Is there any good reference on Riemann Surface and Riemann Theta Function?

In summary, the person is looking for a reference on Integrable System and is struggling with concepts such as Riemann Surface, genus, divisors, and Riemann Theta Functions. They are hoping to find an introduction or pedagogical reference on the topic to read during their winter vacation. They also mention that they learned a lot from a specific chapter in a book and suggest a "college on riemann surfaces" that took place in 1987.
  • #1
yicong2011
75
0
Hi,

Currently, I need to read some reference about Integrable System, but I am stuck in Riemann Surface, genus, divisors, and Riemann Theta Functions. This makes me anxious.

Is there introduction or pedagogical reference on this topic? I think I can spend some time read it during winter vacation.

Thank you very much. Enjoy your Thanksgiving.
 
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  • #2
I learned a lot from chapter 2 of Algebraic geometry, by Griffiths and Harris.

There was also a "college on riemann surfaces" at the ICTP in 1987 that may be useful,

'Lectures on Riemann surfaces
proceedings of the College on Riemann Surfaces, International Centre for Theoretical Physics, Trieste, Italy, 9 Nov.-18 Dec., 1987
editors, M. Cornalba, X. Gomez-Mont, A. Verjovsky.
Published 1989 by World Scientific in Singapore, Teaneck, NJ .
Written in English.
 
  • #3
here is a quick sketch. the theta divisor mentioned on the last page is the zero locus of Riemann's theta function.
 

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1. What is a Riemann Surface?

A Riemann Surface is a type of complex manifold that allows for the extension of complex analysis to multivalued functions. It is a surface that is locally modeled on the complex plane and globally is a one-dimensional complex manifold.

2. What is the Riemann Theta Function?

The Riemann Theta Function is a multivariable function defined on the complex plane. It plays a crucial role in the study of Riemann Surfaces and has applications in number theory, algebraic geometry, and mathematical physics.

3. What are the applications of Riemann Surfaces and Riemann Theta Function?

Riemann Surfaces and Riemann Theta Function have applications in various areas of mathematics such as number theory, algebraic geometry, and mathematical physics. They are also used in the study of modular forms, elliptic curves, and abelian varieties.

4. Are there any prerequisites for studying Riemann Surfaces and Riemann Theta Function?

A basic understanding of complex analysis and algebraic topology is necessary for studying Riemann Surfaces and Riemann Theta Function. Familiarity with concepts such as holomorphic functions, manifolds, and sheaves is also helpful.

5. Can you recommend any good references for learning about Riemann Surfaces and Riemann Theta Function?

Some popular books on the subject include "Riemann Surfaces" by Simon Donaldson, "Riemann Surfaces and Algebraic Curves" by Rick Miranda, and "Theta Functions on Riemann Surfaces" by Serge Lang. There are also many online resources and lecture notes available for self-study.

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