Arithmetic and diophantine properties of elliptic curves with complex multiplication
Publikation: Bog/antologi/afhandling/rapport › Ph.d.-afhandling › Forskning
Dokumenter
- Francesco Campagna_Arithmetic and diophantine properties of elliptic curves with complex multiplication
Forlagets udgivne version, 1,68 MB, PDF-dokument
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.
Originalsprog | Engelsk |
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Forlag | Department of Mathematical Sciences, Faculty of Science, University of Copenhagen |
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Antal sider | 169 |
Status | Udgivet - 2021 |
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