The classical point-counting algorithms by Schoof and Pila for curves compute the number

#X(F_q) of F_q-points on a fixed smooth projective curve X in time polynomial in log q.

A motivating question for this talk is whether such efficient algorithms exist for more general

varieties X. Via the Lefschetz trace formula, it suffices to be able to efficiently compute the

étale cohomology groups on X for suitable coefficients. However, the currently known

algorithms (Poonen-Testa-van Luijk, Madore-Orgogozo) have no known upper bounds on

their running time.

In this talk, as a first step towards a more efficient algorithm, we describe how to (algorithmically)

compute the étale cohomology on a smooth affine curve X with coefficients in a constructible

abelian étale sheaf A on X of torsion coprime to the characteristic of the base field, using the

interpretation of the elements of the first étale cohomology as A-torsors on X. Most of this talk

is based on part of my dissertation, which was supervised by Bas Edixhoven and Lenny Taelman.

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