Theta Characteristics and Their Moduli

Theta characteristics appeared for the first time in the context of characteristic theory of odd and even theta functions in the papers of Göpel [Go] and Rosenhain [Ro] on Jacobi’s inversion formula for genus 2. They were initially considered in connection with Riemann’s bilinear addition relation between the degree two monomials in theta functions with characteristics. Later, in order to systematize the relations between theta constants, Frobenius [Fr1], [Fr2] developed an algebra of characteristics ; he distinguished between period and theta characteristics. The distinction, which in modern terms amounts to the difference between the Prym moduli space Rg and the spin moduli space Sg, played a crucial role in elucidating the transformation law for theta functions under a linear transformation of the moduli, and it ultimately led to a correct definition of the action of symplectic group Sp(F 2 ) on the set of characteristics. An overwiew of the 19th century theory of theta functions can be found in Krazer’s monumental book [Kr]. It is a very analytic treatise in character, with most geometric applications relegated to footnotes. The remarkable book [Cob] by Coble 2 represents a departure from the analytic view towards a more abstract understanding of theta characteristics using configurations in finite geometry. Coble viewed theta characteristics as quadrics in a vector space over F2. In this language Frobenius’ earlier concepts (syzygetic and azygetic triples, fundamental systems of characteristics) have an elegant translation. With fashions in algebraic geometry drastically changing, the work of Coble was forgotten for many decades .