Modular polynomials on Hilbert surfaces

Modular polynomials on Hilbert surfaces

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 o...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Journal of number theory 2020-11, Vol.216, p.403-459
Hauptverfasser: Milio, Enea, Robert, Damien
Format: Artikel
Sprache:eng
Schlagworte:
Abelian surface
Cyclic isogeny
Humbert surface
Mathematics
Modular polynomials
Moduli space of Hilbert and Siegel
Theta constant
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Beschreibung
Zusammenfassung: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.
ISSN:0022-314X
1096-1658
DOI:10.1016/j.jnt.2020.04.014