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Modular polynomials on Hilbert surfaces. (English) Zbl 1456.11078
Summary: 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.
MSC:
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11G18 Arithmetic aspects of modular and Shimura varieties
11Y40 Algebraic number theory computations
14G35 Modular and Shimura varieties
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