It can be found, at Hartshorne exercise 4.2.7 for example, that in the case where $\operatorname{char}(k) \neq 2$ we have a nice correspondence between etale degree 2 covers of a curve $C$ and 2-torsion in $JC$. This correspondence, as constructed in Hartshorne, breaks down in the case where $\operatorname{char}(k) = 2$.
However it has been suggested in Pries and Stevenson – A survey of Galois theory of curves in characteristic $p$, at the bottom of section 4.7, that "remarks following proposition 4.13 in chapter 3 of Milne's Étale cohomology" show that some correspondence between $JC[2](k)$ and etale degree 2 covers of $C$ still exists. Does anyone know how to construct such a correspondence explicitly? I believe it is erroneous (there is no proposition 4.13, it is a lemma!).
Otherwise, do étale degree 2 covers of $C$ correspond to anything else? Artin–Schreier theory suggests that it should be $H^1(C,\mathcal{O}_C)^F$. However from such a nontrivial cover $\pi:X \to C$ we also obtain a nontrivial extension of $\mathcal{O}_C$-algebras
$$0 \to \mathcal{O}_C \to \pi_*\mathcal{O}_X \to \mathcal{O}_C \to 0$$
by looking at affines of $C$ and writing explicit Artin–Schreier extensions, and observing that the cocycles on $\pi_*\mathcal{O}_X$ are unipotent $2\times2$ matrices. Is there any way to relate this to $JC$?
Best Answer
There is a duality between degree 2 coverings and two-torsion points on the Jacobian — i.e. both form elementary abelian 2-groups, and these groups are naturally dual.
This is the Artin–Milne Poincaré duality pairing in fppf cohomology (Corollary 4.9 of Duality in the Flat Cohomology of Curves)
$$ H^1(C_{\overline{k}}, \mathbb Z/2) \times H^1 (C_{\overline{k}},\mu_2) \to H^2(C_{\overline{k}}, \mu_2) \to \mathbb Z/2 $$ with $H^1(C_{\overline{k}}, \mathbb Z/2)$ classifying degree 2 coverings and $H^1 (C_{\overline{k}},\mu_2)$ classifying 2-torsion points of $J$.
Concretely, we can express this by noting that each covering of $C$ comes from a unique covering $A \to J$ of the same degree, and setting the pairing of this covering with a point in $J[2]$ to be zero if and only if that point lies in the image of $A[2]$.
More generally, for $C$ in characteristic $p$ and a $\mathbb Z/p$-covering of $C$ (and thus an isogeny $0 \to \mathbb Z/p \to A \to J\to 0$), we define the pairing with a $p$-torsion point by taking any lift of that point, multiplying by $p$, and obtaining an element of the kernel $\mathbb Z/p$. This doesn't depend on our choice of preimage because two lifts differ by an element of $\mathbb Z/p$ which is killed by multiplication by $p$.
This is linear in the $A[p]$ variable since every step is compatible with multiplication by $p$. This is linear in the covering variable since the addition map on coverings is fiber product followed by adding the kernels, and the product of two lifts is a lift to the fiber product.
To check this pairing has no kernel in the covering variable, the main point is to check that $A[p]$ and $J[p]$ have the same number of elements (since the rank is the multiplicity of the slope 0 of the Newton polygon, which is an isogeny invariant). Since $A \to J$ is a degree $p$ finite étale covering, its kernel contains a single $p$-torsion point, so the image of $A[p] \to J[p]$ has codimension $1$, forcing the pairing to be nontrivial.
To check it has no kernel in the other variable, factor the multiplication by $p$ map $A \to A$ (which kills all the $p$-torsion) into a totally inseparable morphism (which preserves points and thus $p$-torsion and an étale morphism (which therefore must kill all the $p$-torsion). The étale morphism is a fiber product of $\mathbb Z/p$-coverings, so if a point was in the image of $p$-torsion on each $\mathbb Z/p$-covering it would have to be in the image of $p$-torsion on this covering, forcing it to be trivial.
You're correct on your interpretation of Artin–Schreier theory. (I think you meant unipotent instead of unitary.) I don't know how to connect that perspective to the duality with the Jacobian — in particular, I don't think the Poincaré duality pairing above plays nicely with the Artin–Schreier map $\mathbb Z/2 \to \mathcal O_C$.