Effectiveness of the distinguished theta characteristic in characteristic 2 Let $k$ be an algebraically closed field of characteristic 2. Let $C$ be a (smooth projective connected) curve over $k$. Can there exist a rational function on $C$ whose differential is holomorphic but nonzero? (Note that every perfect square has zero differential.)
Alternate formulation: at the end of his paper "Theta characteristics of an algebraic curve", Mumford points out that the canonical sheaf on $C$ has a distinguished square root $\mathcal{L}$: for any rational function $f$ on $C$ which is not a perfect square, the divisor $(df)$ is even and the class of $\frac{1}{2}(df)$ does not depend on $f$. The question is then whether $\mathcal{L}$ can admit a nonzero section.
For example, it is an entertaining exercise to check (from a Weierstrass model) that this can never occur for ordinary elliptic curves.
 A: Let $F:X\to X$ be the relative Frobenius morphism. It is a finite morphism of degree $2$, and we have an exact sequence 
$$0\to \mathcal{O}_X\to F_*\mathcal{O}_X \to L \to 0$$
for some invertible sheaf $L$ on $X$. The standard formula for the canonical class of a finite morphism tells us that 
$\omega_X = F^*(\omega_X\otimes L^{-1}) = \omega_X^{\otimes 2}\otimes 
L^{\otimes -2}.$
This implies that $L = \theta$ is a theta characteristic on $X$. I believe this is the theta characteristic discussed by Mumford. Now apply the cohomology to the first exact sequence to get that $H^1(X,\theta) \cong H^0(X,\theta)^*$ is the cokernel of the Frobenius map $H^1(X,\mathcal{O}_X) \to H^1(X,\mathcal{O}_X)$.
So, $\theta$ is effective if and only if the curve is not ordinary.
A: $ $ Hi, Kiran, welcome to MO. 
Supersingular elliptic curves in characteristic two have exact holomorphic differentials. On $y^2+y=x^3+ax+b$, $dx$ is holomorphic. In general, there are such things if and only if the curve (or its Jacobian) is not ordinary. This is discussed in §3 of my paper with Stohr, "A formula for the Cartier operator on plane algebraic curves" Crelle 377 (1987), 49-64.
