The Faltings-Moriwaki modular height and isogenies of elliptic curves

Abstract

We estimate the variation of the Faltings-Moriwaki modular height under isogenies of elliptic curves. In particular, we reprove the Faltings finiteness theorem for elliptic curves over fields of finite type over $\mathbb{Q}$. Our approach is \lq\lqeffective\rq\rq in the sence that we generalize Raynaud's upper bound to higher dimensional cases.