Campanella, Alessandro (2025) Entanglements of Galois Representations for Elliptic Curves over Q: Foundations and Future Directions. Masters thesis, Concordia University.
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Abstract
This thesis investigates the non-surjectivity of adelic Galois representations associated with elliptic curves over Q, a phenomenon explained by two forms of entanglement. We introduce and differentiate between vertical and horizontal entanglements, providing a group-theoretic perspective on the latter. We also develop the concept of 'entanglement networks', diagrams derived from a theorem on field intersections, which offer a framework for analyzing these phenomena and suggest avenues for future combinatorial and cryptographic study. Computationally, we address the classification of potential mod-n Galois images by providing SageMath code to compute all applicable subgroups of GL_2(Z/nZ) for any n. Further SageMath implementations are presented to compute relevant entanglements for a given elliptic curve, utilizing data from the LMFDB database. Finally, we present a theorem providing conditions for the surjectivity of mod-n Galois representations, linking it to the surjectivity of mod-p representations for prime factors p of n and a constant related to Serre's work.
| Divisions: | Concordia University > Faculty of Arts and Science > Mathematics and Statistics |
|---|---|
| Item Type: | Thesis (Masters) |
| Authors: | Campanella, Alessandro |
| Institution: | Concordia University |
| Degree Name: | M.A. |
| Program: | Mathematics |
| Date: | 20 August 2025 |
| Thesis Supervisor(s): | Rosso, Giovanni and Pagano, Carlo |
| Keywords: | Elliptic Curves Galois Representations Horizontal Entanglement Adelic Representation Division Fields Complex Multiplication Entanglement Networks SageMath |
| ID Code: | 996063 |
| Deposited By: | Alessandro Campanella |
| Deposited On: | 04 Nov 2025 17:04 |
| Last Modified: | 04 Nov 2025 17:04 |
| Related URLs: |
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