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

Stefan Schmid

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arXiv · 1908.11088

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

Stefan Schmid

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Abstract

For a fixed $j$-invariant $j_0$ 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 $j_0$ are isogenous. Our bounds are effective. We also modify the problem slightly by fixing a singular modulus $\alpha$ and looking at all $j$ such that $j-\alpha$ is an algebraic unit and such that elliptic curves corresponding to $j$ and $j_0$ are isogenous. The number of such $j$ can again be bounded effectively.

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Stefan Schmid. 2019-08-29. Integrality properties in the Moduli Space of Elliptic Curves: Isogeny Case. https://arxiv.org/abs/1908.11088

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