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Abelian functions associated with genus three algebraic curves

Part of: Infinite-dimensional Hamiltonian systems Abelian varieties and schemes Curves

Published online by Cambridge University Press:  01 November 2011

J. C. Eilbeck ,
M. England  and
Y. Ônishi
J. C. Eilbeck
Affiliation:
Department of Mathematics and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, United Kingdom (email: j.c.eilbeck@hw.ac.uk)
M. England
Affiliation:
School of Mathematics and Statistics, University of Glasgow, Glasgow, G12 8QQ, United Kingdom (email: matthew.england@glasgow.ac.uk)
Y. Ônishi
Affiliation:
Faculty of Education and Human Sciences, University of Yamanashi, 4-3-11, Takeda, Kofu, 400-8511, Japan (email: yonishi@yamanashi.ac.jp)

Abstract

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We develop the theory of Abelian functions associated with algebraic curves. The growth in computer power and the advancement of efficient symbolic computation techniques have allowed for recent progress in this area. In this paper we focus on the genus three cases, comparing the two canonical classes of hyperelliptic and trigonal curves. We present new addition formulae, derive bases for the spaces of Abelian functions and discuss the differential equations such functions satisfy.

Supplementary materials are available with this article.

MSC classification

Secondary: 37K20: Relations with algebraic geometry, complex analysis, special functions 14H55: Riemann surfaces; Weierstrass points; gap sequences 14K25: Theta functions 14H45: Special curves and curves of low genus 14H40: Jacobians, Prym varieties
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 14 , 2011 , pp. 291 - 326
Copyright
Copyright © London Mathematical Society 2011

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