Modular invariants and isogenies

TL;DR

This paper establishes explicit bounds on the height differences of j-invariants of isogenous elliptic curves over algebraic closures of rationals, utilizing classical and modern inequalities, with applications to modular forms and isogeny computations.

Contribution

It provides new explicit bounds on the height differences of j-invariants of isogenous elliptic curves, combining classical estimates with recent isogeny bounds.

Findings

Explicit bounds on height differences of j-invariants

Applications to Vélú's formulas and modular polynomials

Enhanced understanding of isogeny height relations

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 R\'emond. We give applications to the study of V\'elu's formulas and of modular polynomials.