Effective Artin-Schreier-Witt theory for curves

2025/09/12 by Levrat, Christophe, Muñoz--Bertrand, Rubén
#11G20 #11Y16 #14F20 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2509.10633

Abstract

We present an algorithm which, given a connected smooth projective curve X over an algebraically closed field of characteristic p>0 and its Hasse--Witt matrix, as well as a positive integer n, computes all étale Galois covers of X with group ℤ/pⁿℤ. We compute the complexity of this algorithm when X is defined over a finite field, and provide a complete implementation in SageMath, as well as some explicit examples. We then apply this algorithm to the computation of the cohomology complex of a locally constant sheaf of ℤ/pⁿℤ-modules on such a curve.

Citations

• Faster computation of Witt vectors over polynomial rings
• On the non-special divisors in algebraic function fields defined over \mathbbF_q
• Computing the cohomology of constructible étale sheaves on curves
• On Galois groups of linearized polynomials related to the general linear group of prime degree
• Counting points on curves using a map to P¹, II
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Related

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• Explicit computation of the first étale cohomology on curves
• Computing the cohomology of constructible étale sheaves on curves
• p-torsion étale sheaves on the Jacobian of a curve
• An Infinite Family of Artin-Schreier Curves with Minimal a-number