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

TL;DR

This paper presents a practical algorithm for computing the p-divisible group of a curve over a finite field by leveraging crystalline cohomology and building on Tuitman's p-adic point counting method.

Contribution

It introduces an efficient algorithm to compute the Dieudonné module of the p-divisible group of a curve using crystalline cohomology, extending existing p-adic point counting techniques.

Findings

Algorithm successfully computes the Frobenius and Verschiebung operators.

Implementation demonstrates practical computation for specific curves.

Extends Tuitman's p-adic point counting to p-divisible group analysis.

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 Dieudonn\'{e} 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.