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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On homeomorphisms preserving principal divisors


Author: Yosef Stein
Journal: Proc. Amer. Math. Soc. 83 (1981), 557-562
MSC: Primary 14H15; Secondary 30F20
DOI: https://doi.org/10.1090/S0002-9939-1981-0627691-2
MathSciNet review: 627691
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Abstract: Let $ {S_1}$ and $ {S_2}$ be compact Riemann surfaces of genus $ g > 1$. Let $ \tau :{S_1} \to {S_2}$ be a continuous map. The map $ \tau $ induces a group homomorphism from the group of divisors on $ {S_1}$ into the group of divisors on $ {S_2}$. This group homomorphism will be denoted by the same letter $ \tau $ throughout this paper. If $ D = \sum _{i = 1}^n{m_i}{p_i}$ is a divisor on $ {S_1}$, then $ \tau (D) = \sum _{i = 1}^n{m_i}\tau ({p_i})$. If $ \tau $ is a holomorphic or an anti-holomorphic homeomorphism, then $ \tau (D)$ is a principal divisor on $ {S_2}$ if $ D$ is a principal divisor on $ {S_1}$. To what extent is the converse of this statement true?

The answer to this question is provided by Theorem 1 of this paper: If $ \tau (D)$ is a principal divisor on $ {S_2}$ whenever $ D$ is a principal divisor on $ {S_1}$, then $ \tau $ is either a holomorphic or an anti-holomorphic homeomorphism.


References [Enhancements On Off] (What's this?)

  • [1] R. C. Gunning, Lectures on Riemann surfaces, Princeton Mathematical Notes, Princeton University Press, Princeton, N.J., 1966. MR 0207977
  • [2] R. C. Gunning, Lectures on Riemann surfaces, Jacobi varieties, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. Mathematical Notes, No. 12. MR 0357407

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DOI: https://doi.org/10.1090/S0002-9939-1981-0627691-2
Article copyright: © Copyright 1981 American Mathematical Society