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Hyperelliptic addition law. (English) Zbl 1362.33024
Summary: Given a family of genus \(g\) algebraic curves, with the equation \(f(x,y, \Lambda)=0\), we consider two fiber-bundles \mathsfU and \mathsfX over the space of parameters \(\Lambda\). A fiber of \mathsfU is the Jacobi variety of the curve. \mathsfU is equipped with the natural groupoid structure that induces the canonical addition on a fiber. A fiber of \mathsfX is the \(g\)-th symmetric power of the curve. We describe the algebraic groupoid structure on \mathsfX using the Weierstrass gap theorem to define the ‘addition law’ on its fiber. The addition theorems that are the subject of the present study are represented by the formulas, mostly explicit, determining the isomorphism of groupoids \mathsfU\(\to\)\mathsfX. At \(\mathrm{g}=1\) this gives the classic addition formulas for the elliptic Weierstrass \(\wp\) and \(\wp'\) functions. To illustrate the efficiency of our approach the hyperelliptic curves of the form \(y^2=x^{2g+1}+\Sigma^{2g-1}_{i=0}\,\lambda_{4g+2-2i}x^i\) are considered. We construct the explicit form of the addition law for hyperelliptic Abelian vector functions \(\wp\) and \(\wp'\) (the functions \(\wp\) and \(\wp'\) form a basis in the field of hyperelliptic Abelian functions, i.e., any function from the field can be expressed as a rational function of \(\wp\) and \(\wp'\)). Addition formulas for the higher genera zetafunctions are discussed. The genus 2 result is written in a Hirota-like trilinear form for the sigma-function. We propose a conjecture to describe the general formula in these terms.

MSC:
33E05 Elliptic functions and integrals
14H52 Elliptic curves
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References:
[1] Baker H F Abelian Functions, Cambridge University Press, Cambridge, 1995
[2] Baker H F Multiply Periodic Functions, Cambridge University Press, Cambridge, 1907
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