For that case, the canonical class is trivial (zero in the divisor class group) and so the theta characteristics of an elliptic curve E over the complex numbers are seen to be in 1-1 correspondence with the four points P on E with 2P = 0; this is counting of the solutions is clear from the group structure, a product of two circle groups, when E is treated as a complex torus.
In case of higher genus g, assuming the field over which C is defined does not have characteristic 2, the theta characteristics can be counted as in number if the base field is algebraically closed.
Over a general field, see the theory explained at Hasse-Witt matrix for the counting of the p-rank of an abelian variety.
A theta characteristic Θ will be called even or odd depending on the dimension of its space of global sections
The geometric construction of Q as an intersection form is with modern tools possible algebraically.
In fact the Weil pairing applies, in its abelian variety form.
Triples (θ1, θ2, θ3) of theta characteristics are called syzygetic and asyzygetic depending on whether Arf(θ1)+Arf(θ2)+Arf(θ3)+Arf(θ1+θ2+θ3) is 0 or 1.
Atiyah (1971) showed that, for a compact complex manifold, choices of theta characteristics correspond bijectively to spin structures.