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RESEARCH PRODUCT

Algorithmic approaches to Siegel's fundamental domain

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Siegel determined a fundamental domain using the Minkowski reduction of quadratic forms. He gave all the details concerning this domain for genus 1. It is the determination of the Minkowski fundamental domain presented as the second condition and the maximal height condition, presented as the third condition, which prevents the exact determination of this domain for the general case. The latest results were obtained by Gottschling for the genus 2 in 1959. It has since remained unexplored and poorly understood, in particular the different regions of Minkowski reduction. In order to identify Siegel's fundamental domain for genus 3, we present some results concerning the third condition of this domain. Every abelian function can be written in terms of rational functions of theta functions and their derivatives. This allows the expression of solutions of integrable systems in terms of theta functions. Such solutions are relevant in the description of surface water waves, non linear optics. Because of these applications, Deconinck and Van Hoeij have developed and implemented al-gorithms for computing the Riemann matrix and Deconinck et al. have developed the computation of the corresponding theta functions. Deconinck et al. have used Siegel's algorithm to approximately reach the Siegel fundamental domain and have adopted the LLL reduction algorithm to nd the shortest lattice vector. However, we opt here to use a Minkowski algorithmup to dimension 5 and an exact determination of the shortest lattice vector for greater dimensions.