The importance of this concept was realised first in the analytic theory oftheta functions, and geometrically in the theory of bitangents. In the analytic theory, there are four fundamental theta functions in the theory of Jacobian elliptic functions. Their labels are in effect the theta characteristics of an elliptic curve. For that case, the canonical class is trivial 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 pointsP 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.
Higher genus
For C of genus 0 there is one such divisor class, namely the class of -P, where P is any point on the curve. 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. 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 varietyJ of C. In the complex case, again, the result follows since J is a complex torus of dimension 2g. Over a general field, see the theory explained at Hasse-Witt matrix for the counting of the p-rank of an abelian variety. The answer is the same, provided the characteristic of the field is not 2. A theta characteristic Θ will be called even or odd depending on the dimension of its space of global sections. It turns out that on C there are even and odd theta characteristics.
Classical theory
Classically the theta characteristics were divided into these two kinds, odd and even, according to the value of the Arf invariant of a certain quadratic formQ with values mod 2. Thus in case of g = 3 and a plane quartic curve, there were 28 of one type, and the remaining 36 of the other; this is basic in the question of counting bitangents, as it corresponds to the 28 bitangents of a quartic. 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 of theta characteristics are called syzygetic and asyzygetic depending on whether Arf+Arf+Arf+Arf is 0 or 1.
Spin structures
showed that, for a compact complex manifold, choices of theta characteristics correspond bijectively to spin structures.