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Low degree points on modular curves
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Box, Josha (2021) Low degree points on modular curves. PhD thesis, University of Warwick.
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WRAP_Theses_Box_2021.pdf - Submitted Version - Requires a PDF viewer. Download (1201Kb) | Preview |
Official URL: http://webcat.warwick.ac.uk/record=b3723638
Abstract
In this thesis we study modular curves and their points defined over number fields of degrees 2, 3 and 4. In Chapter 1, we introduce modular curves and describe the moduli interpretation of their morphisms. This is used in Chapter 2 to find an algorithm for computing models of quotients of general modular curves, by finding the equations satisfied by the corresponding cusp forms. We also provide in Chapter 1 an overview of the computational methods used in Chapters 3 and 4 for studying points of fixed degree on modular curves.
In Chapter 3 we examine the modular curves X0(N) for
N ϵ {37; 43; 53; 57; 61; 65; 67; 73g:
These are exactly the modular curves X0(N) of genus between 2 and 5 whose Jacobians have infinite Mordell{Weil group. We determine the (isolated) quadratic points on each of them, the cubic points on those with N _ 53, and the isolated quartic points on X0(65). To analyse their quadratic points, we use Siksek's relative symmetric Chabauty method, while we develop a generalisation to study points of higher degrees. All these points are listed in the tables in Section 3.4.
Finally, in Chapter 4 we focus on the quartic points on various modular curves to prove that elliptic curves over totally real quartic fields not containing p 5 are modular. The algorithm developed in Chapter 2 and the generalised Chabauty method established in Chapter 3 are essential ingredients for the proof.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Modular curves, Morphisms (Mathematics), Modular curves -- Mathematical models | ||||
Official Date: | June 2021 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Siksek, Samir | ||||
Format of File: | |||||
Extent: | viii, 136 leaves : illustrations | ||||
Language: | eng |
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