\contentsline {section}{{\bf Preface}}{1}{chapter*.2}
\contentsline {chapter}{\numberline {1}The Main Objects}{3}{chapter.1}
\contentsline {section}{\numberline {1.1}Torsion points on elliptic curves}{3}{section.1.1}
\contentsline {subsection}{\numberline {1.1.1}The Tate module}{3}{subsection.1.1.1}
\contentsline {section}{\numberline {1.2}Galois representations}{4}{section.1.2}
\contentsline {section}{\numberline {1.3}Modular forms}{5}{section.1.3}
\contentsline {section}{\numberline {1.4}Hecke operators}{6}{section.1.4}
\contentsline {chapter}{\numberline {2}Modular Representations and Algebraic Curves}{7}{chapter.2}
\contentsline {section}{\numberline {2.1}Modular forms and Arithmetic}{7}{section.2.1}
\contentsline {section}{\numberline {2.2}Characters}{9}{section.2.2}
\contentsline {section}{\numberline {2.3}Parity conditions}{9}{section.2.3}
\contentsline {section}{\numberline {2.4}Conjectures of Serre (mod $\ell $ version)}{9}{section.2.4}
\contentsline {section}{\numberline {2.5}General remarks on mod $p$ Galois representations}{10}{section.2.5}
\contentsline {section}{\numberline {2.6}Serre's conjecture}{11}{section.2.6}
\contentsline {section}{\numberline {2.7}Wiles's perspective}{11}{section.2.7}
\contentsline {chapter}{\numberline {3}Modular Forms of Level 1}{13}{chapter.3}
\contentsline {section}{\numberline {3.1}The Definition}{13}{section.3.1}
\contentsline {section}{\numberline {3.2}Some examples and conjectures}{14}{section.3.2}
\contentsline {section}{\numberline {3.3}Modular forms as functions on lattices}{15}{section.3.3}
\contentsline {section}{\numberline {3.4}Hecke operators}{17}{section.3.4}
\contentsline {subsection}{\numberline {3.4.1}Relations Between Hecke Operators}{18}{subsection.3.4.1}
\contentsline {section}{\numberline {3.5}Hecke operators directly on $q$-expansions}{19}{section.3.5}
\contentsline {subsection}{\numberline {3.5.1}Explicit description of sublattices}{20}{subsection.3.5.1}
\contentsline {subsection}{\numberline {3.5.2}Hecke operators on $q$-expansions}{21}{subsection.3.5.2}
\contentsline {subsection}{\numberline {3.5.3}The Hecke algebra and eigenforms}{22}{subsection.3.5.3}
\contentsline {subsection}{\numberline {3.5.4}Examples}{23}{subsection.3.5.4}
\contentsline {section}{\numberline {3.6}Two Conjectures about Hecke operators on level\nobreakspace {}$1$ modular forms}{24}{section.3.6}
\contentsline {subsection}{\numberline {3.6.1}Maeda's conjecture}{24}{subsection.3.6.1}
\contentsline {subsection}{\numberline {3.6.2}The Gouvea-Mazur conjecture}{24}{subsection.3.6.2}
\contentsline {section}{\numberline {3.7}An Algorithm for computing characteristic polynomials of Hecke operators}{25}{section.3.7}
\contentsline {subsection}{\numberline {3.7.1}Review of basic facts about modular forms}{26}{subsection.3.7.1}
\contentsline {subsection}{\numberline {3.7.2}The Naive approach}{26}{subsection.3.7.2}
\contentsline {subsection}{\numberline {3.7.3}The Eigenform method}{27}{subsection.3.7.3}
\contentsline {subsection}{\numberline {3.7.4}How to write down an eigenvector over an extension field}{28}{subsection.3.7.4}
\contentsline {subsection}{\numberline {3.7.5}Simple example: weight $36$, $p=3$}{29}{subsection.3.7.5}
\contentsline {chapter}{\numberline {4}Duality, Rationality, and Integrality}{31}{chapter.4}
\contentsline {section}{\numberline {4.1}Modular forms for $\SL _2(\mathbf {Z})$ and Eisenstein series}{31}{section.4.1}
\contentsline {section}{\numberline {4.2}Pairings between Hecke algebras and modular forms}{32}{section.4.2}
\contentsline {section}{\numberline {4.3}Eigenforms}{33}{section.4.3}
\contentsline {section}{\numberline {4.4}Integrality}{34}{section.4.4}
\contentsline {section}{\numberline {4.5}A Result from Victor Miller's thesis}{34}{section.4.5}
\contentsline {section}{\numberline {4.6}The Petersson inner product}{35}{section.4.6}
\contentsline {chapter}{\numberline {5}Analytic Theory of Modular Curves}{39}{chapter.5}
\contentsline {section}{\numberline {5.1}The Modular group}{39}{section.5.1}
\contentsline {subsection}{\numberline {5.1.1}The Upper half plane}{39}{subsection.5.1.1}
\contentsline {section}{\numberline {5.2}Points on modular curves parameterize elliptic curves with extra structure}{40}{section.5.2}
\contentsline {section}{\numberline {5.3}The Genus of $X(N)$}{43}{section.5.3}
\contentsline {chapter}{\numberline {6}Modular Curves}{47}{chapter.6}
\contentsline {section}{\numberline {6.1}Cusp Forms}{47}{section.6.1}
\contentsline {section}{\numberline {6.2}Modular curves}{47}{section.6.2}
\contentsline {section}{\numberline {6.3}Classifying $\Gamma (N)$-structures}{48}{section.6.3}
\contentsline {section}{\numberline {6.4}More on integral Hecke operators}{49}{section.6.4}
\contentsline {section}{\numberline {6.5}Complex conjugation}{49}{section.6.5}
\contentsline {section}{\numberline {6.6}Isomorphism in the real case}{49}{section.6.6}
\contentsline {section}{\numberline {6.7}The Eichler-Shimura isomorphism}{50}{section.6.7}
\contentsline {chapter}{\numberline {7}Modular Symbols}{53}{chapter.7}
\contentsline {section}{\numberline {7.1}Modular symbols}{53}{section.7.1}
\contentsline {section}{\numberline {7.2}Manin symbols}{54}{section.7.2}
\contentsline {subsection}{\numberline {7.2.1}Using continued fractions to obtain surjectivity}{55}{subsection.7.2.1}
\contentsline {subsection}{\numberline {7.2.2}Triangulating $X(G)$ to obtain injectivity}{56}{subsection.7.2.2}
\contentsline {section}{\numberline {7.3}Hecke operators}{59}{section.7.3}
\contentsline {section}{\numberline {7.4}Modular symbols and rational homology}{60}{section.7.4}
\contentsline {section}{\numberline {7.5}Special values of $L$-functions}{61}{section.7.5}
\contentsline {chapter}{\numberline {8}Modular Forms of Higher Level}{65}{chapter.8}
\contentsline {section}{\numberline {8.1}Modular Forms on $\Gamma _1(N)$}{65}{section.8.1}
\contentsline {section}{\numberline {8.2}Diamond bracket and Hecke operators}{66}{section.8.2}
\contentsline {subsection}{\numberline {8.2.1}Diamond bracket operators}{66}{subsection.8.2.1}
\contentsline {subsection}{\numberline {8.2.2}Hecke operators on $q$-expansions}{68}{subsection.8.2.2}
\contentsline {section}{\numberline {8.3}Old and new subspaces}{68}{section.8.3}
\contentsline {chapter}{\numberline {9}Newforms and Euler Products}{71}{chapter.9}
\contentsline {section}{\numberline {9.1}Atkin-Lehner-Li theory}{71}{section.9.1}
\contentsline {section}{\numberline {9.2}The $U_p$ operator}{75}{section.9.2}
\contentsline {subsection}{\numberline {9.2.1}A Connection with Galois representations}{76}{subsection.9.2.1}
\contentsline {subsection}{\numberline {9.2.2}When is $U_p$ semisimple?}{77}{subsection.9.2.2}
\contentsline {subsection}{\numberline {9.2.3}An Example of non-semisimple $U_p$}{77}{subsection.9.2.3}
\contentsline {section}{\numberline {9.3}The Cusp forms are free of rank $1$ over $\mathbf {T}_\mathbf {C}$}{77}{section.9.3}
\contentsline {subsection}{\numberline {9.3.1}Level $1$}{77}{subsection.9.3.1}
\contentsline {subsection}{\numberline {9.3.2}General level}{78}{subsection.9.3.2}
\contentsline {section}{\numberline {9.4}Decomposing the anemic Hecke algebra}{80}{section.9.4}
\contentsline {chapter}{\numberline {10}Some Explicit Genus Computations}{83}{chapter.10}
\contentsline {section}{\numberline {10.1}Computing the dimension of $S_2(\Gamma )$}{83}{section.10.1}
\contentsline {section}{\numberline {10.2}Application of Riemann-Hurwitz}{84}{section.10.2}
\contentsline {section}{\numberline {10.3}The Genus of $X(N)$}{85}{section.10.3}
\contentsline {section}{\numberline {10.4}The Genus of $X_0(N)$, for $N$ prime}{86}{section.10.4}
\contentsline {section}{\numberline {10.5}Modular forms mod $p$}{86}{section.10.5}
\contentsline {chapter}{\numberline {11}The Field of Moduli}{89}{chapter.11}
\contentsline {section}{\numberline {11.1}Algebraic definition of $X(N)$}{89}{section.11.1}
\contentsline {section}{\numberline {11.2}Digression on moduli}{90}{section.11.2}
\contentsline {section}{\numberline {11.3}When is $\rho _E$ surjective?}{91}{section.11.3}
\contentsline {section}{\numberline {11.4}Observations}{92}{section.11.4}
\contentsline {section}{\numberline {11.5}A descent problem}{93}{section.11.5}
\contentsline {section}{\numberline {11.6}Second look at the descent exercise}{94}{section.11.6}
\contentsline {section}{\numberline {11.7}Action of $\GL _2$}{95}{section.11.7}
\contentsline {chapter}{\numberline {12}Hecke Operators as Correspondences}{97}{chapter.12}
\contentsline {section}{\numberline {12.1}The Definition}{97}{section.12.1}
\contentsline {section}{\numberline {12.2}Maps induced by correspondences}{99}{section.12.2}
\contentsline {section}{\numberline {12.3}Induced maps on Jacobians of curves}{100}{section.12.3}
\contentsline {section}{\numberline {12.4}More on Hecke operators}{100}{section.12.4}
\contentsline {section}{\numberline {12.5}Hecke operators acting on Jacobians}{101}{section.12.5}
\contentsline {subsection}{\numberline {12.5.1}The Albanese Map}{102}{subsection.12.5.1}
\contentsline {subsection}{\numberline {12.5.2}The Hecke algebra}{103}{subsection.12.5.2}
\contentsline {section}{\numberline {12.6}The Eichler-Shimura relation}{103}{section.12.6}
\contentsline {section}{\numberline {12.7}Applications of the Eichler-Shimura relation}{107}{section.12.7}
\contentsline {subsection}{\numberline {12.7.1}The Characteristic polynomial of Frobenius}{107}{subsection.12.7.1}
\contentsline {subsection}{\numberline {12.7.2}The Cardinality of $J_0(N)(\mathbf {F}_{p})$}{109}{subsection.12.7.2}
\contentsline {chapter}{\numberline {13}Abelian Varieties}{111}{chapter.13}
\contentsline {section}{\numberline {13.1}Abelian varieties}{111}{section.13.1}
\contentsline {section}{\numberline {13.2}Complex tori}{112}{section.13.2}
\contentsline {subsection}{\numberline {13.2.1}Homomorphisms}{112}{subsection.13.2.1}
\contentsline {subsection}{\numberline {13.2.2}Isogenies}{114}{subsection.13.2.2}
\contentsline {subsection}{\numberline {13.2.3}Endomorphisms}{115}{subsection.13.2.3}
\contentsline {section}{\numberline {13.3}Abelian varieties as complex tori}{115}{section.13.3}
\contentsline {subsection}{\numberline {13.3.1}Hermitian and Riemann forms}{116}{subsection.13.3.1}
\contentsline {subsection}{\numberline {13.3.2}Complements, quotients, and semisimplicity of the endomorphism algebra}{117}{subsection.13.3.2}
\contentsline {subsection}{\numberline {13.3.3}Theta functions}{119}{subsection.13.3.3}
\contentsline {section}{\numberline {13.4}A Summary of duality and polarizations}{119}{section.13.4}
\contentsline {subsection}{\numberline {13.4.1}Sheaves}{120}{subsection.13.4.1}
\contentsline {subsection}{\numberline {13.4.2}The Picard group}{120}{subsection.13.4.2}
\contentsline {subsection}{\numberline {13.4.3}The Dual as a complex torus}{120}{subsection.13.4.3}
\contentsline {subsection}{\numberline {13.4.4}The N\'eron-Severi group and polarizations}{121}{subsection.13.4.4}
\contentsline {subsection}{\numberline {13.4.5}The Dual is functorial}{121}{subsection.13.4.5}
\contentsline {section}{\numberline {13.5}Jacobians of curves}{121}{section.13.5}
\contentsline {subsection}{\numberline {13.5.1}Divisors on curves and linear equivalence}{122}{subsection.13.5.1}
\contentsline {subsection}{\numberline {13.5.2}Algebraic definition of the Jacobian}{123}{subsection.13.5.2}
\contentsline {subsection}{\numberline {13.5.3}The Abel-Jacobi theorem}{124}{subsection.13.5.3}
\contentsline {subsection}{\numberline {13.5.4}Every abelian variety is a quotient of a Jacobian}{125}{subsection.13.5.4}
\contentsline {section}{\numberline {13.6}N\'eron models}{127}{section.13.6}
\contentsline {subsection}{\numberline {13.6.1}What are N\'eron models?}{127}{subsection.13.6.1}
\contentsline {subsection}{\numberline {13.6.2}The Birch and Swinnerton-Dyer conjecture and N\'eron models}{129}{subsection.13.6.2}
\contentsline {subsection}{\numberline {13.6.3}Functorial properties of Neron models}{131}{subsection.13.6.3}
\contentsline {chapter}{\numberline {14}Abelian Varieties Attached to Modular Forms}{133}{chapter.14}
\contentsline {section}{\numberline {14.1}Decomposition of the Hecke algebra}{133}{section.14.1}
\contentsline {subsection}{\numberline {14.1.1}The Dimension of the algebras $L_f$}{134}{subsection.14.1.1}
\contentsline {section}{\numberline {14.2}Decomposition of $J_1(N)$}{135}{section.14.2}
\contentsline {subsection}{\numberline {14.2.1}Aside: intersections and congruences}{136}{subsection.14.2.1}
\contentsline {section}{\numberline {14.3}Galois representations attached to $A_f$}{137}{section.14.3}
\contentsline {subsection}{\numberline {14.3.1}The Weil pairing}{138}{subsection.14.3.1}
\contentsline {subsection}{\numberline {14.3.2}The Determinant}{140}{subsection.14.3.2}
\contentsline {section}{\numberline {14.4}Remarks about the modular polarization}{141}{section.14.4}
\contentsline {chapter}{\numberline {15}Modularity of Abelian Varieties}{143}{chapter.15}
\contentsline {section}{\numberline {15.1}Modularity over\nobreakspace {}$\mathbf {Q}$}{143}{section.15.1}
\contentsline {section}{\numberline {15.2}Modularity of elliptic curves over $\overline {\mathbf {Q}}$}{146}{section.15.2}
\contentsline {section}{\numberline {15.3}Modularity of abelian varieties over $\overline {\mathbf {Q}}$}{146}{section.15.3}
\contentsline {chapter}{\numberline {16}$L$-functions}{149}{chapter.16}
\contentsline {section}{\numberline {16.1}$L$-functions attached to modular forms}{149}{section.16.1}
\contentsline {subsection}{\numberline {16.1.1}Analytic continuation and functional equations}{150}{subsection.16.1.1}
\contentsline {subsection}{\numberline {16.1.2}A Conjecture about nonvanishing of $L(f,k/2)$}{152}{subsection.16.1.2}
\contentsline {subsection}{\numberline {16.1.3}Euler products}{152}{subsection.16.1.3}
\contentsline {subsection}{\numberline {16.1.4}Visualizing $L$-function}{153}{subsection.16.1.4}
\contentsline {chapter}{\numberline {17}The Birch and Swinnerton-Dyer Conjecture}{155}{chapter.17}
\contentsline {section}{\numberline {17.1}The Rank conjecture}{155}{section.17.1}
\contentsline {section}{\numberline {17.2}Refined rank zero conjecture}{157}{section.17.2}
\contentsline {subsection}{\numberline {17.2.1}The Number of real components}{158}{subsection.17.2.1}
\contentsline {subsection}{\numberline {17.2.2}The Manin index}{158}{subsection.17.2.2}
\contentsline {subsection}{\numberline {17.2.3}The Real volume $\Omega _A$}{159}{subsection.17.2.3}
\contentsline {subsection}{\numberline {17.2.4}The Period mapping}{160}{subsection.17.2.4}
\contentsline {subsection}{\numberline {17.2.5}The Manin-Drinfeld theorem}{160}{subsection.17.2.5}
\contentsline {subsection}{\numberline {17.2.6}The Period lattice}{160}{subsection.17.2.6}
\contentsline {subsection}{\numberline {17.2.7}The Special value $L(A,1)$}{161}{subsection.17.2.7}
\contentsline {subsection}{\numberline {17.2.8}Rationality of $L(A,1)/\Omega _A$}{161}{subsection.17.2.8}
\contentsline {section}{\numberline {17.3}General refined conjecture}{163}{section.17.3}
\contentsline {section}{\numberline {17.4}The Conjecture for non-modular abelian varieties}{163}{section.17.4}
\contentsline {section}{\numberline {17.5}Visibility of Shafarevich-Tate groups}{164}{section.17.5}
\contentsline {subsection}{\numberline {17.5.1}Definitions}{165}{subsection.17.5.1}
\contentsline {subsection}{\numberline {17.5.2}Every element of $\HH ^1(K,A)$ is visible somewhere}{166}{subsection.17.5.2}
\contentsline {subsection}{\numberline {17.5.3}Visibility in the context of modularity}{166}{subsection.17.5.3}
\contentsline {subsection}{\numberline {17.5.4}Future directions}{168}{subsection.17.5.4}
\contentsline {section}{\numberline {17.6}Kolyvagin's Euler system of Heegner points}{169}{section.17.6}
\contentsline {subsection}{\numberline {17.6.1}A Heegner point when $N=11$}{178}{subsection.17.6.1}
\contentsline {subsection}{\numberline {17.6.2}Kolyvagin's Euler system for curves of rank at least $2$}{179}{subsection.17.6.2}
\contentsline {chapter}{\numberline {18}The Gorenstein Property for Hecke Algebras}{181}{chapter.18}
\contentsline {section}{\numberline {18.1}Mod $\ell $ representations associated to modular forms}{181}{section.18.1}
\contentsline {section}{\numberline {18.2}The Gorenstein property}{184}{section.18.2}
\contentsline {section}{\numberline {18.3}Proof of the Gorenstein property}{186}{section.18.3}
\contentsline {subsection}{\numberline {18.3.1}Vague comments}{189}{subsection.18.3.1}
\contentsline {section}{\numberline {18.4}Finite flat group schemes}{190}{section.18.4}
\contentsline {section}{\numberline {18.5}Reformulation of $V=W$ problem}{190}{section.18.5}
\contentsline {section}{\numberline {18.6}Dieudonn\'{e} theory}{191}{section.18.6}
\contentsline {section}{\numberline {18.7}The proof: part II}{192}{section.18.7}
\contentsline {section}{\numberline {18.8}Key result of Boston-Lenstra-Ribet}{194}{section.18.8}
\contentsline {chapter}{\numberline {19}Local Properties of $\rho _{\lambda }$}{197}{chapter.19}
\contentsline {section}{\numberline {19.1}Definitions}{197}{section.19.1}
\contentsline {section}{\numberline {19.2}Local properties at primes $p\nmid N$}{198}{section.19.2}
\contentsline {section}{\numberline {19.3}Weil-Deligne Groups}{198}{section.19.3}
\contentsline {section}{\numberline {19.4}Local properties at primes $p\mid {}N$}{198}{section.19.4}
\contentsline {section}{\numberline {19.5}Definition of the reduced conductor}{199}{section.19.5}
\contentsline {chapter}{\numberline {20}Adelic Representations}{201}{chapter.20}
\contentsline {section}{\numberline {20.1}Adelic representations associated to modular forms}{201}{section.20.1}
\contentsline {section}{\numberline {20.2}More local properties of the $\rho _{\lambda }$.}{204}{section.20.2}
\contentsline {subsection}{\numberline {20.2.1}Possibilities for $\pi _p$}{205}{subsection.20.2.1}
\contentsline {subsection}{\numberline {20.2.2}The case $\ell =p$}{206}{subsection.20.2.2}
\contentsline {subsection}{\numberline {20.2.3}Tate curves}{207}{subsection.20.2.3}
\contentsline {chapter}{\numberline {21}Serre's Conjecture}{209}{chapter.21}
\contentsline {section}{\numberline {21.1}The Family of $\lambda $-adic representations attached to a newform}{210}{section.21.1}
\contentsline {section}{\numberline {21.2}Serre's Conjecture A}{210}{section.21.2}
\contentsline {subsection}{\numberline {21.2.1}The Field of definition of $\rho $}{211}{subsection.21.2.1}
\contentsline {section}{\numberline {21.3}Serre's Conjecture B}{212}{section.21.3}
\contentsline {section}{\numberline {21.4}The Level}{212}{section.21.4}
\contentsline {subsection}{\numberline {21.4.1}Remark on the case $N(\rho )=1$}{213}{subsection.21.4.1}
\contentsline {subsection}{\numberline {21.4.2}Remark on the proof of Conjecture B}{214}{subsection.21.4.2}
\contentsline {section}{\numberline {21.5}The Weight}{215}{section.21.5}
\contentsline {subsection}{\numberline {21.5.1}The Weight modulo $\ell -1$}{215}{subsection.21.5.1}
\contentsline {subsection}{\numberline {21.5.2}Tameness at $\ell $}{215}{subsection.21.5.2}
\contentsline {subsection}{\numberline {21.5.3}Fundamental characters of the tame extension}{216}{subsection.21.5.3}
\contentsline {subsection}{\numberline {21.5.4}The Pair of characters associated to $\rho $}{217}{subsection.21.5.4}
\contentsline {subsection}{\numberline {21.5.5}Recipe for the weight}{218}{subsection.21.5.5}
\contentsline {subsection}{\numberline {21.5.6}The World's first view of fundamental characters}{219}{subsection.21.5.6}
\contentsline {subsection}{\numberline {21.5.7}Fontaine's theorem}{219}{subsection.21.5.7}
\contentsline {subsection}{\numberline {21.5.8}Guessing the weight (level 2 case)}{219}{subsection.21.5.8}
\contentsline {subsection}{\numberline {21.5.9}$\theta $-cycles}{220}{subsection.21.5.9}
\contentsline {subsection}{\numberline {21.5.10}Edixhoven's paper}{222}{subsection.21.5.10}
\contentsline {section}{\numberline {21.6}The Character}{222}{section.21.6}
\contentsline {subsection}{\numberline {21.6.1}A Counterexample}{224}{subsection.21.6.1}
\contentsline {section}{\numberline {21.7}The Weight revisited: level 1 case}{225}{section.21.7}
\contentsline {subsection}{\numberline {21.7.1}Companion forms}{225}{subsection.21.7.1}
\contentsline {subsection}{\numberline {21.7.2}The Weight: the remaining level 1 case}{226}{subsection.21.7.2}
\contentsline {subsection}{\numberline {21.7.3}Finiteness}{227}{subsection.21.7.3}
\contentsline {chapter}{\numberline {22}Fermat's Last Theorem}{229}{chapter.22}
\contentsline {section}{\numberline {22.1}The application to Fermat}{229}{section.22.1}
\contentsline {section}{\numberline {22.2}Modular elliptic curves}{231}{section.22.2}
\contentsline {chapter}{\numberline {23}Deformations}{233}{chapter.23}
\contentsline {section}{\numberline {23.1}Introduction}{233}{section.23.1}
\contentsline {section}{\numberline {23.2}Condition $(*)$}{234}{section.23.2}
\contentsline {subsection}{\numberline {23.2.1}Finite flat representations}{235}{subsection.23.2.1}
\contentsline {section}{\numberline {23.3}Classes of liftings}{235}{section.23.3}
\contentsline {subsection}{\numberline {23.3.1}The case $p\not =\ell $}{235}{subsection.23.3.1}
\contentsline {subsection}{\numberline {23.3.2}The case $p=\ell $}{236}{subsection.23.3.2}
\contentsline {section}{\numberline {23.4}Wiles's Hecke algebra}{237}{section.23.4}
\contentsline {chapter}{\numberline {24}The Hecke Algebra $T_{\Sigma }$}{239}{chapter.24}
\contentsline {section}{\numberline {24.1}The Hecke algebra}{239}{section.24.1}
\contentsline {section}{\numberline {24.2}The Maximal ideal in $R$}{241}{section.24.2}
\contentsline {subsection}{\numberline {24.2.1}Strip away certain Euler factors}{241}{subsection.24.2.1}
\contentsline {subsection}{\numberline {24.2.2}Make into an eigenform for $U_{\ell }$}{242}{subsection.24.2.2}
\contentsline {section}{\numberline {24.3}The Galois representation}{243}{section.24.3}
\contentsline {subsection}{\numberline {24.3.1}The Structure of $\mathbf {T}_{\mathfrak {m}}$}{244}{subsection.24.3.1}
\contentsline {subsection}{\numberline {24.3.2}The Philosophy in this picture}{244}{subsection.24.3.2}
\contentsline {subsection}{\numberline {24.3.3}Massage $\rho $}{244}{subsection.24.3.3}
\contentsline {subsection}{\numberline {24.3.4}Massage $\rho '$}{245}{subsection.24.3.4}
\contentsline {subsection}{\numberline {24.3.5}Representations from modular forms mod $\ell $}{246}{subsection.24.3.5}
\contentsline {subsection}{\numberline {24.3.6}Representations from modular forms mod $\ell ^n$}{246}{subsection.24.3.6}
\contentsline {section}{\numberline {24.4}$\rho '$ is of type $\Sigma $}{247}{section.24.4}
\contentsline {section}{\numberline {24.5}Isomorphism between $\mathbf {T}_{\mathfrak {m}}$ and $R_{\mathfrak {m}_R}$}{248}{section.24.5}
\contentsline {section}{\numberline {24.6}Deformations}{249}{section.24.6}
\contentsline {section}{\numberline {24.7}Wiles's main conjecture}{250}{section.24.7}
\contentsline {section}{\numberline {24.8}$\mathbf {T}_{\Sigma }$ is a complete intersection}{252}{section.24.8}
\contentsline {section}{\numberline {24.9}The Inequality $\#\mathcal {O}/\eta \leq \#\wp _T/\wp _T^2\leq \# \wp _R/\wp _R^2$}{252}{section.24.9}
\contentsline {subsection}{\numberline {24.9.1}The Definitions of the ideals}{253}{subsection.24.9.1}
\contentsline {subsection}{\numberline {24.9.2}Aside: Selmer groups}{254}{subsection.24.9.2}
\contentsline {subsection}{\numberline {24.9.3}Outline of some proofs}{254}{subsection.24.9.3}
\contentsline {subsubsection}{Step 1: $\Sigma =\emptyset $}{254}{section*.6}
\contentsline {subsubsection}{Step 2: Passage from $\Sigma =\emptyset $ to $\sigma $ general}{255}{section*.7}
\contentsline {chapter}{\numberline {25}Computing with Modular Forms and Abelian Varieties}{257}{chapter.25}
\contentsline {chapter}{\numberline {26}The Modular Curve $X_0(389)$}{259}{chapter.26}
\contentsline {section}{\numberline {26.1}Factors of $J_0(389)$}{260}{section.26.1}
\contentsline {subsection}{\numberline {26.1.1}Newforms of level $389$}{260}{subsection.26.1.1}
\contentsline {subsubsection}{Congruences}{261}{section*.8}
\contentsline {subsection}{\numberline {26.1.2}Isogeny structure}{261}{subsection.26.1.2}
\contentsline {subsection}{\numberline {26.1.3}Mordell-Weil ranks}{261}{subsection.26.1.3}
\contentsline {section}{\numberline {26.2}The Hecke algebra}{262}{section.26.2}
\contentsline {subsection}{\numberline {26.2.1}The Discriminant is divisible by $p$}{262}{subsection.26.2.1}
\contentsline {subsection}{\numberline {26.2.2}Congruences primes in $S_{p+1}(\Gamma _0(1))$}{263}{subsection.26.2.2}
\contentsline {section}{\numberline {26.3}Supersingular points in characteristic $389$}{264}{section.26.3}
\contentsline {subsection}{\numberline {26.3.1}The Supersingular $j$-invariants in characteristic $389$}{264}{subsection.26.3.1}
\contentsline {section}{\numberline {26.4}Miscellaneous}{264}{section.26.4}
\contentsline {subsection}{\numberline {26.4.1}The Shafarevich-Tate group}{264}{subsection.26.4.1}
\contentsline {subsection}{\numberline {26.4.2}Weierstrass points on $X_0^+(p)$}{265}{subsection.26.4.2}
\contentsline {subsection}{\numberline {26.4.3}A Property of the plus part of the integral homology}{265}{subsection.26.4.3}
\contentsline {subsection}{\numberline {26.4.4}The Field generated by points of small prime order on an elliptic curve}{265}{subsection.26.4.4}
\contentsline {chapter}{\numberline {}References}{267}{chapter*.9}