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Journal Articles Journal of Number Theory Year : 2020

Modular polynomials on Hilbert surfaces

Abstract

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.
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Dates and versions

hal-01520262 , version 1 (10-05-2017)
hal-01520262 , version 2 (09-09-2017)
hal-01520262 , version 3 (09-01-2020)

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Enea Milio, Damien Robert. Modular polynomials on Hilbert surfaces. Journal of Number Theory, 2020, 216, pp.403-459. ⟨10.1016/j.jnt.2020.04.014⟩. ⟨hal-01520262v3⟩
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