Modular invariants and isogenies

Abstract

We provide explicit bounds on the difference of heights of the $j$-invariants of isogenous elliptic curves defined over $\overline{\mathbb{Q}}$. The first one is reminiscent of a classical estimate for the Faltings height of isogenous abelian varieties, which is indeed used in the proof. We also use an explicit version of Silverman's inequality and isogeny estimates by Gaudron and Rémond. We give applications to the study of Vélu's formulas and of modular polynomials.

Cited by in corpus (3)

• Degree and height estimates for modular equations on PEL Shimura varieties
• Torsion points on isogenous abelian varieties
• Effective bounds on differences of singular moduli that are S-units