论文标题
评估Abelian表面的模块化方程
Evaluating modular equations for abelian surfaces
论文作者
论文摘要
我们设计有效的算法来评估使用复杂近似值在数字字段或有限场上的Abelian表面的Siegel和Hilbert Type的模块化方程。当相关的模块化形式在z上明确知道时，它们的输出是正确的。这包括Siegel案件，以及Hilbert的二次判别5和8的案例。作为证据的一部分，我们为2属的某些关键数值算法建立了新的正确性和复杂性结果，即在属2属的时期矩阵，即，对基本域的减少算法，AGM方法和计算型构建和计算大型构建。
We design efficient algorithms to evaluate modular equations of Siegel and Hilbert type for abelian surfaces over number fields or finite fields using complex approximations. Their output is provably correct when the associated graded ring of modular forms over Z is explicitly known; this includes the Siegel case, and the Hilbert case for the quadratic fields of discriminant 5 and 8. As part of the proofs, we establish new correctness and complexity results for certain key numerical algorithms on period matrices in genus 2, namely the reduction algorithm to the fundamental domain, the AGM method, and computing big period matrices and RM structures.
