 This comes about because the solutions of the equation on the divisor class level will form a single coset of the solutions of

In other words, with "K" the canonical class and Θ any given solution of

any other solution will be of form

This reduces counting the theta characteristics to finding the 2-rank of the Jacobian variety "J"("C") of "C".In the complex case, again, the result follows since "J"("C") is a complex torus of dimension 2"g".Over a general field, see the theory explained at Hasse-Witt matrix for the counting of the p-rank of an abelian variety.