多维theta函数的高效计算
Efficient computation of multidimensional theta functions
摘要：
有效计算多维θ函数的一个重要步骤是为一个给定的黎曼矩阵构造适当的辛变换，以保证θ级数的快速收敛。给出了一种将Riemann矩阵近似映射到Siegel基本域的算法。该算法准确地确定了由黎曼矩阵生成的格的最短向量，并保证其长度大于$\sqrt{3}/2$。该方法是基于Deconinck等人的先前算法。使用LLL算法进行格约简。在这里，LLL算法被替换为精确的Minkowski减少小属和精确识别的最短格向量的属的较大值。
英文摘要：
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.
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