Arithmetic and diophantine properties of elliptic curves with complex multiplication
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- Francesco Campagna_Arithmetic and diophantine properties of elliptic curves with complex multiplication
Final published version, 1.68 MB, PDF document
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.
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.
Original language | English |
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Publisher | Department of Mathematical Sciences, Faculty of Science, University of Copenhagen |
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Number of pages | 169 |
Publication status | Published - 2021 |
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