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.