Archiv der Mathematik

, Volume 107, Issue 5, pp 499–509 | Cite as

The Riemann constant for a non-symmetric Weierstrass semigroup

  • Jiryo Komeda
  • Shigeki Matsutani
  • Emma Previato


The zero divisor of the theta function of a compact Riemann surface X of genus g is the canonical theta divisor of Pic\({^{(g-1)}}\) up to translation by the Riemann constant \({\Delta}\) for a base point P of X. The complement of the Weierstrass gaps at the base point P gives a numerical semigroup, called the Weierstrass semigroup. It is classically known that the Riemann constant \({\Delta}\) is a half period, namely an element of \({\frac{1}{2} \Gamma_\tau}\) , for the Jacobi variety \({\mathcal{J}(X)=\mathbb{C}^{g}/\Gamma_\tau}\) of X if and only if the Weierstrass semigroup at P is symmetric. In this article, we analyze the non-symmetric case. Using a semi-canonical divisor D 0, we express the relation between the Riemann constant \({\Delta}\) and a half period in the non-symmetric case. We point out an application to an algebraic expression for the Jacobi inversion problem. We also identify the semi-canonical divisor D 0 for trigonal pointed curves, namely with total ramification at P.


Riemann constant Non-symmetric Weierstrass semigroup Theta function Abel map Sigma function 

Mathematics Subject Classification

Primary 14H55 Secondary 14H50 14K25 14H40 


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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Jiryo Komeda
    • 1
  • Shigeki Matsutani
    • 2
  • Emma Previato
    • 3
  1. 1.Department of Mathematics, Center for Basic Education and Integrated LearningKanagawa Institute of TechnologyAtsugiJapan
  2. 2.Industrial MathematicsNational Institute of Technology, Sasebo CollegeSaseboJapan
  3. 3.Department of Mathematics and StatisticsBoston UniversityBostonU.S.A.

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