\indexentry{Hecke operators}{1}
\indexentry{Hecke operators}{1}
\indexentry{Hecke operators}{1}
\indexentry{Birch and Swinnerton-Dyer conjecture|see{BSD conjecture}}{2}
\indexentry{Conjecture!Birch and Swinnerton-Dyer|see{BSD conjecture}}{2}
\indexentry{Shimura-Taniyama conjecture}{2}
\indexentry{Conjecture!Shimura and Taniyama}{2}
\indexentry{BSD conjecture!is still unknown}{2}
\indexentry{BSD conjecture}{2}
\indexentry{Congruences!and BSD conjecture}{2}
\indexentry{Modular forms!and BSD}{2}
\indexentry{BSD conjecture!in higher dimensions}{2}
\indexentry{Agashe}{2}
\indexentry{Mazur}{2}
\indexentry{Congruences!and BSD conjecture}{2}
\indexentry{BSD conjecture|textit}{2}
\indexentry{Jacobian}{2}
\indexentry{Mazur}{2}
\indexentry{Optimal quotient|textit}{2}
\indexentry{BSD conjecture!statement of}{3}
\indexentry{Shafarevich-Tate group}{3}
\indexentry{Local-to-global principle}{3}
\indexentry{Tamagawa numbers}{3}
\indexentry{Agashe}{3}
\indexentry{Modular symbols}{3}
\indexentry{Mazur}{3}
\indexentry{Manin constant}{3}
\indexentry{Lattice index}{3}
\indexentry{Hecke algebra}{3}
\indexentry{Manin constant}{3}
\indexentry{Torsion subgroup}{3}
\indexentry{Tamagawa numbers}{4}
\indexentry{Mazur}{4}
\indexentry{Mazur}{4}
\indexentry{Ribet}{4}
\indexentry{Merel}{4}
\indexentry{Tamagawa numbers}{4}
\indexentry{Motifs}{4}
\indexentry{Conjecture!Bloch and Kato}{4}
\indexentry{Bloch and Kato conjecture}{4}
\indexentry{BSD conjecture!generalization of}{4}
\indexentry{Tamagawa numbers}{4}
\indexentry{Motifs}{4}
\indexentry{Euler system}{4}
\indexentry{BSD conjecture!verification of}{4}
\indexentry{Heegner points}{5}
\indexentry{Bound of!Kolyvagin}{5}
\indexentry{Bound of!Kato}{5}
\indexentry{Euler system}{5}
\indexentry{BSD conjecture!and $\Sha$}{5}
\indexentry{Mazur}{5}
\indexentry{Visibility!of $\Sha$|textit}{5}
\indexentry{Shafarevich-Tate group!visible part of}{5}
\indexentry{Mazur}{5}
\indexentry{Mazur}{5}
\indexentry{Modular forms!congruences between}{6}
\indexentry{Congruences!and lower bounds on $\Sha$}{6}
\indexentry{Shafarevich-Tate group!invisible elements of}{6}
\indexentry{Invisible elements of $\Sha$}{6}
\indexentry{BSD conjecture!predicts invisible elements}{6}
\indexentry{Shafarevich-Tate group!visibility at higher level}{6}
\indexentry{Visibility!at higher level}{6}
\indexentry{Ribet}{6}
\indexentry{Visibility!in some Jacobian}{6}
\indexentry{Jacobian!visibility in}{6}
\indexentry{Congruences!between elliptic curves}{6}
\indexentry{Kani}{6}
\indexentry{Conjecture!Kani}{6}
\indexentry{Shafarevich-Tate group}{6}
\indexentry{Jacobian}{6}
\indexentry{Visibility!existence theorem}{7}
\indexentry{Component group!geometric}{7}
\indexentry{Purely toric reduction}{8}
\indexentry{BSD conjecture!predicted order of $\Sha$}{10}
\indexentry{Explanatory factor}{10}
\indexentry{Invisible elements of $\Sha$}{10}
\indexentry{Table of!odd invisible $\Sha_E$}{12}
\indexentry{BSD conjecture}{12}
\indexentry{Mazur}{13}
\indexentry{Logan}{13}
\indexentry{Shafarevich-Tate group!first invisible example}{13}
\indexentry{Ribet}{13}
\indexentry{Mestre}{14}
\indexentry{Table of!$\Sha$ at prime level}{15}
\indexentry{Table of!new visible $\Sha$}{16}
\indexentry{Table of!explanatory factors}{17}
\indexentry{Table of!factorizations}{17}
\indexentry{Table of!component groups of explanatory factors}{18}
\indexentry{Modular symbols}{19}
\indexentry{Cremona}{19}
\indexentry{Merel}{19}
\indexentry{Merel}{19}
\indexentry{Level of modular symbols|textit}{20}
\indexentry{Weight of modular symbols|textit}{20}
\indexentry{Dirichlet character|textit}{20}
\indexentry{Modular symbols!relations satisfied by}{20}
\indexentry{Modular symbols|textit}{20}
\indexentry{Cuspidal modular symbols|textit}{21}
\indexentry{Boundary modular symbols|textit}{21}
\indexentry{Cuspidal modular symbols|textit}{21}
\indexentry{Modular symbols!duality with modular forms}{21}
\indexentry{Modular forms!duality with modular symbols}{21}
\indexentry{Integration pairing}{21}
\indexentry{Cusp forms|textit}{21}
\indexentry{Cusp forms!antiholomorphic|textit}{21}
\indexentry{Antiholomorphic cusp forms|textit}{21}
\indexentry{Merel}{22}
\indexentry{Hecke operators}{22}
\indexentry{Operators!Hecke}{22}
\indexentry{Hecke algebra|textit}{22}
\indexentry{Hecke operators!respect pairing}{22}
\indexentry{Star involution|textit}{23}
\indexentry{Operators!$*$-involution|textit}{23}
\indexentry{Star involution!is well defined}{23}
\indexentry{Modular forms}{23}
\indexentry{Fourier coefficients}{23}
\indexentry{Modular forms}{23}
\indexentry{Star involution!and integration pairing}{23}
\indexentry{Plus-one quotient|textit}{23}
\indexentry{Modular symbols!plus-one quotient of}{23}
\indexentry{Modular symbols!minus-one quotient of}{23}
\indexentry{Minus-one quotient}{23}
\indexentry{Operators!Atkin-Lehner|textit}{23}
\indexentry{Atkin-Lehner involution|textit}{23}
\indexentry{Modular forms!and Atkin-Lehner involution}{24}
\indexentry{Atkin-Lehner involution!and integration pairing}{24}
\indexentry{Degeneracy maps}{24}
\indexentry{Dirichlet character}{24}
\indexentry{New modular symbols|textit}{25}
\indexentry{Old modular symbols|textit}{25}
\indexentry{Modular symbols!new and old subspace of|textit}{25}
\indexentry{Coset representatives}{26}
\indexentry{Projective line modulo~$N$|textit}{26}
\indexentry{Algorithm for computing!coset representatives}{26}
\indexentry{Degeneracy maps!compatibility}{27}
\indexentry{Modular forms}{27}
\indexentry{Manin symbols}{27}
\indexentry{Modular symbols!finite presentation of}{27}
\indexentry{Manin symbols|textit}{27}
\indexentry{Manin's trick}{28}
\indexentry{Manin symbols!and Manin's trick}{28}
\indexentry{Manin symbols!conversion to modular symbols}{28}
\indexentry{Modular symbols!conversion to Manin symbols}{28}
\indexentry{Atkin-Lehner involution}{28}
\indexentry{Manin's trick|textit}{28}
\indexentry{Manin!trick of|textit}{28}
\indexentry{Continued fractions}{28}
\indexentry{Manin symbols}{28}
\indexentry{Modular symbols}{28}
\indexentry{Hecke operators!on Manin symbols}{29}
\indexentry{Manin symbols!and Hecke operators}{29}
\indexentry{Merel}{29}
\indexentry{Cremona}{29}
\indexentry{Manin symbols!and cuspidal subspace}{30}
\indexentry{Manin symbols!and boundary space}{30}
\indexentry{Cuspidal modular symbols!and Manin symbols}{30}
\indexentry{Boundary modular symbols!and Manin symbols}{30}
\indexentry{Merel}{30}
\indexentry{Manin symbols}{30}
\indexentry{Boundary map}{30}
\indexentry{Cusps!and boundary map}{30}
\indexentry{Algorithm for computing!cusps}{31}
\indexentry{Cremona}{31}
\indexentry{Algorithm for computing!equivalent cusps}{31}
\indexentry{Cusps!criterion for vanishing}{32}
\indexentry{Dirichlet character!and cusps}{32}
\indexentry{Plus-one quotient}{32}
\indexentry{Minus-one quotient}{32}
\indexentry{Complex torus}{33}
\indexentry{Modular forms!associated complex torus}{33}
\indexentry{Dirichlet character}{33}
\indexentry{Hecke operators}{33}
\indexentry{Shimura}{33}
\indexentry{Petersson pairing}{33}
\indexentry{Conjecture!Bloch and Kato}{33}
\indexentry{Bloch and Kato conjecture}{33}
\indexentry{Motifs}{33}
\indexentry{Modular forms}{33}
\indexentry{Hecke algebra!and integration pairing}{33}
\indexentry{Integration pairing!and complex torus}{33}
\indexentry{Period mapping|textit}{33}
\indexentry{Star involution}{33}
\indexentry{Period mapping!is injective}{33}
\indexentry{Complex torus!dual of}{34}
\indexentry{Modular forms!associated subtorus}{34}
\indexentry{Hecke algebra}{34}
\indexentry{Complex torus!in weight two}{34}
\indexentry{Shimura}{34}
\indexentry{Modular forms}{34}
\indexentry{Optimal quotient|textit}{34}
\indexentry{BSD conjecture!and $L(A,j)/\Omega_j$}{35}
\indexentry{BSD conjecture}{35}
\indexentry{Manin}{35}
\indexentry{Modular symbols!computing}{35}
\indexentry{Dirichlet character}{36}
\indexentry{Algorithm for computing!space of modular symbols}{36}
\indexentry{Dirichlet character}{36}
\indexentry{Manin symbols}{36}
\indexentry{Manin}{36}
\indexentry{Manin symbols}{36}
\indexentry{Manin symbols}{36}
\indexentry{Algorithm for computing!integral modular symbols}{37}
\indexentry{Dirichlet character}{37}
\indexentry{Hecke algebra!computation of}{37}
\indexentry{Hecke algebra!generators as module}{37}
\indexentry{Hecke operators}{37}
\indexentry{Ribet}{37}
\indexentry{Agashe}{37}
\indexentry{Hecke algebra!generators as ring}{37}
\indexentry{Hecke operators}{37}
\indexentry{Bound of!Sturm}{38}
\indexentry{Sturm bound}{38}
\indexentry{Dirichlet character}{38}
\indexentry{Dirichlet character!conductor of|textit}{39}
\indexentry{Conductor of Dirichlet character|textit}{39}
\indexentry{Dimension of $S_k(N,\eps)$}{40}
\indexentry{Cremona}{40}
\indexentry{Modular symbols}{40}
\indexentry{Degeneracy maps}{41}
\indexentry{Hecke operators!and degeneracy maps}{41}
\indexentry{New subspace}{41}
\indexentry{Old subspace}{41}
\indexentry{Algorithm for computing!decomposition of space of modular symbols}{41}
\indexentry{Hecke operators}{41}
\indexentry{Hecke operators}{41}
\indexentry{Sturm bound}{42}
\indexentry{Bound of!Sturm}{42}
\indexentry{Hecke operators!computation on subspace of dual}{42}
\indexentry{Algorithm for computing!Hecke operators on the dual}{42}
\indexentry{Manin symbols}{42}
\indexentry{Algorithm for computing!Hecke operators on the dual}{43}
\indexentry{Algorithm for computing!an eigenvector}{43}
\indexentry{Eigenforms!computing}{44}
\indexentry{Algorithm for computing!eigenvalues}{44}
\indexentry{Hecke operators}{44}
\indexentry{Hecke operators}{44}
\indexentry{Hecke operators}{44}
\indexentry{Manin symbols}{44}
\indexentry{Heilbronn matrices}{44}
\indexentry{Ordering of eigenforms}{44}
\indexentry{Eigenforms!sorting and labeling}{44}
\indexentry{Atkin-Lehner involution!and ordering eigenforms}{45}
\indexentry{Congruences!computing}{45}
\indexentry{Snake lemma}{46}
\indexentry{T-shirt design}{46}
\indexentry{Congruences!between $q$-expansions}{47}
\indexentry{Algorithm for computing!congruences}{47}
\indexentry{Congruences!computed using homology}{47}
\indexentry{Fourier coefficients}{47}
\indexentry{Hecke algebra!and congruences}{47}
\indexentry{Hecke algebra!and rational period mapping}{47}
\indexentry{Rational period mapping|textit}{47}
\indexentry{Algorithm for computing!rational period mapping}{48}
\indexentry{Hecke operators}{48}
\indexentry{Cuspidal points}{49}
\indexentry{Manin}{49}
\indexentry{Algorithm for computing!cuspidal subgroup}{49}
\indexentry{Hecke algebra!and cuspidal subgroup}{49}
\indexentry{Manin}{49}
\indexentry{Torsion subgroup!lower bounds on|textit}{50}
\indexentry{Torsion subgroup!upper bounds on|textit}{50}
\indexentry{Hecke algebra!bounds torsion}{50}
\indexentry{Eichler-Shimura relation}{50}
\indexentry{Modular degree}{50}
\indexentry{Modular map}{50}
\indexentry{Modular degree}{50}
\indexentry{Optimal quotient}{51}
\indexentry{Algorithm for computing!modular kernel}{51}
\indexentry{Hecke algebra}{51}
\indexentry{Snake lemma}{51}
\indexentry{Rational part of $L(A,j)$}{52}
\indexentry{Special value $L(A,j)$!rational part of}{52}
\indexentry{Agashe}{52}
\indexentry{$L$-series|textit}{52}
\indexentry{Hecke}{52}
\indexentry{$L$-series}{52}
\indexentry{Mazur}{52}
\indexentry{Merel}{52}
\indexentry{Winding element}{52}
\indexentry{Real volume|textit}{53}
\indexentry{Minus volume|textit}{53}
\indexentry{Agashe}{53}
\indexentry{Cremona}{53}
\indexentry{Manin}{53}
\indexentry{Shimura}{53}
\indexentry{Algorithm for computing!rational part of $L(A,j)$}{53}
\indexentry{BSD conjecture!and $\Omega_A$}{53}
\indexentry{Lattice|textit}{54}
\indexentry{Lattice index|textit}{54}
\indexentry{Index of lattices|textit}{54}
\indexentry{Period mapping}{54}
\indexentry{Hecke operators}{54}
\indexentry{Hecke operators}{56}
\indexentry{Manin!comment on BSD}{56}
\indexentry{BSD conjecture!Manin's remarks on}{56}
\indexentry{Agashe}{57}
\indexentry{Manin constant}{57}
\indexentry{Hecke algebra}{57}
\indexentry{Mazur}{57}
\indexentry{Manin constant}{57}
\indexentry{Mazur}{57}
\indexentry{Manin constant}{57}
\indexentry{N\'eron model}{57}
\indexentry{Mazur}{57}
\indexentry{Saturated}{57}
\indexentry{Snake lemma}{58}
\indexentry{$q$-expansion principle}{58}
\indexentry{Manin constant!conjecture about|textit}{58}
\indexentry{Conjecture!that Manin constant equals $1$|textit}{58}
\indexentry{Optimal quotient}{58}
\indexentry{Conjecture!Agashe and Stein}{58}
\indexentry{Agashe!conjecture of}{58}
\indexentry{Agashe}{58}
\indexentry{Manin constant}{58}
\indexentry{Analytic invariants}{59}
\indexentry{Complex torus}{59}
\indexentry{Integration pairing}{59}
\indexentry{Extended modular symbols|textit}{59}
\indexentry{Modular symbols}{59}
\indexentry{Integration pairing!and extended modular symbols}{59}
\indexentry{Extended modular symbols}{60}
\indexentry{Algorithm for computing!period integrals}{60}
\indexentry{Period integrals!algorithm for computing}{60}
\indexentry{Period lattice}{61}
\indexentry{Verrill}{61}
\indexentry{Manin}{61}
\indexentry{Modular symbols}{62}
\indexentry{Modular forms}{62}
\indexentry{Atkin-Lehner involution}{63}
\indexentry{Period mapping!computation of}{64}
\indexentry{Rational period mapping}{64}
\indexentry{Special value $L(A,j)$!computing}{64}
\indexentry{Winding element|textit}{65}
\indexentry{Period mapping}{65}
\indexentry{Hecke algebra}{65}
\indexentry{Real volume|textit}{65}
\indexentry{Minus volume|textit}{65}
\indexentry{Algorithm for computing!real and minus volumes}{65}
\indexentry{BSD conjecture!and $\Omega_A$}{66}
\indexentry{Component group!archimedean|textit}{66}
\indexentry{Jacobian!of genus-two curve}{67}
\indexentry{Genus-two curves}{67}
\indexentry{Manin constant}{67}
\indexentry{Manin constant}{67}
\indexentry{Table of!volumes of level one cusp forms}{68}
\indexentry{Real volume}{68}
\indexentry{Minus volume}{68}
\indexentry{Table of!CM elliptic curves of weight $>2$}{69}
\indexentry{Table of!volumes of higher dimensional abelian varieties}{69}
\indexentry{Component group}{70}
\indexentry{Optimal quotient!component groups of}{70}
\indexentry{BSD conjecture!and component groups}{70}
\indexentry{Tamagawa numbers|textit}{70}
\indexentry{Optimal quotient!Tamagawa numbers of}{70}
\indexentry{Component group!rational points of}{70}
\indexentry{N\'eron model}{70}
\indexentry{Jacobian!of $X_0(N)$}{70}
\indexentry{Ribet}{70}
\indexentry{Mestre}{70}
\indexentry{N\'eron model|textit}{71}
\indexentry{Component group|textit}{71}
\indexentry{Jacobian!of $X_0(N)$}{71}
\indexentry{Grothendieck}{71}
\indexentry{Mumford}{71}
\indexentry{Jacobian!semistable}{71}
\indexentry{Component group}{71}
\indexentry{Monodromy pairing}{71}
\indexentry{Canonical polarization}{71}
\indexentry{Character group of torus}{71}
\indexentry{Toric part}{71}
\indexentry{Monodromy pairing}{71}
\indexentry{Snake lemma}{72}
\indexentry{Jacobian}{72}
\indexentry{Optimal quotient|textit}{72}
\indexentry{Optimal quotient!dual map is injective}{72}
\indexentry{Cartier dual}{72}
\indexentry{Polarization}{72}
\indexentry{Modular degree|textit}{72}
\indexentry{N\'eron model!closed fiber of}{73}
\indexentry{Closed fiber of N\'eron model|textit}{73}
\indexentry{N\'eron model}{73}
\indexentry{\'Etale group scheme|textit}{73}
\indexentry{Purely toric reduction}{73}
\indexentry{Semistable reduction}{73}
\indexentry{Character group of torus|textit}{73}
\indexentry{Rigid uniformization|textit}{73}
\indexentry{Semistable reduction!and uniformization}{73}
\indexentry{Purely toric reduction}{73}
\indexentry{Symmetric isogeny}{73}
\indexentry{Character group of torus}{73}
\indexentry{Raynaud}{74}
\indexentry{Uniformization cross}{74}
\indexentry{One-motif dual}{74}
\indexentry{Tate curve!uniformization of}{74}
\indexentry{Optimal quotient!and semistable reduction}{74}
\indexentry{Semistable reduction}{74}
\indexentry{Purely toric reduction}{74}
\indexentry{Snake lemma}{75}
\indexentry{Purely toric reduction}{75}
\indexentry{Uniformization cross}{75}
\indexentry{Symmetric isogeny|textit}{75}
\indexentry{Snake lemma}{76}
\indexentry{Optimal quotient}{76}
\indexentry{Jacobian!semistable}{76}
\indexentry{Character group of torus}{76}
\indexentry{Monodromy pairing}{76}
\indexentry{Jacobian!is principally polarized}{76}
\indexentry{Tate curve!and monodromy pairing}{76}
\indexentry{Tate curve!component group of}{76}
\indexentry{Monodromy pairing}{77}
\indexentry{Modular degree!and character group|textit}{78}
\indexentry{Component group!and character group|textit}{78}
\indexentry{Optimal quotient!of $J_0(N)$}{79}
\indexentry{Modular curve}{79}
\indexentry{Jacobian!of $X_0(N)$}{80}
\indexentry{Hecke algebra|textit}{80}
\indexentry{Optimal quotient}{80}
\indexentry{Purely toric reduction}{80}
\indexentry{Integration pairing}{80}
\indexentry{Atkin-Lehner involution}{81}
\indexentry{Method of graphs}{81}
\indexentry{Quaternion algebras}{81}
\indexentry{Monodromy pairing}{81}
\indexentry{Conjecture!about modular degree}{82}
\indexentry{Hecke algebra}{82}
\indexentry{Conjecture!refined Eisenstein}{82}
\indexentry{BSD conjecture!predicts large component groups}{83}
\indexentry{Shafarevich-Tate group}{83}
\indexentry{Table of!component groups at low level}{84}
\indexentry{Component group!table of}{84}
\indexentry{Table of!big $L(A,1)/\Omega_A$}{85}
\indexentry{Component group!table of}{85}
\indexentry{Table of!big component groups}{85}
\indexentry{Component group!table of}{85}
\indexentry{Table of!component groups of quotients}{86}
\indexentry{Component group!table of}{86}
\indexentry{Table of!component groups at prime level}{87}
\indexentry{Component group!table of}{87}