\begin{theindex}

  \item $L$-series, 52, \textit{52}
  \item $q$-expansion principle, 58
  \item \'Etale group scheme, \textit{73}

  \indexspace

  \item Agashe, 2, 3, 37, 52, 53, 57, 58
    \subitem conjecture of, 58
  \item Algorithm for computing
    \subitem an eigenvector, 43
    \subitem congruences, 47
    \subitem coset representatives, 26
    \subitem cuspidal subgroup, 49
    \subitem cusps, 30
    \subitem decomposition of space of modular symbols, 41
    \subitem eigenvalues, 44
    \subitem equivalent cusps, 31
    \subitem Hecke operators on the dual, 42, 43
    \subitem integral modular symbols, 37
    \subitem modular kernel, 51
    \subitem period integrals, 60
    \subitem rational part of $L(A,j)$, 53
    \subitem rational period mapping, 48
    \subitem real and minus volumes, 65
    \subitem space of modular symbols, 36
  \item Analytic invariants, 59
  \item Antiholomorphic cusp forms, \textit{21}
  \item Atkin-Lehner involution, \textit{23}, 28, 63, 81
    \subitem and integration pairing, 24
    \subitem and ordering eigenforms, 45

  \indexspace

  \item Birch and Swinnerton-Dyer conjecture, \see{BSD conjecture}{2}
  \item Bloch and Kato conjecture, 4, 33
  \item Bound of
    \subitem Kato, 5
    \subitem Kolyvagin, 5
    \subitem Sturm, 38, 42
  \item Boundary map, 30
  \item Boundary modular symbols, \textit{21}
    \subitem and Manin symbols, 30
  \item BSD conjecture, 2, \textit{2}, 12, 35
    \subitem and $\Omega_A$, 53, 66
    \subitem and $\Sha$, 5
    \subitem and $L(A,j)/\Omega_j$, 35
    \subitem and component groups, 70
    \subitem generalization of, 4
    \subitem in higher dimensions, 2
    \subitem is still unknown, 2
    \subitem Manin's remarks on, 56
    \subitem predicted order of $\Sha$, 10
    \subitem predicts invisible elements, 6
    \subitem predicts large component groups, 83
    \subitem statement of, 3
    \subitem verification of, 4

  \indexspace

  \item Canonical polarization, 71
  \item Cartier dual, 72
  \item Character group of torus, 71, 73, \textit{73}, 76
  \item Closed fiber of N\'eron model, \textit{73}
  \item Complex torus, 32, 59
    \subitem dual of, 33
    \subitem in weight two, 34
  \item Component group, 70, 71, \textit{71}
    \subitem and character group, \textit{78}
    \subitem archimedean, \textit{66}
    \subitem geometric, 7
    \subitem rational points of, 70
    \subitem table of, 84--87
  \item Conductor of Dirichlet character, \textit{39}
  \item Congruences
    \subitem and BSD conjecture, 2
    \subitem and lower bounds on $\Sha$, 6
    \subitem between $q$-expansions, 47
    \subitem between elliptic curves, 6
    \subitem computed using homology, 47
    \subitem computing, 45
  \item Conjecture
    \subitem about modular degree, 82
    \subitem Agashe and Stein, 58
    \subitem Birch and Swinnerton-Dyer, \see{BSD conjecture}{2}
    \subitem Bloch and Kato, 4, 33
    \subitem Kani, 6
    \subitem refined Eisenstein, 82
    \subitem Shimura and Taniyama, 2
    \subitem that Manin constant equals $1$, \textit{58}
  \item Continued fractions, 28
  \item Coset representatives, 26
  \item Cremona, 19, 29, 30, 40, 53
  \item Cusp forms, \textit{21}
    \subitem antiholomorphic, \textit{21}
  \item Cuspidal modular symbols, \textit{21}
    \subitem and Manin symbols, 30
  \item Cuspidal points, 49
  \item Cusps
    \subitem and boundary map, 30
    \subitem criterion for vanishing, 31

  \indexspace

  \item Degeneracy maps, 24, 41
    \subitem compatibility, 27
  \item Dimension of $S_k(N,\eps)$, 40
  \item Dirichlet character, \textit{20}, 24, 32, 36--38
    \subitem and cusps, 31
    \subitem conductor of, \textit{39}

  \indexspace

  \item Eichler-Shimura relation, 50
  \item Eigenforms
    \subitem computing, 44
    \subitem sorting and labeling, 44
  \item Euler system, 4, 5
  \item Explanatory factor, 10
  \item Extended modular symbols, \textit{59}, 60

  \indexspace

  \item Fourier coefficients, 23, 47

  \indexspace

  \item Genus-two curves, 67
  \item Grothendieck, 71

  \indexspace

  \item Hecke, 52
  \item Hecke algebra, 3, \textit{22}, 33, 51, 57, 65, \textit{80}, 82
    \subitem and congruences, 47
    \subitem and cuspidal subgroup, 49
    \subitem and integration pairing, 33
    \subitem and rational period mapping, 47
    \subitem bounds torsion, 50
    \subitem computation of, 37
    \subitem generators as module, 37
    \subitem generators as ring, 37
  \item Hecke operators, 1, 22, 32, 37, 41, 44, 48, 54, 56
    \subitem and degeneracy maps, 41
    \subitem computation on subspace of dual, 42
    \subitem on Manin symbols, 29
    \subitem respect pairing, 22
  \item Heegner points, 5
  \item Heilbronn matrices, 44

  \indexspace

  \item Index of lattices, \textit{54}
  \item Integration pairing, 21, 59, 80
    \subitem and complex torus, 33
    \subitem and extended modular symbols, 59
  \item Invisible elements of $\Sha$, 6, 10

  \indexspace

  \item Jacobian, 2, 6, 72
    \subitem is principally polarized, 76
    \subitem of $X_0(N)$, 70, 71, 80
    \subitem of genus-two curve, 67
    \subitem semistable, 71, 76
    \subitem visibility in, 6

  \indexspace

  \item Kani, 6

  \indexspace

  \item Lattice, \textit{54}
  \item Lattice index, 3, \textit{54}
  \item Level of modular symbols, \textit{20}
  \item Local-to-global principle, 3
  \item Logan, 13

  \indexspace

  \item Manin, 35, 36, 49, 53, 61
    \subitem comment on BSD, 56
    \subitem trick of, \textit{28}
  \item Manin constant, 3, 57, 58, 67
    \subitem conjecture about, \textit{58}
  \item Manin symbols, 27, \textit{27}, 28, 30, 36, 42, 44
    \subitem and boundary space, 30
    \subitem and cuspidal subspace, 30
    \subitem and Hecke operators, 29
    \subitem and Manin's trick, 27
    \subitem conversion to modular symbols, 28
  \item Manin's trick, 27, \textit{28}
  \item Mazur, 2--5, 13, 52, 57
  \item Merel, 4, 19, 22, 29, 30, 52
  \item Mestre, 14, 70
  \item Method of graphs, 82
  \item Minus volume, \textit{53}, \textit{65}, 68
  \item Minus-one quotient, 23, 32
  \item Modular curve, 80
  \item Modular degree, 50, \textit{72}
    \subitem and character group, \textit{78}
  \item Modular forms, 23, 27, 33, 34, 62
    \subitem and Atkin-Lehner involution, 24
    \subitem and BSD, 2
    \subitem associated complex torus, 32
    \subitem associated subtorus, 33
    \subitem congruences between, 6
    \subitem duality with modular symbols, 21
  \item Modular map, 50
  \item Modular symbols, 3, 19, \textit{20}, 28, 40, 59, 62
    \subitem computing, 35
    \subitem conversion to Manin symbols, 28
    \subitem duality with modular forms, 21
    \subitem finite presentation of, 27
    \subitem minus-one quotient of, 23
    \subitem new and old subspace of, \textit{25}
    \subitem plus-one quotient of, 23
    \subitem relations satisfied by, 20
  \item Monodromy pairing, 71, 76, 77, 82
  \item Motifs, 4, 33
  \item Mumford, 71

  \indexspace

  \item N\'eron model, 57, 70, \textit{71}, 73
    \subitem closed fiber of, 73
  \item New modular symbols, \textit{25}
  \item New subspace, 41

  \indexspace

  \item Old modular symbols, \textit{25}
  \item Old subspace, 41
  \item One-motif dual, 74
  \item Operators
    \subitem $*$-involution, \textit{23}
    \subitem Atkin-Lehner, \textit{23}
    \subitem Hecke, 22
  \item Optimal quotient, \textit{2}, \textit{34}, 51, 58, \textit{72}, 
		76, 80
    \subitem and semistable reduction, 74
    \subitem component groups of, 70
    \subitem dual map is injective, 72
    \subitem of $J_0(N)$, 80
    \subitem Tamagawa numbers of, 70
  \item Ordering of eigenforms, 44

  \indexspace

  \item Period integrals
    \subitem algorithm for computing, 60
  \item Period lattice, 61
  \item Period mapping, \textit{33}, 54, 65
    \subitem computation of, 64
    \subitem is injective, 33
  \item Petersson pairing, 32
  \item Plus-one quotient, \textit{23}, 32
  \item Polarization, 72
  \item Projective line modulo~$N$, \textit{26}
  \item Purely toric reduction, 8, 73--75, 80

  \indexspace

  \item Quaternion algebras, 82

  \indexspace

  \item Rational part of $L(A,j)$, 52
  \item Rational period mapping, \textit{47}, 64
  \item Raynaud, 74
  \item Real volume, \textit{53}, \textit{65}, 68
  \item Ribet, 4, 6, 13, 37, 70
  \item Rigid uniformization, \textit{73}

  \indexspace

  \item Saturated, 57
  \item Semistable reduction, 73, 74
    \subitem and uniformization, 73
  \item Shafarevich-Tate group, 3, 6, 83
    \subitem first invisible example, 13
    \subitem invisible elements of, 6
    \subitem visibility at higher level, 6
    \subitem visible part of, 5
  \item Shimura, 32, 34, 53
  \item Shimura-Taniyama conjecture, 2
  \item Snake lemma, 46, 51, 58, 72, 75, 76
  \item Special value $L(A,j)$
    \subitem computing, 64
    \subitem rational part of, 52
  \item Star involution, \textit{23}, 33
    \subitem and integration pairing, 23
    \subitem is well defined, 23
  \item Sturm bound, 38, 42
  \item Symmetric isogeny, 73, \textit{75}

  \indexspace

  \item T-shirt design, 46
  \item Table of
    \subitem $\Sha$ at prime level, 15
    \subitem big $L(A,1)/\Omega_A$, 85
    \subitem big component groups, 85
    \subitem CM elliptic curves of weight $>2$, 69
    \subitem component groups at low level, 84
    \subitem component groups at prime level, 87
    \subitem component groups of explanatory factors, 18
    \subitem component groups of quotients, 86
    \subitem explanatory factors, 17
    \subitem factorizations, 17
    \subitem new visible $\Sha$, 16
    \subitem odd invisible $\Sha_E$, 12
    \subitem volumes of higher dimensional abelian varieties, 69
    \subitem volumes of level one cusp forms, 68
  \item Tamagawa numbers, 3, 4, \textit{70}
  \item Tate curve
    \subitem and monodromy pairing, 77
    \subitem component group of, 77
    \subitem uniformization of, 74
  \item Toric part, 71
  \item Torsion subgroup, 3
    \subitem lower bounds on, \textit{50}
    \subitem upper bounds on, \textit{50}

  \indexspace

  \item Uniformization cross, 74, 75

  \indexspace

  \item Verrill, 61
  \item Visibility
    \subitem at higher level, 6
    \subitem existence theorem, 7
    \subitem in some Jacobian, 6
    \subitem of $\Sha$, \textit{5}

  \indexspace

  \item Weight of modular symbols, \textit{20}
  \item Winding element, 52, \textit{65}

\end{theindex}