Modular polynomials on Hilbert surfaces

JOURNAL OF NUMBER THEORY

卷:216

Article

Milio, Enea³  Robert, Damien^1,2 

[1] INRIA Bordeaux Sud Ouest, 200 Ave Vieille Tour, F-33405 Talence, France

[2] Inst Math Bordeaux, 351 Cours Liberat, F-33400 Talence, France

[3] Ecole Polytech Fed Lausanne, EPFL SB MathGeom GR JET, Lausanne, Switzerland

关键词: Modular polynomials;    Cyclic isogeny;    Abelian surface;    Humbert surface;    Moduli space of Hilbert and Siegel;    Theta constant;   

DOI  :  10.1016/j.jnt.2020.04.014

来源: Elsevier

【 摘 要 】

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. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free