In computing the etale cohomology of curves, one of the key facts one needs is the torsion in $Pic(X) = H^1(X\_{et}, \mathcal{O}\_X^\*)$ for a smooth projective curve $X$. Namely, one shows that the $n$-torsion (for $n$ prime to the characteristic, at least) is given by $(\mathbb{Z}/n\mathbb{Z})^{2g}$ where $g$ is the genus by invoking the theory of the Jacobian. However, I don't really know anything about the Jacobian (and pretty much all that I do know is in characteristic zero, in which case all you need is Abel's theorem for the aforementioned fact), so I am curious: can this fact, describing the torsion in $Pic(X)$, be proved directly in an elementary fashion?



An elementary description of the torsion in $Pic(X)$ for a smooth projective curve $X$
As far as I know, the standard computation of the torsion in $Pic(X$) proceeds as follows:
one considers the exact sequence $0 \to Pic^0(X) \to Pic(X) \to \mathbb Z \to 0,$
the map to $\mathbb Z$ being the degree map. This shows that the torsion subgroups
of $Pic(X)$ and $Pic^0(X)$ coincide. One then shows that $Pic^0(X)$ is
an abelian variety of dimension $g$ (this is the algebraic theory of the Jacobian), 
and one applies the general theory of abelian varieties to compute its torsion.


An (apparently) alternative approach is as follows: one uses an $n$-torsion line bundle on $X$ to construct a degree $n$ cyclic cover of $X$, and thus classifies the $n$-torsion
of the Jacobian in terms of the maps from the etale $\pi\_1(X)$ to $\mathbb Z/n\mathbb Z$.
But I don't know how (in positive characteristics, where one can't resort to analytic
arguments) to concretely compute the group of such maps without going back to the
computation in terms of $Pic^0(X)$ and the theory of abelian varieties.


