Abstract
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.
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© 1980 Springer Science+Business Media New York
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Farkas, H.M., Kra, I. (1980). Theta Functions. In: Riemann Surfaces. Graduate Texts in Mathematics, vol 71. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-9930-8_7
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DOI: https://doi.org/10.1007/978-1-4684-9930-8_7
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4684-9932-2
Online ISBN: 978-1-4684-9930-8
eBook Packages: Springer Book Archive