Integrality Properties in the Moduli Space of Elliptic Curves: Isogeny Case

Abstract

Abstract For a fixed j–invariant j0 of an elliptic curve without complex multiplication we bound the number of j–invariants j that are algebraic units and such that elliptic curves corresponding to j and j0 are isogenous. Our bounds are explicit. We also modify the problem slightly by fixing a singular modulus α and looking at all j such that $j-\alpha$ is an algebraic unit and such that elliptic curves corresponding to j and j0 are isogenous. The number of such j can again be bounded explicitly.