Title: Efficient computation of multidimensional theta functions,
Abstract: 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 $\sqrt{3}/2$. 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.