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The Vanishing of the Theta Function in the KP Direction: A Geometric Approach

Published online by Cambridge University Press:  04 December 2007

Christina Birkenhake  and
Pol Vanhaecke
Christina Birkenhake
Affiliation:
Mathematisches Institut, Bismarkstrasse 1 1/2, D-91054 Erlangen. Germany. e-mail: birken@mi.uni-erlangen.de
Pol Vanhaecke
Affiliation:
Université de Poitiers, Département de Poitiers, Téléport 2, Boulevard Marie et Pierre Curie, BP 30179, F-86962 Futuroscope Chasseneuil Cedex, France. e-mail: Pol.Vanhaecke@mathlabo.univ-poitiers.fr
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Abstract

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We give a geometric proof of a formula, due to Segal and Wilson, which describes the order of vanishing of the Riemann theta function in the direction which corresponds to the direction of the tangent space of a Riemann surface at a marked point. While this formula appears in the work of Segal and Wilson as a by-product of some nontrivial constructions from the theory of integrable systems (loop groups, infinite-dimensional Grassmannians, tau functions, Schur polynomials, etc.) our proof only uses the classical theory of linear systems on Riemann surfaces.

Keywords

gap sequencesJacobianstheta functions
Type
Research Article
Information
Compositio Mathematica , Volume 135 , Issue 3 , February 2003 , pp. 323 - 330
Copyright
© 2003 Kluwer Academic Publishers