The Converse to Abel’s Theorem

  • Jorge Vitório Pereira
  • Luc Pirio
Part of the IMPA Monographs book series (IMPA, volume 2)


The main result of this chapter is a converse to Abel’s addition Theorem stated in Sect. 4.1. It ensures the algebraicity of local datum satisfying the hypotheses of Abel’s addition Theorem. Its first version was established by Sophus Lie in the context of double-translation surfaces. Lie’s arguments consisted in a tour-de-force analysis of an overdetermined system of PDEs. Later Poincaré introduced a geometrical method to handle the problem solved analytically by Lie. Poincaré’s approach was later revisited, and made more precise by Darboux, to whom the approach presented in Sect. 4.2 can be traced back. By the way, those willing to take for granted the validity of the converse of Abel’s Theorem can safely skip Sect. 4.2.


Abelian Variety Maximal Rank Projective Curve Translation Surface Distinct Parametrization 
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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Jorge Vitório Pereira
    • 1
  • Luc Pirio
    • 2
  1. 1.Instituto de Matemática Pura e AplicadaRio de JaneiroBrazil
  2. 2.Institut de Recherches Mathématiques de Rennes IRMAR, UMR 6625 du CNRSUniversité Rennes 1RennesFrance

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