manuscripta math. 78, 369- 380 (1993) manuscripta mathematica Spfinger-~rlag 1993 PERIOD MATRICES OF HYPERELLIPTIC CURVES Bernhard Schindler 1 Introduction To fix notation let me first recall the definition of a period matrix of a hyperelliptic curve. Let C be a hyperelliptic curve over C of genus g > 1. Then H~ flv) ~ C g and Hi(C, Z) ~ 7rl(C, p0) ~b TM Z 2g and we get an embedding H,(C,Z) ~ n~ * 3' ~ f,y" such that the image is a lattice in H~ ~c)'. The quotient Jac(C) = H~ ~c)*/H~(C, Z) is a complex toms, allowing a principal polarisation defined by a hermitian form on H~ ~c) /, ('1,'2) ~ ]'1 A&2 or equivalently by a symplectic form on Hi(C, Z) (71,72) ~ 71 "72, the intersection product. The importance of Jac(C) is underlined by the following theorem.