We have seen in Chapters III and IV how to construct meromorphic functions on Riemann surfaces. In this chapter, we construct holomorphic functions on the Jacobian variety of a compact surface, and via the embedding of the Riemann surface into its Jacobian variety, multivalued holomorphic functions on the surface. The high point of our present development is the Riemann vanishing theorem (Theorem VI.3.5). Along the way, we will reprove the Jacobi inversion theorem.
KeywordsRiemann Surface Theta Function Compact Riemann Surface Jacobian Variety Theta Divisor
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