Deriving Bases for Abelian Functions Matthew England

We consider multiply periodic functions, sometimes called Abelian functions, defined with respect to the period matrices associated with classes of algebraic curves. We realise them as generalisations of the Weierstraß ℘-function using two different approaches. These functions arise naturally as solutions to some of the important equations of mathematical physics and their differential equations, addition formulae, and applications have all been recent topics of study.The first approach discussed sees the functions defined as logarithmic derivatives of the σ-function, a modified Riemann θ-function. We can make use of known properties of the sigma function to derive power series expansions and in turn the properties mentioned above. This approach has been extended to a wide range of non hyperelliptic and higher genus curves and an overview of recent results is given.The second approach defines the functions algebraically, after first modifying the curve into its equivariant form. This approach allows the use of representation theory to derive a range of results at lower computational cost. We discuss the development of this theory for hyperelliptic curves and how it may be extended in the future.

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“…The first example of these automorphism addition formulae was given in [17] with further examples recently published in [19] and [21].…”

Section: Deriving Differential Equations and Addition Formulaementioning

confidence: 99%

Generalised elliptic functions

England

Athorne

2012

Open Mathematics

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“…This new function is the difference ℘ 11 ℘ 22 − ℘ 2 12 , in which the poles of order 4 in each term cancel. In [16,19] the basis problem was solved in various cases, by a group including one of the present authors, through the introduction of analogous functions; polynomials in ℘ with coefficients chosen to cancel poles. However for curves with g > 2 hyperelliptic and g > 3 non-hyperelliptic the bases cannot be finitely generated by differentiation and this approach does not generalise easily to give an infinite number of functions.…”

Section: The Basis Problem For Abelian Functionsmentioning

confidence: 99%

“…The existence of such formulae helped motivate the new Abelian functions constructed in Section 6. For examples of using bases to calculate differential equations and addition formulae see [9,16,18,19,21]. While the ℘-functions may be defined algebraically using the curve, an alternative definition using the σ-function of the curve can be easier to work with.…”

Section: The Basis Problem For Abelian Functionsmentioning

confidence: 99%

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Building Abelian Functions with Generalised Baker-Hirota Operators

England

Athorne

2012

SIGMA

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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.

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“…In the case of trigonal curves, the authors found that further addition formula after making specialisations of the curve parameters in analogy with the equianharmonic case. Results for genus three were given in Theorem 10.1 of [5] and Theorem 5.4 in [6]), and for genus four in Theorem 8 in [7].…”

Section: Introductionmentioning

confidence: 99%

Some new addition formulae for Weierstrass elliptic functions

Eilbeck¹,

England

Ônishi

2014

Proc. R. Soc. A.

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We present new addition formulae for the Weierstrass functions associated with a general elliptic curve. We prove the structure of the formulae in n-variables and give the explicit addition formulae for the 2-and 3-variable cases. These new results were inspired by new addition formulae found in the case of an equianharmonic curve, which we can now observe as a specialization of the results here. The new formulae, and the techniques used to find them, also follow the recent work for the generalization of Weierstrass functions to curves of higher genus.

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Deriving Bases for Abelian Functions Matthew England

Cited by 3 publications

References 22 publications

Generalised elliptic functions

Generalised elliptic functions

Building Abelian Functions with Generalised Baker-Hirota Operators

Some new addition formulae for Weierstrass elliptic functions

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