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

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.

💡 Research Summary

The paper presents a practical algorithm for computing the p‑divisible group of the Jacobian of a smooth projective curve X over a finite field F_q of characteristic p, by explicitly determining its Dieudonné module. The authors start by recalling that the first crystalline cohomology H^1_crys(X) is a free Z_q‑module of rank 2g equipped with Frobenius (F) and Verschiebung (V) operators satisfying FV = VF = p. This module is canonically isomorphic to the Dieudonné module of the p‑divisible group Jac(X)