Sharedwww / ribetnotes.auxOpen in CoCalc
\relax 
\@writefile{toc}{\contentsline {chapter}{Preface}{iii}}
\@writefile{lof}{\addvspace {10\[email protected] }}
\@writefile{lot}{\addvspace {10\[email protected] }}
\citation{silverman1}
\@writefile{toc}{\contentsline {chapter}{\numberline {1}Introduction}{1}}
\@writefile{lof}{\addvspace {10\[email protected] }}
\@writefile{lot}{\addvspace {10\[email protected] }}
\@writefile{toc}{\contentsline {section}{\numberline {1.1}Two Dimensional Galois Representations}{1}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.1.1}Finite Fields (Weil, Tate)}{1}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.1.2}Galois Representations (Taniyama, Shimura, Mumford-Tate)}{2}}
\@writefile{toc}{\contentsline {section}{\numberline {1.2}Modular Forms and Galois Representations}{2}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.2.1}Cusp Forms}{2}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.2.2}Hecke Operators (Mordell)}{2}}
\citation{shimura1}
\@writefile{toc}{\contentsline {chapter}{\numberline {2}Modular Representations and Curves}{5}}
\@writefile{lof}{\addvspace {10\[email protected] }}
\@writefile{lot}{\addvspace {10\[email protected] }}
\@writefile{toc}{\contentsline {section}{\numberline {2.1}Arithmetic of Modular Forms}{5}}
\@writefile{toc}{\contentsline {section}{\numberline {2.2}Characters}{6}}
\@writefile{toc}{\contentsline {section}{\numberline {2.3}Parity Conditions}{6}}
\@writefile{toc}{\contentsline {section}{\numberline {2.4}Conjectures of Serre (mod $\ell $ version)}{7}}
\@writefile{toc}{\contentsline {section}{\numberline {2.5}General remarks on mod $p$ Galois representations}{7}}
\@writefile{toc}{\contentsline {section}{\numberline {2.6}Serre's Conjecture}{8}}
\@writefile{toc}{\contentsline {section}{\numberline {2.7}Wiles' Perspective}{8}}
\citation{serre2}
\citation{antwerp}
\@writefile{toc}{\contentsline {chapter}{\numberline {3}Modular Forms}{9}}
\@writefile{lof}{\addvspace {10\[email protected] }}
\@writefile{lot}{\addvspace {10\[email protected] }}
\@writefile{toc}{\contentsline {section}{\numberline {3.1}Cusp Forms}{9}}
\@writefile{toc}{\contentsline {section}{\numberline {3.2}Lattices}{9}}
\@writefile{toc}{\contentsline {section}{\numberline {3.3}Relationship With Elliptic Curves}{9}}
\@writefile{toc}{\contentsline {section}{\numberline {3.4}Hecke Operators}{10}}
\@writefile{toc}{\contentsline {section}{\numberline {3.5}Explicit Description of Sublattices}{11}}
\@writefile{toc}{\contentsline {section}{\numberline {3.6}Action of Hecke Operators on Modular Forms}{12}}
\citation{serre2}
\citation{lang1}
\@writefile{toc}{\contentsline {chapter}{\numberline {4}Embedding Hecke Operators in the Dual}{15}}
\@writefile{lof}{\addvspace {10\[email protected] }}
\@writefile{lot}{\addvspace {10\[email protected] }}
\@writefile{toc}{\contentsline {section}{\numberline {4.1}The Space of Modular Forms}{15}}
\@writefile{toc}{\contentsline {section}{\numberline {4.2}Inner Product}{16}}
\@writefile{toc}{\contentsline {section}{\numberline {4.3}Eigenforms}{17}}
\citation{lang2}
\@writefile{toc}{\contentsline {chapter}{\numberline {5}Rationality and Integrality Questions}{19}}
\@writefile{lof}{\addvspace {10\[email protected] }}
\@writefile{lot}{\addvspace {10\[email protected] }}
\@writefile{toc}{\contentsline {section}{\numberline {5.1}Review}{19}}
\@writefile{toc}{\contentsline {section}{\numberline {5.2}Integrality}{19}}
\@writefile{toc}{\contentsline {section}{\numberline {5.3}Victor Miller's Thesis}{20}}
\@writefile{toc}{\contentsline {section}{\numberline {5.4}Petersson Inner Product}{20}}
\citation{serre2}
\@writefile{toc}{\contentsline {chapter}{\numberline {6}Modular Curves}{23}}
\@writefile{lof}{\addvspace {10\[email protected] }}
\@writefile{lot}{\addvspace {10\[email protected] }}
\@writefile{toc}{\contentsline {section}{\numberline {6.1}Cusp Forms}{23}}
\@writefile{toc}{\contentsline {section}{\numberline {6.2}Modular Curves}{23}}
\citation{hartshorne}
\citation{katzmazur}
\citation{silverman1}
\@writefile{toc}{\contentsline {section}{\numberline {6.3}Classifying $\Gamma (N)$-structures}{24}}
\@writefile{toc}{\contentsline {section}{\numberline {6.4}More on Integral Hecke Operators}{24}}
\citation{lang2}
\citation{shimura2}
\citation{shimura1}
\@writefile{toc}{\contentsline {section}{\numberline {6.5}Complex Conjugation}{25}}
\@writefile{toc}{\contentsline {section}{\numberline {6.6}Isomorphism in the Real Case}{25}}
\@writefile{toc}{\contentsline {section}{\numberline {6.7}The Eichler-Shimura Isomorphism}{25}}
\citation{lang2}
\@writefile{toc}{\contentsline {section}{\numberline {6.8}The Petterson Inner Product is Hecke Compatible}{27}}
\citation{shimura1}
\@writefile{toc}{\contentsline {chapter}{\numberline {7}Higher Weight Modular Forms}{29}}
\@writefile{lof}{\addvspace {10\[email protected] }}
\@writefile{lot}{\addvspace {10\[email protected] }}
\@writefile{toc}{\contentsline {section}{\numberline {7.1}Definitions of $\@mathbf {T}$}{29}}
\@writefile{toc}{\contentsline {section}{\numberline {7.2}Double Cosets}{29}}
\@writefile{toc}{\contentsline {section}{\numberline {7.3}More General Congruence Subgroups}{30}}
\@writefile{toc}{\contentsline {section}{\numberline {7.4}Explicit Formulas}{31}}
\@writefile{toc}{\contentsline {section}{\numberline {7.5}Old and New Forms}{31}}
\@writefile{toc}{\contentsline {chapter}{\numberline {8}New Forms}{33}}
\@writefile{lof}{\addvspace {10\[email protected] }}
\@writefile{lot}{\addvspace {10\[email protected] }}
\citation{edixcole}
\@writefile{toc}{\contentsline {section}{\numberline {8.1}Connection With Galois Representations}{34}}
\@writefile{toc}{\contentsline {section}{\numberline {8.2}Semisimplicity of $U_p$}{34}}
\@writefile{toc}{\contentsline {section}{\numberline {8.3}Shimura's Example of Nonsemisimple $U_p$}{34}}
\@writefile{toc}{\contentsline {section}{\numberline {8.4}An Interesting Duality}{35}}
\citation{lang1}
\@writefile{toc}{\contentsline {section}{\numberline {8.5}Observations on $T_n$}{36}}
\@writefile{toc}{\contentsline {chapter}{\numberline {9}Some Explicit Genus Computations}{37}}
\@writefile{lof}{\addvspace {10\[email protected] }}
\@writefile{lot}{\addvspace {10\[email protected] }}
\@writefile{toc}{\contentsline {section}{\numberline {9.1}Computing the Dimension of $S_k(\Gamma )$}{37}}
\@writefile{toc}{\contentsline {section}{\numberline {9.2}Application of Riemann-Hurwitz}{37}}
\@writefile{toc}{\contentsline {section}{\numberline {9.3}Explicit Genus Computations}{38}}
\@writefile{toc}{\contentsline {section}{\numberline {9.4}The Genus of $X(N)$}{38}}
\@writefile{toc}{\contentsline {section}{\numberline {9.5}The Genus of $X_0(N)$}{39}}
\@writefile{toc}{\contentsline {section}{\numberline {9.6}Modular Forms mod $p$}{40}}
\citation{lnm349}
\@writefile{toc}{\contentsline {chapter}{\numberline {10}The Field of Moduli}{41}}
\@writefile{lof}{\addvspace {10\[email protected] }}
\@writefile{lot}{\addvspace {10\[email protected] }}
\newlabel{PropModSurj}{{10.0.2}{41}}
\@writefile{toc}{\contentsline {section}{\numberline {10.1}Digression on Moduli}{41}}
\citation{ig1}
\@writefile{toc}{\contentsline {section}{\numberline {10.2}When is $\rho _E$ Surjective?}{42}}
\@writefile{toc}{\contentsline {section}{\numberline {10.3}Observations}{43}}
\citation{serre1}
\citation{silverman1}
\@writefile{toc}{\contentsline {section}{\numberline {10.4}A Descent Problem}{44}}
\@writefile{toc}{\contentsline {section}{\numberline {10.5}Second Look at the Descent Exercise}{44}}
\citation{serre3}
\@writefile{toc}{\contentsline {section}{\numberline {10.6}Action of $\GL  _2$}{45}}
\@writefile{toc}{\contentsline {chapter}{\numberline {11}Hecke Operators as Correspondences}{47}}
\@writefile{lof}{\addvspace {10\[email protected] }}
\@writefile{lot}{\addvspace {10\[email protected] }}
\@writefile{toc}{\contentsline {section}{\numberline {11.1}Some Philosophy}{47}}
\@writefile{toc}{\contentsline {section}{\numberline {11.2}Hecke Operators as Correspondences}{48}}
\@writefile{toc}{\contentsline {section}{\numberline {11.3}Generalities on Correspondences}{49}}
\@writefile{toc}{\contentsline {section}{\numberline {11.4}Jacobians of Curves}{50}}
\@writefile{toc}{\contentsline {section}{\numberline {11.5}More on Hecke Operators}{51}}
\@writefile{toc}{\contentsline {section}{\numberline {11.6}Hecke Operators acting on Jacobians}{51}}
\@writefile{toc}{\contentsline {subsection}{\numberline {11.6.1}The Albanese Map}{52}}
\citation{mumford1}
\citation{mumford2}
\citation{schilling}
\@writefile{toc}{\contentsline {subsection}{\numberline {11.6.2}The Hecke Algebra}{53}}
\@writefile{toc}{\contentsline {section}{\numberline {11.7}The Eichler-Shimura Relation: Part I}{53}}
\@writefile{toc}{\contentsline {section}{\numberline {11.8}The Eichler-Shimura Relation: Part II}{54}}
\@writefile{toc}{\contentsline {section}{\numberline {11.9}Applications}{56}}
\@writefile{toc}{\contentsline {section}{\numberline {11.10}More on Eichler-Shimura}{57}}
\citation{shimura1}
\citation{ddt}
\@writefile{toc}{\contentsline {chapter}{\numberline {12}Abelian Varieties from Modular Forms}{59}}
\@writefile{lof}{\addvspace {10\[email protected] }}
\@writefile{lot}{\addvspace {10\[email protected] }}
\citation{silverman1}
\@writefile{toc}{\contentsline {section}{\numberline {12.1}Computing the Determinent of $\rho _{\lambda }$}{61}}
\citation{hartshorne}
\citation{milne1}
\citation{murty}
\@writefile{toc}{\contentsline {section}{\numberline {12.2}Duality and Polarizations}{62}}
\citation{milne1}
\@writefile{toc}{\contentsline {section}{\numberline {12.3}The Weil Pairing}{63}}
\@writefile{toc}{\contentsline {section}{\numberline {12.4}The Fancy Proof}{63}}
\@writefile{toc}{\contentsline {section}{\numberline {12.5}The Concrete Proof}{64}}
\@writefile{toc}{\contentsline {section}{\numberline {12.6}The Construction for $X_1(N)$}{64}}
\@writefile{toc}{\contentsline {chapter}{\numberline {13}The Gorenstein Property}{67}}
\@writefile{lof}{\addvspace {10\[email protected] }}
\@writefile{lot}{\addvspace {10\[email protected] }}
\citation{curtisreiner}
\@writefile{toc}{\contentsline {section}{\numberline {13.1}The Gorenstein Property}{69}}
\citation{blr}
\citation{mazur1}
\@writefile{toc}{\contentsline {section}{\numberline {13.2}Proof the Gorenstein Property}{72}}
\@writefile{toc}{\contentsline {subsection}{\numberline {13.2.1}Vague Comments}{75}}
\@writefile{toc}{\contentsline {section}{\numberline {13.3}Finite Flat Group Schemes}{75}}
\@writefile{toc}{\contentsline {section}{\numberline {13.4}Reformulation of $V=W$ problem}{75}}
\@writefile{toc}{\contentsline {section}{\numberline {13.5}Dieudonn\'{e} Theory}{76}}
\@writefile{toc}{\contentsline {section}{\numberline {13.6}The Proof: Part II}{77}}
\@writefile{toc}{\contentsline {section}{\numberline {13.7}Key Result of Boston-Lenstra-Ribet}{79}}
\@writefile{toc}{\contentsline {chapter}{\numberline {14}Local Properties of $\rho _{\lambda }$}{81}}
\@writefile{lof}{\addvspace {10\[email protected] }}
\@writefile{lot}{\addvspace {10\[email protected] }}
\@writefile{toc}{\contentsline {section}{\numberline {14.1}Definitions}{81}}
\@writefile{toc}{\contentsline {section}{\numberline {14.2}Local Properties when $p\tmspace  -\thinmuskip {.1667em}\tmspace  -\thinmuskip {.1667em}\not |N$}{81}}
\@writefile{toc}{\contentsline {section}{\numberline {14.3}Weil-Deligne Groups}{82}}
\@writefile{toc}{\contentsline {section}{\numberline {14.4}Local Properties when $p|N$}{82}}
\@writefile{toc}{\contentsline {section}{\numberline {14.5}Definition of the Reduced Conductor}{83}}
\@writefile{toc}{\contentsline {section}{\numberline {14.6}Introduction}{84}}
\@writefile{toc}{\contentsline {section}{\numberline {14.7}Adelic Representations Associated to Modular Forms}{84}}
\citation{st68}
\@writefile{toc}{\contentsline {section}{\numberline {14.8}More Local Properties of the $\rho _{\lambda }$.}{87}}
\@writefile{toc}{\contentsline {subsection}{\numberline {14.8.1}Possibilities for $\pi _p$}{87}}
\citation{ribet1}
\citation{serre4}
\@writefile{toc}{\contentsline {subsection}{\numberline {14.8.2}The case $\ell =p$}{88}}
\citation{silverman2}
\@writefile{toc}{\contentsline {subsection}{\numberline {14.8.3}Tate Curves}{89}}
\citation{serre5}
\@writefile{toc}{\contentsline {chapter}{\numberline {15}The Weight and Serre's Conjectures}{91}}
\@writefile{lof}{\addvspace {10\[email protected] }}
\@writefile{lot}{\addvspace {10\[email protected] }}
\@writefile{toc}{\contentsline {section}{\numberline {15.1}Introduction}{91}}
\@writefile{toc}{\contentsline {section}{\numberline {15.2}Review of the $\lambda $-adic case}{91}}
\@writefile{toc}{\contentsline {section}{\numberline {15.3}Serre's conjecture 0}{91}}
\citation{serre5}
\@writefile{toc}{\contentsline {subsection}{\numberline {15.3.1}Problems}{92}}
\@writefile{toc}{\contentsline {section}{\numberline {15.4}Serre's conjecture 1}{92}}
\citation{queen}
\citation{ribet2}
\citation{edixhoven}
\@writefile{toc}{\contentsline {subsection}{\numberline {15.4.1}Key background points}{93}}
\@writefile{toc}{\contentsline {section}{\numberline {15.5}The weight and fundamental characters}{94}}
\citation{edixhoven}
\@writefile{toc}{\contentsline {section}{\numberline {15.6}The weight in Serre's conjectures on modular representations}{98}}
\@writefile{toc}{\contentsline {subsection}{\numberline {15.6.1}$\theta $-series}{98}}
\@writefile{toc}{\contentsline {subsection}{\numberline {15.6.2}Edixhoven's paper}{100}}
\@writefile{toc}{\contentsline {section}{\numberline {15.7}The extra assumption}{100}}
\@writefile{toc}{\contentsline {subsection}{\numberline {15.7.1}Companion Forms}{102}}
\citation{serre5}
\@writefile{toc}{\contentsline {section}{\numberline {15.8}The exceptional level 1 case}{103}}
\citation{serre5}
\@writefile{toc}{\contentsline {chapter}{\numberline {16}Fermat's Last Theorem}{105}}
\@writefile{lof}{\addvspace {10\[email protected] }}
\@writefile{lot}{\addvspace {10\[email protected] }}
\@writefile{toc}{\contentsline {section}{\numberline {16.1}The application to Fermat}{105}}
\@writefile{toc}{\contentsline {section}{\numberline {16.2}Modular Elliptic Curves}{106}}
\@writefile{toc}{\contentsline {chapter}{\numberline {17}Deformations}{109}}
\@writefile{lof}{\addvspace {10\[email protected] }}
\@writefile{lot}{\addvspace {10\[email protected] }}
\@writefile{toc}{\contentsline {section}{\numberline {17.1}Introduction}{109}}
\@writefile{toc}{\contentsline {section}{\numberline {17.2}Condition $(*)$}{110}}
\@writefile{toc}{\contentsline {subsection}{\numberline {17.2.1}Finite flat representations}{111}}
\@writefile{toc}{\contentsline {section}{\numberline {17.3}Classes of Liftings}{111}}
\@writefile{toc}{\contentsline {subsection}{\numberline {17.3.1}The case $p\not =\ell $}{111}}
\@writefile{toc}{\contentsline {subsection}{\numberline {17.3.2}The case $p=\ell $}{112}}
\@writefile{toc}{\contentsline {section}{\numberline {17.4}Wiles' Hecke algebra}{112}}
\@writefile{toc}{\contentsline {chapter}{\numberline {18}The Hecke Algebra $T_{\Sigma }$}{115}}
\@writefile{lof}{\addvspace {10\[email protected] }}
\@writefile{lot}{\addvspace {10\[email protected] }}
\@writefile{toc}{\contentsline {section}{\numberline {18.1}The Hecke Algebra}{115}}
\@writefile{toc}{\contentsline {section}{\numberline {18.2}The maximal ideal in $R$}{117}}
\@writefile{toc}{\contentsline {subsection}{\numberline {18.2.1}Strip away certain Euler factors}{117}}
\@writefile{toc}{\contentsline {subsection}{\numberline {18.2.2}Make into an eigenform for $U_{\ell }$}{118}}
\@writefile{toc}{\contentsline {section}{\numberline {18.3}The Galois Representation}{118}}
\@writefile{toc}{\contentsline {subsection}{\numberline {18.3.1}The structure of $\@mathbf {T}_{\@mathbf {m}}$}{120}}
\@writefile{toc}{\contentsline {subsection}{\numberline {18.3.2}The philosophy in this picture}{120}}
\@writefile{toc}{\contentsline {subsection}{\numberline {18.3.3}Massage $\rho $}{120}}
\@writefile{toc}{\contentsline {subsection}{\numberline {18.3.4}Massage $\rho '$}{121}}
\@writefile{toc}{\contentsline {subsection}{\numberline {18.3.5}Representations from modular forms mod $\ell $}{121}}
\@writefile{toc}{\contentsline {subsection}{\numberline {18.3.6}Representations from modular forms mod $\ell ^n$}{122}}
\@writefile{toc}{\contentsline {section}{\numberline {18.4}$\rho '$ is of type $\Sigma $}{122}}
\citation{ddt}
\@writefile{toc}{\contentsline {section}{\numberline {18.5}Isomorphism between $\@mathbf {T}_{\@mathbf {m}}$ and $R_{\@mathbf {m}_R}$}{123}}
\citation{mazur2}
\@writefile{toc}{\contentsline {section}{\numberline {18.6}Deformations}{124}}
\citation{lenstra}
\@writefile{toc}{\contentsline {section}{\numberline {18.7}Wiles Main Conjecture}{125}}
\@writefile{toc}{\contentsline {section}{\numberline {18.8}$\@mathbf {T}_{\Sigma }$ is a complete intersection}{127}}
\@writefile{toc}{\contentsline {section}{\numberline {18.9}The inequality $\#\@mathcal {O}/\eta \leq \#\wp _T/\wp _T^2\leq \wp _R/\wp _R^2$}{127}}
\@writefile{toc}{\contentsline {subsection}{\numberline {18.9.1}The definitions of the ideals}{128}}
\citation{tw}
\citation{wiles}
\citation{tw}
\@writefile{toc}{\contentsline {subsection}{\numberline {18.9.2}Aside: Selmer Groups}{129}}
\@writefile{toc}{\contentsline {subsection}{\numberline {18.9.3}Outline of some proofs}{129}}
\@writefile{toc}{\contentsline {subsubsection}{Step 1: $\Sigma =\emptyset $}{129}}
\@writefile{toc}{\contentsline {subsubsection}{Step 2: Passage from $\Sigma =\emptyset $ to $\sigma $ general}{129}}
\bibcite{antwerp}{1}
\bibcite{blr}{2}
\bibcite{curtisreiner}{3}
\bibcite{ddt}{4}
\bibcite{edixcole}{5}
\bibcite{edixhoven}{6}
\bibcite{hartshorne}{7}
\bibcite{ig1}{8}
\bibcite{katzmazur}{9}
\bibcite{lang1}{10}
\bibcite{lang2}{11}
\bibcite{lenstra}{12}
\bibcite{lnm349}{13}
\bibcite{mazur1}{14}
\bibcite{mazur2}{15}
\bibcite{milne1}{16}
\bibcite{mumford1}{17}
\bibcite{mumford2}{18}
\bibcite{murty}{19}
\bibcite{queen}{20}
\bibcite{ribet1}{21}
\bibcite{ribet2}{22}
\bibcite{serre1}{23}
\bibcite{serre2}{24}
\bibcite{serre3}{25}
\bibcite{serre4}{26}
\bibcite{serre5}{27}
\bibcite{serre6}{28}
\bibcite{st68}{29}
\bibcite{schilling}{30}
\bibcite{shimura1}{31}
\bibcite{shimura2}{32}
\bibcite{silverman1}{33}
\bibcite{silverman2}{34}
\bibcite{tw}{35}
\bibcite{wiles}{36}