In this thesis we consider elliptic curves with complex multiplication from three different angles:
diophantine, algebraic and arithmetic statistical.
•Diophantine point of view: We study certain integrality properties of singular moduli i.e. of j-invariants of elliptic curves with complex multiplication. We prove various effective finiteness statements concerning differences of singular moduli that are S-units (Chapter 2).
•Algebraic point of view: For every CM elliptic curve E defined over a number field F, we analyze the Galois representation associated to the action of the absolute Galois group of F on the torsion points of E. This includes an investigation of the entanglement in the family of p∞-division fields of E for p prime (Chapter 3 and Chapter 4).
•Arithmetic statistical point of view: Given an elliptic curve E over a number field F, we look at the density of the set of primes p ⊆ F of good reduction for which the point group on the reduced elliptic curve E mod p is cyclic. We detail both the CM and the non-CM case, outlining differences and similarities (Chapter 5).
Some of the material contained in the thesis has been used to write the following manuscripts: [Cam21b], [CS19], [CP21] and [Cam21a]. The article [CS19] has been written in collaboration with Peter Stevenhagen while the article [CP21] has been written in collaboration with Riccardo Pengo.