Integrality properties in the moduli space of elliptic curves

Schmid, Stefan. Integrality properties in the moduli space of elliptic curves. 2019, Doctoral Thesis, University of Basel, Faculty of Science.

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Official URL: http://edoc.unibas.ch/diss/DissB_13452

In the thesis at hand we discuss two problems of integral points in the moduli space of elliptic curves. The first problem can be described as follows. We fix an algebraic number $\alpha$ that is the $j$-invariant of an elliptic curve without complex multiplication. We prove that the number of $j$-invariants with complex multiplication such that $j-\alpha$ is an algebraic unit can be bounded by a computable number.
The second problem is of similar nature. For this we fix $j_0$ the $j$-invariant of an elliptic curve without complex multiplication defined over some number field. We show that there are only finitely many algebraic units $j$ such that elliptic curves with $j$-invariants $j$ and $j_0$ are isogenous. A slight modification shows that only finitely $j$-invariants exist such that $j$ and $j_0$ are isogenous and such that $j-\alpha$ is a unit, where $\alpha$ is an arbitrary but fixed $j$-invariant of an elliptic curve with complex multiplication.