We describe an evaluation/interpolation approach to compute modular polynomials on a Hilbert surface, which parametrizes abelian surfaces with maximal real multiplication. Under some heuristics we obtain a quasi-linear algorithm. The corresponding modular polynomials are much smaller than the ones on the Siegel threefold. We explain how to compute even smaller polynomials by using pullbacks of theta functions to the Hilbert surface, and give an application to the CRT method to construct class polynomials.

中文翻译：

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希尔伯特曲面上的模多项式

我们描述了一种计算 Hilbert 曲面上的模多项式的评估/插值方法，该方法使用最大实数乘法对阿贝尔曲面进行参数化。在一些启发下，我们获得了一个拟线性算法。相应的模多项式比 Siegel 三倍上的要小得多。我们解释了如何通过将 theta 函数回退到 Hilbert 曲面来计算更小的多项式，并应用 CRT 方法来构造类多项式。