Efficient computation of multidimensional theta functions

Jörg Frauendiener (1) , Carine Jaber (2) , Christian Klein (2)

1 Department of Mathematics and Statistics [Dunedin]
2 IMB - Institut de Mathématiques de Bourgogne [Dijon]

An important step in the efficient computation of multi-dimensional theta functions is the construction of appropriate symplectic transformations for a given Riemann matrix assuring a rapid convergence of the theta series. An algorithm is presented to approximately map the Riemann matrix to the Siegel fundamental domain. The shortest vector of the lattice generated by the Riemann matrix is identified exactly, and the algorithm ensures that its length is larger than root 3/2. The approach is based on a previous algorithm by Deconinck et al.. using the LLL algorithm for lattice reductions. Here, the LLL algorithm is replaced by exact Minkowski reductions for small genus and an exact identification of the shortest lattice vector for larger values of the genus

Mots clés: Multidimensional theta functions; Siegel fundamental domain; Minkowski reduction; LLL reduction; riemann; reduction; algorithms

Domaines: Mathématiques [math]; Physique mathématique [math-ph]

Jörg Frauendiener, Carine Jaber, Christian Klein. Efficient computation of multidimensional theta functions. Journal of Geometry and Physics, 2019, 141, pp.147-158. ⟨10.1016/j.geomphys.2019.03.011⟩. ⟨hal-02173551⟩