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

2019/08/29 by Schmid, Stefan
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1908.11088

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 effective. We also modify the problem slightly by fixing a singular modulus α and looking at all j such that j-α 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 effectively.

Related

• Integrality properties in the Moduli Space of Elliptic Curves: CM Case
• Computing Isogenies at Singular Points of the Modular Polynomial
• No singular modulus is a unit
• Local invariants of isogenous elliptic curves
• Distinct differences of singular moduli