\contentsline {chapter}{Table of Contents}{iii}
\contentsline {chapter}{Acknowledgements}{i}
\contentsline {chapter}{\numberline {1}Mordell-Weil Theorem, Shafarevich-Tate Group and Selmer Groups for Elliptic Curves.}{3}
\contentsline {section}{\numberline {1.1}Weak Mordell-Weil Group and Kummer Pairing via Galois Cohomology}{3}
\contentsline {section}{\numberline {1.2}Properties of $L = K([m]^{-1}E(K))/K$}{5}
\contentsline {section}{\numberline {1.3}Computing the Weak Mordell-Weil Group and Principal Homogeneous Spaces}{7}
\contentsline {section}{\numberline {1.4}Applications and Complete 2-descent}{9}
\contentsline {section}{\numberline {1.5}Definition of the $\phi $-Selmer and Shafarevich-Tate Groups}{11}
\contentsline {section}{\numberline {1.6}Finiteness of the Selmer Group}{14}
\contentsline {chapter}{\numberline {2}Abelian Varieties}{16}
\contentsline {section}{\numberline {2.1}Abelian Varieties over Arbitrary Fields}{16}
\contentsline {section}{\numberline {2.2}The Dual Abelian Variety in Characteristic Zero}{17}
\contentsline {section}{\numberline {2.3}The Dual Isogeny and the Dual Exact Sequence}{19}
\contentsline {section}{\numberline {2.4}Jacobians of Curves Over $\@mathbb {C}$. The Analytic Construction.}{20}
\contentsline {section}{\numberline {2.5}Jacobians of Curves Over Arbitrary Fields. Weil's Construction.}{22}
\contentsline {chapter}{\numberline {3}Modular Abelian Varieties Attached to Newforms}{24}
\contentsline {section}{\numberline {3.1}Hecke Operators as Correspondences}{24}
\contentsline {section}{\numberline {3.2}Constructing an Abelian Variety $A_f$ as a Quotient of $J_0(N)$}{26}
\contentsline {section}{\numberline {3.3}The Dual Abelian Variety as a Subvariety of $J_0(N)$}{28}
\contentsline {chapter}{\numberline {4}Visibility Theory}{31}
\contentsline {section}{\numberline {4.1}Visible Subgroups of $H^1(K, A)$ and ${\unhbox \voidb@x \hbox {{\fontencoding {OT2}\fontfamily {wncyr}\fontseries {m}\fontshape {n}\selectfont Sh}}}(A / K)$}{31}
\contentsline {section}{\numberline {4.2}The First Property of Visibility}{32}
\contentsline {section}{\numberline {4.3}Producing Upper Bound on the Visibility Dimension}{35}
\contentsline {section}{\numberline {4.4}Smooth and Surjective Morphisms}{36}
\contentsline {subsection}{\numberline {4.4.1}Flat, Smooth and \'Etale Morphisms}{36}
\contentsline {subsection}{\numberline {4.4.2}Henselian Rings and Strictly Henselian Rings}{37}
\contentsline {subsection}{\numberline {4.4.3}Surjectivity of $[n] : G(R) \rightarrow G(R)$}{38}
\contentsline {subsection}{\numberline {4.4.4}Surjectivity of the Induced Map on Generic Fibers}{39}
\contentsline {section}{\numberline {4.5}Producing Visible Elements of the Shafarevich-Tate Group}{40}
\contentsline {chapter}{\numberline {5}Computational Examples and Algorithms}{45}
\contentsline {section}{\numberline {5.1}Algorithms for Computing with Modular Abelian Varieties}{46}
\contentsline {subsection}{\numberline {5.1.1}Computing the Modular Degree}{46}
\contentsline {subsection}{\numberline {5.1.2}Intersecting Complex Tori}{49}
\contentsline {subsection}{\numberline {5.1.3}Producing a Multiple of the Order of the Torsion Subgroup}{50}
\contentsline {subsection}{\numberline {5.1.4}Producing a Divisor of the Order of the Torsion Subgroup}{52}
\contentsline {subsection}{\numberline {5.1.5}Computation of the Tamagawa Numbers}{53}
\contentsline {subsection}{\numberline {5.1.6}Computing the $L$-Ratio}{54}
\contentsline {section}{\numberline {5.2}Examples of Visible Elements.}{56}
\contentsline {subsection}{\numberline {5.2.1}A 20-Dimensional Quotient of $J_0(389)$.}{56}
\contentsline {subsection}{\numberline {5.2.2}Evidence for the Birch and Swinnerton-Dyer Conjecture for an 18-Dimensional Quotient of $J_0(551)$.}{58}
\contentsline {chapter}{Bibliography}{64}