the cohomology of a sheaf on $\mathbb{P}^1$ with a ramified cover.
   • The example explicitly demonstrates the construction of the cover $V\langle n \rangle$, the calculation of inertia groups at ramification points, and the resulting cohomology groups ($H^0, H^1, H^2$) with their Galois actions.
2. Sheaves on Subsets of an Elliptic Curve:
   • A more complex case involving a degree 2 cover of an elliptic curve by a genus 2 curve over a finite field ($\mathbb{F}_{11}$).
   • The example illustrates the computation of $H^1(C, \mu_2)$ using Jacobian torsion points, the construction of the Galois group of the trivializing cover, and the explicit calculation of the cohomology complex using Magma.
   • It highlights the optimization of choosing a subcover defined over a smaller field to reduce complexity.

5. Significance and Future Applications

• Point Counting on Surfaces: The paper argues that this algorithm is a crucial step toward a polynomial-time point counting algorithm for surfaces over finite fields. By using the Lefschetz trace formula, counting points on a surface reduces to computing the trace of Frobenius on the cohomology of a sheaf on a curve (the generic fiber of a Lefschetz pencil). The proposed method makes this step feasible.
• Practical Computability: Unlike previous theoretical existence proofs, this work provides a framework that can be implemented in computer algebra systems (like Magma or SageMath) to compute étale cohomology for non-trivial sheaves and singular curves.
• Theoretical Unification: It unifies the treatment of smooth and singular curves, and locally constant and constructible sheaves, under a single explicit cohomological framework.

In summary, Levrat's paper bridges the gap between the theoretical computability of étale cohomology and practical algorithmic implementation, offering a robust tool for arithmetic geometry and number theory.