\contentsline {chapter}{List of Figures}{v}
\contentsline {chapter}{List of Tables}{vi}
\contentsline {chapter}{List of Symbols}{vii}
\contentsline {chapter}{Preface}{1}
\contentsline {chapter}{\numberline {1}The Birch and Swinnerton-Dyer conjecture}{2}
\contentsline {section}{\numberline {1.1}The BSD conjecture}{2}
\contentsline {subsection}{\numberline {1.1.1}The ratio $L(A,1)/\Omega _A$}{3}
\contentsline {subsection}{\numberline {1.1.2}Torsion subgroup}{3}
\contentsline {subsection}{\numberline {1.1.3}Tamagawa numbers}{4}
\contentsline {subsection}{\numberline {1.1.4}Upper bounds on $\#\unhbox \voidb@x \hbox {\cyr X}(A)$}{4}
\contentsline {subsubsection}{Kolyvagin's bounds}{5}
\contentsline {subsubsection}{Kato's bounds}{5}
\contentsline {subsection}{\numberline {1.1.5}Lower bounds on $\#\unhbox \voidb@x \hbox {\cyr X}(A)$}{5}
\contentsline {subsubsection}{Invisible elements of $\#\unhbox \voidb@x \hbox {\cyr X}(A^{\vee })$}{6}
\contentsline {subsubsection}{Visibility at higher level}{6}
\contentsline {subsubsection}{Visibility in some Jacobian}{6}
\contentsline {subsection}{\numberline {1.1.6}Motivation for considering abelian varieties}{6}
\contentsline {section}{\numberline {1.2}Existence of nontrivial visible elements of $\unhbox \voidb@x \hbox {\cyr X}(A)$}{6}
\contentsline {section}{\numberline {1.3}Description of tables}{9}
\contentsline {subsection}{\numberline {1.3.1}Notation}{9}
\contentsline {subsection}{\numberline {1.3.2}Table\nobreakspace {}1.2\hbox {}: Shafarevich-Tate groups at prime level}{10}
\contentsline {subsubsection}{Notation}{10}
\contentsline {subsubsection}{Ranks of the explanatory factors}{10}
\contentsline {subsubsection}{Discussion of the data}{10}
\contentsline {subsubsection}{Errata to Brumer's paper}{11}
\contentsline {subsection}{\numberline {1.3.3}Tables\nobreakspace {}1.3\hbox {}--1.6\hbox {}: New visible Shafarevich-Tate groups}{11}
\contentsline {subsubsection}{Notation}{11}
\contentsline {subsubsection}{Remarks on the data}{11}
\contentsline {section}{\numberline {1.4}Further visibility computations}{12}
\contentsline {subsection}{\numberline {1.4.1}Does $\unhbox \voidb@x \hbox {\cyr X}$ become visible at higher level?}{12}
\contentsline {subsubsection}{How we found the explanatory curves}{12}
\contentsline {subsubsection}{2849A}{13}
\contentsline {subsubsection}{4343B}{13}
\contentsline {subsubsection}{5389A}{13}
\contentsline {subsubsection}{3364C, 4229A, 5073D}{14}
\contentsline {subsubsection}{4194N, 5054C}{14}
\contentsline {subsection}{\numberline {1.4.2}Positive rank example}{14}
\contentsline {chapter}{\numberline {2}Modular symbols}{19}
\contentsline {section}{\numberline {2.1}The definition of modular symbols}{19}
\contentsline {section}{\numberline {2.2}Cuspidal modular symbols}{21}
\contentsline {section}{\numberline {2.3}Duality between modular symbols and modular forms}{21}
\contentsline {section}{\numberline {2.4}Linear operators}{22}
\contentsline {subsection}{\numberline {2.4.1}Hecke operators}{22}
\contentsline {subsection}{\numberline {2.4.2}The $*$-involution}{23}
\contentsline {subsection}{\numberline {2.4.3}The Atkin-Lehner involutions}{23}
\contentsline {section}{\numberline {2.5}Degeneracy maps}{24}
\contentsline {subsection}{\numberline {2.5.1}Computing coset representatives}{26}
\contentsline {subsection}{\numberline {2.5.2}Compatibility with modular forms}{27}
\contentsline {section}{\numberline {2.6}Manin symbols}{27}
\contentsline {subsection}{\numberline {2.6.1}Conversion between modular and Manin symbols}{28}
\contentsline {subsection}{\numberline {2.6.2}Hecke operators on Manin symbols}{29}
\contentsline {subsection}{\numberline {2.6.3}The cuspidal and boundary spaces in terms of Manin symbols}{30}
\contentsline {subsection}{\numberline {2.6.4}Computing the boundary map}{30}
\contentsline {section}{\numberline {2.7}The complex torus attached to a modular form}{33}
\contentsline {subsection}{\numberline {2.7.1}The case when the weight is $2$}{34}
\contentsline {chapter}{\numberline {3}Applications of modular symbols}{35}
\contentsline {section}{\numberline {3.1}Computing the space of modular symbols}{35}
\contentsline {section}{\numberline {3.2}Computing the Hecke algebra}{37}
\contentsline {section}{\numberline {3.3}Representing and enumerating Dirichlet characters}{38}
\contentsline {section}{\numberline {3.4}The dimension of $S_k(N,\varepsilon )$}{40}
\contentsline {section}{\numberline {3.5}Decomposing the space of modular symbols}{40}
\contentsline {subsection}{\numberline {3.5.1}Duality}{41}
\contentsline {subsection}{\numberline {3.5.2}Efficient computation of Hecke operators on the dual space}{42}
\contentsline {subsection}{\numberline {3.5.3}Eigenvectors}{43}
\contentsline {subsection}{\numberline {3.5.4}Eigenvalues}{44}
\contentsline {subsection}{\numberline {3.5.5}Sorting and labeling eigenforms}{44}
\contentsline {section}{\numberline {3.6}Intersections and congruences}{45}
\contentsline {subsection}{\numberline {3.6.1}A strategy for computing congruences}{47}
\contentsline {section}{\numberline {3.7}The rational period mapping}{47}
\contentsline {section}{\numberline {3.8}The images of cuspidal points}{49}
\contentsline {subsection}{\numberline {3.8.1}Rational torsion}{49}
\contentsline {subsection}{\numberline {3.8.2}Upper bound on torsion: Counting points mod\nobreakspace {}$p$}{50}
\contentsline {section}{\numberline {3.9}The modular degree}{50}
\contentsline {section}{\numberline {3.10}The rational part of $L(A_f,j)$}{52}
\contentsline {subsection}{\numberline {3.10.1}$L$-functions}{52}
\contentsline {subsection}{\numberline {3.10.2}Winding elements}{52}
\contentsline {subsection}{\numberline {3.10.3}Real and minus volumes}{53}
\contentsline {subsection}{\numberline {3.10.4}The theorem}{53}
\contentsline {subsection}{\numberline {3.10.5}Bounding the denominator of the ratio}{55}
\contentsline {section}{\numberline {3.11}The Manin constant}{57}
\contentsline {subsection}{\numberline {3.11.1}The primes that might divide\nobreakspace {}$c_A$}{57}
\contentsline {subsection}{\numberline {3.11.2}Numerical evidence for the $c_A=1$ conjecture}{58}
\contentsline {section}{\numberline {3.12}Analytic invariants}{59}
\contentsline {subsection}{\numberline {3.12.1}Extended modular symbols}{59}
\contentsline {subsection}{\numberline {3.12.2}Numerically computing period integrals}{60}
\contentsline {subsection}{\numberline {3.12.3}The $W_N$-trick}{62}
\contentsline {subsection}{\numberline {3.12.4}Computing the period mapping}{64}
\contentsline {subsection}{\numberline {3.12.5}Computing special values}{64}
\contentsline {subsection}{\numberline {3.12.6}The real and minus volume associated to $A_f$}{65}
\contentsline {subsection}{\numberline {3.12.7}The component groups $c_{\infty }^+$ and $c_{\infty }^-$}{66}
\contentsline {subsection}{\numberline {3.12.8}Examples}{67}
\contentsline {subsubsection}{Jacobians of genus-two curves}{67}
\contentsline {subsubsection}{Level one cusp forms}{68}
\contentsline {subsubsection}{CM elliptic curves of weight greater than two}{68}
\contentsline {subsubsection}{Some abelian varieties of large dimension}{68}
\contentsline {chapter}{\numberline {4}Component groups of optimal quotients}{70}
\contentsline {section}{\numberline {4.1}Main results}{70}
\contentsline {subsection}{\numberline {4.1.1}N\'eron models and component groups}{70}
\contentsline {subsection}{\numberline {4.1.2}Motivating problem}{71}
\contentsline {subsection}{\numberline {4.1.3}The main result}{71}
\contentsline {section}{\numberline {4.2}Optimal quotients of Jacobians}{72}
\contentsline {section}{\numberline {4.3}The closed fiber of the N\'{e}ron model}{73}
\contentsline {section}{\numberline {4.4}Rigid uniformization}{73}
\contentsline {subsection}{\numberline {4.4.1}Raynaud's uniformization}{74}
\contentsline {subsection}{\numberline {4.4.2}Some lemmas}{74}
\contentsline {subsubsection}{Abelian varieties with purely toric reduction}{75}
\contentsline {section}{\numberline {4.5}The main theorem}{76}
\contentsline {subsection}{\numberline {4.5.1}Description of the component group in terms of the monodromy pairing}{76}
\contentsline {subsubsection}{Proof of the main theorem}{77}
\contentsline {section}{\numberline {4.6}Optimal quotients of $J_0(N)$}{79}
\contentsline {subsection}{\numberline {4.6.1}Modular curves and semistability}{79}
\contentsline {subsection}{\numberline {4.6.2}Newforms and optimal quotients}{80}
\contentsline {subsection}{\numberline {4.6.3}Homology and the modular degree}{80}
\contentsline {subsection}{\numberline {4.6.4}Rational points of the component group (Tamagawa numbers)}{81}
\contentsline {section}{\numberline {4.7}Computations}{81}
\contentsline {subsection}{\numberline {4.7.1}Conjectures and questions}{82}
\contentsline {subsection}{\numberline {4.7.2}Tables}{82}
\contentsline {subsubsection}{Table\nobreakspace {}4.1\hbox {}: Component groups at low level}{83}
\contentsline {subsubsection}{Table\nobreakspace {}4.2\hbox {}--4.3\hbox {}: Big component groups}{83}
\contentsline {subsubsection}{Table\nobreakspace {}4.4\hbox {}: Quotients of $J_0(N)$}{83}
\contentsline {subsubsection}{Table\nobreakspace {}4.5\hbox {}: Quotients of $J_0(p)^-$}{83}
\contentsline {chapter}{Bibliography}{88}