Isogenies of Elliptic Curves Over Function Fields

We prove two theorems concerning isogenies of elliptic curves over function fields. The first one describes the variation of the height of the j-invariant in an isogeny class. The second one is an “isogeny estimate,” providing an explicit bound on the degree of a minimal isogeny between two isogenous elliptic curves. We also give several corollaries of these two results.