### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 037, 36 pages      arXiv:1203.3409      https://doi.org/10.3842/SIGMA.2012.037

### Building Abelian Functions with Generalised Baker-Hirota Operators

Matthew England a and Chris Athorne b
a) Department of Computer Science, University of Bath, Bath, BA2 7AY, UK
b) School of Mathematics and Statistics, University of Glasgow, G12 8QQ, UK

Received March 16, 2012, in final form June 18, 2012; Published online June 26, 2012

Abstract
We present a new systematic method to construct Abelian functions on Jacobian varieties of plane, algebraic curves. The main tool used is a symmetric generalisation of the bilinear operator defined in the work of Baker and Hirota. We give explicit formulae for the multiple applications of the operators, use them to define infinite sequences of Abelian functions of a prescribed pole structure and deduce the key properties of these functions. We apply the theory on the two canonical curves of genus three, presenting new explicit examples of vector space bases of Abelian functions. These reveal previously unseen similarities between the theories of functions associated to curves of the same genus.

Key words: Baker-Hirota operator; $\mathcal{R}$-function; Abelian function; Kleinian function.

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