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\contentsline {chapter}{Preface}{iii}
\contentsline {chapter}{\numberline {1}Introduction}{1}
\contentsline {section}{\numberline {1.1}Two Dimensional Galois Representations}{1}
\contentsline {subsection}{\numberline {1.1.1}Finite Fields (Weil, Tate)}{1}
\contentsline {subsection}{\numberline {1.1.2}Galois Representations (Taniyama, Shimura, Mumford-Tate)}{2}
\contentsline {section}{\numberline {1.2}Modular Forms and Galois Representations}{2}
\contentsline {subsection}{\numberline {1.2.1}Cusp Forms}{2}
\contentsline {subsection}{\numberline {1.2.2}Hecke Operators (Mordell)}{2}
\contentsline {chapter}{\numberline {2}Modular Representations and Curves}{5}
\contentsline {section}{\numberline {2.1}Arithmetic of Modular Forms}{5}
\contentsline {section}{\numberline {2.2}Characters}{6}
\contentsline {section}{\numberline {2.3}Parity Conditions}{6}
\contentsline {section}{\numberline {2.4}Conjectures of Serre (mod $\ell $ version)}{7}
\contentsline {section}{\numberline {2.5}General remarks on mod $p$ Galois representations}{7}
\contentsline {section}{\numberline {2.6}Serre's Conjecture}{8}
\contentsline {section}{\numberline {2.7}Wiles' Perspective}{8}
\contentsline {chapter}{\numberline {3}Modular Forms}{9}
\contentsline {section}{\numberline {3.1}Cusp Forms}{9}
\contentsline {section}{\numberline {3.2}Lattices}{9}
\contentsline {section}{\numberline {3.3}Relationship With Elliptic Curves}{9}
\contentsline {section}{\numberline {3.4}Hecke Operators}{10}
\contentsline {section}{\numberline {3.5}Explicit Description of Sublattices}{11}
\contentsline {section}{\numberline {3.6}Action of Hecke Operators on Modular Forms}{12}
\contentsline {chapter}{\numberline {4}Embedding Hecke Operators in the Dual}{15}
\contentsline {section}{\numberline {4.1}The Space of Modular Forms}{15}
\contentsline {section}{\numberline {4.2}Inner Product}{16}
\contentsline {section}{\numberline {4.3}Eigenforms}{17}
\contentsline {chapter}{\numberline {5}Rationality and Integrality Questions}{19}
\contentsline {section}{\numberline {5.1}Review}{19}
\contentsline {section}{\numberline {5.2}Integrality}{19}
\contentsline {section}{\numberline {5.3}Victor Miller's Thesis}{20}
\contentsline {section}{\numberline {5.4}Petersson Inner Product}{20}
\contentsline {chapter}{\numberline {6}Modular Curves}{23}
\contentsline {section}{\numberline {6.1}Cusp Forms}{23}
\contentsline {section}{\numberline {6.2}Modular Curves}{23}
\contentsline {section}{\numberline {6.3}Classifying $\Gamma (N)$-structures}{24}
\contentsline {section}{\numberline {6.4}More on Integral Hecke Operators}{24}
\contentsline {section}{\numberline {6.5}Complex Conjugation}{25}
\contentsline {section}{\numberline {6.6}Isomorphism in the Real Case}{25}
\contentsline {section}{\numberline {6.7}The Eichler-Shimura Isomorphism}{25}
\contentsline {section}{\numberline {6.8}The Petterson Inner Product is Hecke Compatible}{27}
\contentsline {chapter}{\numberline {7}Higher Weight Modular Forms}{29}
\contentsline {section}{\numberline {7.1}Definitions of $\@mathbf {T}$}{29}
\contentsline {section}{\numberline {7.2}Double Cosets}{29}
\contentsline {section}{\numberline {7.3}More General Congruence Subgroups}{30}
\contentsline {section}{\numberline {7.4}Explicit Formulas}{31}
\contentsline {section}{\numberline {7.5}Old and New Forms}{31}
\contentsline {chapter}{\numberline {8}New Forms}{33}
\contentsline {section}{\numberline {8.1}Connection With Galois Representations}{34}
\contentsline {section}{\numberline {8.2}Semisimplicity of $U_p$}{34}
\contentsline {section}{\numberline {8.3}Shimura's Example of Nonsemisimple $U_p$}{34}
\contentsline {section}{\numberline {8.4}An Interesting Duality}{35}
\contentsline {section}{\numberline {8.5}Observations on $T_n$}{36}
\contentsline {chapter}{\numberline {9}Some Explicit Genus Computations}{37}
\contentsline {section}{\numberline {9.1}Computing the Dimension of $S_k(\Gamma )$}{37}
\contentsline {section}{\numberline {9.2}Application of Riemann-Hurwitz}{37}
\contentsline {section}{\numberline {9.3}Explicit Genus Computations}{38}
\contentsline {section}{\numberline {9.4}The Genus of $X(N)$}{38}
\contentsline {section}{\numberline {9.5}The Genus of $X_0(N)$}{39}
\contentsline {section}{\numberline {9.6}Modular Forms mod $p$}{40}
\contentsline {chapter}{\numberline {10}The Field of Moduli}{41}
\contentsline {section}{\numberline {10.1}Digression on Moduli}{41}
\contentsline {section}{\numberline {10.2}When is $\rho _E$ Surjective?}{42}
\contentsline {section}{\numberline {10.3}Observations}{43}
\contentsline {section}{\numberline {10.4}A Descent Problem}{44}
\contentsline {section}{\numberline {10.5}Second Look at the Descent Exercise}{44}
\contentsline {section}{\numberline {10.6}Action of $\GL _2$}{45}
\contentsline {chapter}{\numberline {11}Hecke Operators as Correspondences}{47}
\contentsline {section}{\numberline {11.1}Some Philosophy}{47}
\contentsline {section}{\numberline {11.2}Hecke Operators as Correspondences}{48}
\contentsline {section}{\numberline {11.3}Generalities on Correspondences}{49}
\contentsline {section}{\numberline {11.4}Jacobians of Curves}{50}
\contentsline {section}{\numberline {11.5}More on Hecke Operators}{51}
\contentsline {section}{\numberline {11.6}Hecke Operators acting on Jacobians}{51}
\contentsline {subsection}{\numberline {11.6.1}The Albanese Map}{52}
\contentsline {subsection}{\numberline {11.6.2}The Hecke Algebra}{53}
\contentsline {section}{\numberline {11.7}The Eichler-Shimura Relation: Part I}{53}
\contentsline {section}{\numberline {11.8}The Eichler-Shimura Relation: Part II}{54}
\contentsline {section}{\numberline {11.9}Applications}{56}
\contentsline {section}{\numberline {11.10}More on Eichler-Shimura}{57}
\contentsline {chapter}{\numberline {12}Abelian Varieties from Modular Forms}{59}
\contentsline {section}{\numberline {12.1}Computing the Determinent of $\rho _{\lambda }$}{61}
\contentsline {section}{\numberline {12.2}Duality and Polarizations}{62}
\contentsline {section}{\numberline {12.3}The Weil Pairing}{63}
\contentsline {section}{\numberline {12.4}The Fancy Proof}{63}
\contentsline {section}{\numberline {12.5}The Concrete Proof}{64}
\contentsline {section}{\numberline {12.6}The Construction for $X_1(N)$}{64}
\contentsline {chapter}{\numberline {13}The Gorenstein Property}{67}
\contentsline {section}{\numberline {13.1}The Gorenstein Property}{69}
\contentsline {section}{\numberline {13.2}Proof the Gorenstein Property}{72}
\contentsline {subsection}{\numberline {13.2.1}Vague Comments}{75}
\contentsline {section}{\numberline {13.3}Finite Flat Group Schemes}{75}
\contentsline {section}{\numberline {13.4}Reformulation of $V=W$ problem}{75}
\contentsline {section}{\numberline {13.5}Dieudonn\'{e} Theory}{76}
\contentsline {section}{\numberline {13.6}The Proof: Part II}{77}
\contentsline {section}{\numberline {13.7}Key Result of Boston-Lenstra-Ribet}{79}
\contentsline {chapter}{\numberline {14}Local Properties of $\rho _{\lambda }$}{81}
\contentsline {section}{\numberline {14.1}Definitions}{81}
\contentsline {section}{\numberline {14.2}Local Properties when $p\tmspace -\thinmuskip {.1667em}\tmspace -\thinmuskip {.1667em}\not |N$}{81}
\contentsline {section}{\numberline {14.3}Weil-Deligne Groups}{82}
\contentsline {section}{\numberline {14.4}Local Properties when $p|N$}{82}
\contentsline {section}{\numberline {14.5}Definition of the Reduced Conductor}{83}
\contentsline {section}{\numberline {14.6}Introduction}{84}
\contentsline {section}{\numberline {14.7}Adelic Representations Associated to Modular Forms}{84}
\contentsline {section}{\numberline {14.8}More Local Properties of the $\rho _{\lambda }$.}{87}
\contentsline {subsection}{\numberline {14.8.1}Possibilities for $\pi _p$}{87}
\contentsline {subsection}{\numberline {14.8.2}The case $\ell =p$}{88}
\contentsline {subsection}{\numberline {14.8.3}Tate Curves}{89}
\contentsline {chapter}{\numberline {15}The Weight and Serre's Conjectures}{91}
\contentsline {section}{\numberline {15.1}Introduction}{91}
\contentsline {section}{\numberline {15.2}Review of the $\lambda $-adic case}{91}
\contentsline {section}{\numberline {15.3}Serre's conjecture 0}{91}
\contentsline {subsection}{\numberline {15.3.1}Problems}{92}
\contentsline {section}{\numberline {15.4}Serre's conjecture 1}{92}
\contentsline {subsection}{\numberline {15.4.1}Key background points}{93}
\contentsline {section}{\numberline {15.5}The weight and fundamental characters}{94}
\contentsline {section}{\numberline {15.6}The weight in Serre's conjectures on modular representations}{98}
\contentsline {subsection}{\numberline {15.6.1}$\theta $-series}{98}
\contentsline {subsection}{\numberline {15.6.2}Edixhoven's paper}{100}
\contentsline {section}{\numberline {15.7}The extra assumption}{100}
\contentsline {subsection}{\numberline {15.7.1}Companion Forms}{102}
\contentsline {section}{\numberline {15.8}The exceptional level 1 case}{103}
\contentsline {chapter}{\numberline {16}Fermat's Last Theorem}{105}
\contentsline {section}{\numberline {16.1}The application to Fermat}{105}
\contentsline {section}{\numberline {16.2}Modular Elliptic Curves}{106}
\contentsline {chapter}{\numberline {17}Deformations}{109}
\contentsline {section}{\numberline {17.1}Introduction}{109}
\contentsline {section}{\numberline {17.2}Condition $(*)$}{110}
\contentsline {subsection}{\numberline {17.2.1}Finite flat representations}{111}
\contentsline {section}{\numberline {17.3}Classes of Liftings}{111}
\contentsline {subsection}{\numberline {17.3.1}The case $p\not =\ell $}{111}
\contentsline {subsection}{\numberline {17.3.2}The case $p=\ell $}{112}
\contentsline {section}{\numberline {17.4}Wiles' Hecke algebra}{112}
\contentsline {chapter}{\numberline {18}The Hecke Algebra $T_{\Sigma }$}{115}
\contentsline {section}{\numberline {18.1}The Hecke Algebra}{115}
\contentsline {section}{\numberline {18.2}The maximal ideal in $R$}{117}
\contentsline {subsection}{\numberline {18.2.1}Strip away certain Euler factors}{117}
\contentsline {subsection}{\numberline {18.2.2}Make into an eigenform for $U_{\ell }$}{118}
\contentsline {section}{\numberline {18.3}The Galois Representation}{118}
\contentsline {subsection}{\numberline {18.3.1}The structure of $\@mathbf {T}_{\@mathbf {m}}$}{120}
\contentsline {subsection}{\numberline {18.3.2}The philosophy in this picture}{120}
\contentsline {subsection}{\numberline {18.3.3}Massage $\rho $}{120}
\contentsline {subsection}{\numberline {18.3.4}Massage $\rho '$}{121}
\contentsline {subsection}{\numberline {18.3.5}Representations from modular forms mod $\ell $}{121}
\contentsline {subsection}{\numberline {18.3.6}Representations from modular forms mod $\ell ^n$}{122}
\contentsline {section}{\numberline {18.4}$\rho '$ is of type $\Sigma $}{122}
\contentsline {section}{\numberline {18.5}Isomorphism between $\@mathbf {T}_{\@mathbf {m}}$ and $R_{\@mathbf {m}_R}$}{123}
\contentsline {section}{\numberline {18.6}Deformations}{124}
\contentsline {section}{\numberline {18.7}Wiles Main Conjecture}{125}
\contentsline {section}{\numberline {18.8}$\@mathbf {T}_{\Sigma }$ is a complete intersection}{127}
\contentsline {section}{\numberline {18.9}The inequality $\#\@mathcal {O}/\eta \leq \#\wp _T/\wp _T^2\leq \wp _R/\wp _R^2$}{127}
\contentsline {subsection}{\numberline {18.9.1}The definitions of the ideals}{128}
\contentsline {subsection}{\numberline {18.9.2}Aside: Selmer Groups}{129}
\contentsline {subsection}{\numberline {18.9.3}Outline of some proofs}{129}
\contentsline {subsubsection}{Step 1: $\Sigma =\emptyset $}{129}
\contentsline {subsubsection}{Step 2: Passage from $\Sigma =\emptyset $ to $\sigma $ general}{129}