ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

4. Algebraic Geometry

1991 MS Classification: :14, 30 Yoshida, Katsuaki, College of Science and Technology, Department of General Education, Nihon university, Tokyo, Japan Klein’s surface of genus three and associated theta constants Klein’s surface of genus three is the closed Riemann surface R defined by y^{7} = x(x − 1)^{2}. This surface is famous because the automorphism group Aut(R) has order 168, the maximum possible. We represent R as a 7-sheets covering surface over P^{1} and calculate the period matrix. The aim of our work is to decide the proportionalities of associated theta constants of R. First we show that 36 theta constants are distinguished in 6 orbits by the action of rhe representation of an automorphism of order 7. Next, using the theta identities, we determine the proportionalities of theta constants explicitely. The proportionalities mean the homogeneous coordinate of the projective image of the Jacobian variety associated to R by means of theta constants.