Degree and height estimates for modular equations on PEL Shimura varieties

Jean Kieffer (1, 2)

1 LFANT - Lithe and fast algorithmic number theory
2 IMB - Institut de Mathématiques de Bordeaux

We define modular equations in the setting of PEL Shimura varieties as equations describing Hecke correspondences, and prove upper bounds on their degrees and heights. This extends known results about elliptic modular polynomials, and implies complexity bounds for number-theoretic algorithms using these modular equations. In particular, we obtain tight degree bounds for modular equations of Siegel and Hilbert type for abelian surfaces.

Mots clés: Hecke correspondences; Shimura varieties; Heights; Abelian varieties; Modular equations

Domaines: Géométrie algébrique [math.AG]; Théorie des nombres [math.NT]

Jean Kieffer. Degree and height estimates for modular equations on PEL Shimura varieties. Journal of the London Mathematical Society, 2022, 105 (2), pp.1314-1361. ⟨10.1112/jlms.12540⟩. ⟨hal-02436057v4⟩