Explicit computation of the first étale cohomology on curves

2017/07/27 by Jin, Jinbi
#14F20 #14Q05 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1707.08825

Abstract

In this paper, we describe an algorithm that, for a smooth connected curve X over a field k with normal completion having arithmetic genus p_a(X), a finite locally constant sheaf \mathcal A on X_et of abelian groups of torsion invertible in k, represented by a smooth curve with normal completion having arithmetic genus p_a(\mathcal A) and degree n over X, computes the first étale cohomology H¹(X_k^sep,et,\mathcal A) and the first étale cohomology with proper support H¹_c(X_k^sep,et,\mathcal A) as sets of torsors, in arithmetic complexity exponential in n^{log n}, p_a(X), and p_a(\mathcal A). This is done via the computation of a groupoid scheme classifying the relevant torsors (with extra rigidifying data).

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