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Equilibrium point of Green's function for the annulus and Eisenstein series


Authors: Ahmed Sebbar and Thérèse Falliero
Journal: Proc. Amer. Math. Soc. 135 (2007), 313-328
MSC (2000): Primary 11F03, 11F11, 30C40, 34B30
DOI: https://doi.org/10.1090/S0002-9939-06-08353-5
Published electronically: September 11, 2006
MathSciNet review: 2255277
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Abstract: We study the motion of the equilibrium point of Green's function and give an explicit parametrization of the unique zero of the Bergman kernel of the annulus. This problem is reduced to solving the equation $ \wp(z,\tau)= -\frac{\pi^2}{3}E_2(\tau)$, where $ E_2(\tau)$ is the usual Eisenstein series.


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Additional Information

Ahmed Sebbar
Affiliation: LABAG, Laboratoire Bordelais d’Analyse et Géométrie, Institut de Mathématiques, Université Bordeaux I, 33405 Talence, France
Email: sebbar@math.u-bordeaux.fr

Thérèse Falliero
Affiliation: Faculté des Sciences, Université d’Avignon, 84000 Avignon, France
Email: Therese.Falliero@univ-avignon.fr

DOI: https://doi.org/10.1090/S0002-9939-06-08353-5
Keywords: Equilibrium points, Eisenstein series, Bergman kernel
Received by editor(s): January 25, 2005
Received by editor(s) in revised form: June 1, 2005
Published electronically: September 11, 2006
Additional Notes: We are grateful to Henri Cohen and Don Zagier for teaching us some facts about the zeros of the Eisenstein series $E_{2}$.
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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