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Shafarevich-Tate groups for some Modular Abelian Varieties

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Shafarevich-Tate groups for some Modular Abelian Varieties

Barendrecht, Casper (2020) Shafarevich-Tate groups for some Modular Abelian Varieties. Masters thesis, Concordia University.

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Abstract

Let f be a weight 2 newform of level N, and let A be the associated modular abelian variety.
Let Kbe an imaginary quadratic field of discriminant D different from -3 and -4, and let p be a prime of the endomorphism ring of A outside a finite set S.
If A admits a principal polarization, and the Heegner point associated to K has infinite order in A(K), then the Shafarevich-Tate group is finite and its p-primary part is a perfect square.
Generalizing the work of Kolyvagin and McCallum, we give an explicit structure of the p-primary part of the Shafarevich-Tate group.

This thesis aims to provide an accessible proof of this statement for those with restricted knowledge on the subject.
The first three chapters offer an introduction to the basic notion of arithmetic geometry. Chapters 4 and 5 expand on the theory spefic to the thesis. Finally chapter 6 combines the developed theory to proof this structure theorem for Shafarevich-Tate groups.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Thesis (Masters)
Authors:Barendrecht, Casper
Institution:Concordia University
Degree Name:M.A. Sc.
Program:Mathematics
Date:September 2020
Thesis Supervisor(s):Iovita, Adrian and Longo, Matteo
ID Code:987474
Deposited By: Casper Barendrecht
Deposited On:27 Oct 2022 13:51
Last Modified:27 Oct 2022 13:51
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