\contentsline {chapter}{\hbox to\@tempdima {\hfil }Preface}{7}{chapter*.2}
\contentsline {chapter}{\numberline {1}Modular Forms of Level One}{9}{chapter.1}
\contentsline {section}{\numberline {1.1}Basic Definitions}{9}{section.1.1}
\contentsline {section}{\numberline {1.2}Eisenstein Series and Delta}{11}{section.1.2}
\contentsline {section}{\numberline {1.3}Structure Theorem}{13}{section.1.3}
\contentsline {section}{\numberline {1.4}Hecke Operators}{15}{section.1.4}
\contentsline {section}{\numberline {1.5}The Victor Miller Basis}{19}{section.1.5}
\contentsline {section}{\numberline {1.6}Can One Compute the Coefficients of $\Delta $ in Polynomial Time?}{21}{section.1.6}
\contentsline {chapter}{\numberline {2}Dirichlet Characters}{23}{chapter.2}
\contentsline {section}{\numberline {2.1}Representation and Arithmetic}{24}{section.2.1}
\contentsline {section}{\numberline {2.2}Algorithms}{30}{section.2.2}
\contentsline {section}{\numberline {2.3}Alternative Representations of Characters}{34}{section.2.3}
\contentsline {section}{\numberline {2.4}Exercises}{35}{section.2.4}
\contentsline {chapter}{\numberline {3}Modular Forms and Eisenstein Series of\\Higher Level}{37}{chapter.3}
\contentsline {section}{\numberline {3.1}Modular Forms of Higher Level}{37}{section.3.1}
\contentsline {section}{\numberline {3.2}Generalized Bernoulli Numbers}{40}{section.3.2}
\contentsline {section}{\numberline {3.3}Explicit Basis for the Eisenstein Subspace}{42}{section.3.3}
\contentsline {section}{\numberline {3.4}Exercises}{44}{section.3.4}
\contentsline {chapter}{\numberline {4}Computing Dimensions of Spaces of Modular Forms}{45}{chapter.4}
\contentsline {section}{\numberline {4.1}Modular Forms for $\Gamma _0(N)$}{46}{section.4.1}
\contentsline {subsection}{\numberline {4.1.1}New and Old Subspaces}{47}{subsection.4.1.1}
\contentsline {section}{\numberline {4.2}Modular Forms for $\Gamma _1(N)$}{50}{section.4.2}
\contentsline {section}{\numberline {4.3}Modular Forms with Character}{51}{section.4.3}
\contentsline {section}{\numberline {4.4}Exercises}{54}{section.4.4}
\contentsline {chapter}{\numberline {5}Linear Algebra}{55}{chapter.5}
\contentsline {section}{\numberline {5.1}Echelon Form}{55}{section.5.1}
\contentsline {section}{\numberline {5.2}Echelon Forms over $\@mathbb {Q}$}{57}{section.5.2}
\contentsline {section}{\numberline {5.3}Polynomials}{63}{section.5.3}
\contentsline {chapter}{\numberline {6}Modular Symbols}{65}{chapter.6}
\contentsline {section}{\numberline {6.1}Modular Symbols}{66}{section.6.1}
\contentsline {section}{\numberline {6.2}Manin Symbols}{67}{section.6.2}
\contentsline {subsection}{\numberline {6.2.1}Coset Representatives and Manin Symbols}{71}{subsection.6.2.1}
\contentsline {subsection}{\numberline {6.2.2}Modular Symbols With Character}{72}{subsection.6.2.2}
\contentsline {section}{\numberline {6.3}Hecke Operators}{72}{section.6.3}
\contentsline {subsection}{\numberline {6.3.1}General Definition of Hecke Operators}{73}{subsection.6.3.1}
\contentsline {subsection}{\numberline {6.3.2}Hecke Operators on Manin Symbols}{75}{subsection.6.3.2}
\contentsline {subsection}{\numberline {6.3.3}Remarks on Complexity}{76}{subsection.6.3.3}
\contentsline {section}{\numberline {6.4}Cuspidal Modular Symbols}{77}{section.6.4}
\contentsline {section}{\numberline {6.5}The Pairing Between Modular Symbols and Modular Forms}{78}{section.6.5}
\contentsline {section}{\numberline {6.6}Explicitly Computing $\@mathbb {M}_k(\Gamma _0(N)$}{83}{section.6.6}
\contentsline {subsection}{\numberline {6.6.1}Computing $\@mathbb {P}^1(\@mathbb {Z}/N\@mathbb {Z})$}{83}{subsection.6.6.1}
\contentsline {subsection}{\numberline {6.6.2}Examples of Computation of $\@mathbb {M}_k(\Gamma _0(N))$}{86}{subsection.6.6.2}
\contentsline {subsection}{\numberline {6.6.3}Refined Algorithm For Computing Presentation}{94}{subsection.6.6.3}
\contentsline {section}{\numberline {6.7}Applications}{97}{section.6.7}
\contentsline {subsection}{\numberline {6.7.1}Later in this Book}{97}{subsection.6.7.1}
\contentsline {subsection}{\numberline {6.7.2}Discussion of the Literature and Research}{97}{subsection.6.7.2}
\contentsline {section}{\numberline {6.8}Exercises}{98}{section.6.8}
\contentsline {chapter}{\numberline {7}Using Modular Symbols to Compute Spaces of Modular Forms}{101}{chapter.7}
\contentsline {section}{\numberline {7.1}$q$-expansions of Newforms}{101}{section.7.1}
\contentsline {section}{\numberline {7.2}Decomposing Spaces of Modular Symbols}{104}{section.7.2}
\contentsline {subsection}{\numberline {7.2.1}Weiedemann's Minimal Polynomial Algorithm}{105}{subsection.7.2.1}
\contentsline {subsection}{\numberline {7.2.2}Polynomial Factorization}{106}{subsection.7.2.2}
\contentsline {subsection}{\numberline {7.2.3}Decomposition Using Kernels}{106}{subsection.7.2.3}
\contentsline {subsection}{\numberline {7.2.4}A Multi-Modular Decomposition Algorithm}{107}{subsection.7.2.4}
\contentsline {section}{\numberline {7.3}Computing Systems of Eigenvalues}{108}{section.7.3}
\contentsline {subsection}{\numberline {7.3.1}Computing Projection Onto a Subspace}{108}{subsection.7.3.1}
\contentsline {subsection}{\numberline {7.3.2}Systems of Eigenvalues Algorithm}{108}{subsection.7.3.2}
\contentsline {chapter}{\numberline {8}Period Mappings Associated to Newforms}{111}{chapter.8}
\contentsline {section}{\numberline {8.1}Complex Period Mapping}{111}{section.8.1}
\contentsline {section}{\numberline {8.2}Rational and Integral Period Mapping}{111}{section.8.2}
\contentsline {section}{\numberline {8.3}Special Values of $L$-Functions}{111}{section.8.3}
\contentsline {chapter}{\numberline {9}Modular Curves and Modular Abelian Varieties}{113}{chapter.9}
\contentsline {section}{\numberline {9.1}Modular Curves}{113}{section.9.1}
\contentsline {section}{\numberline {9.2}Modular Abelian Varieties}{113}{section.9.2}
\contentsline {section}{\numberline {9.3}The Birch and Swinnerton-Dyer Conjecture}{113}{section.9.3}
\contentsline {section}{\numberline {9.4}How Cremona Computes all Elliptic Curves of Conductor\nobreakspace {}$N$}{113}{section.9.4}
\contentsline {chapter}{\numberline {10}Application: Serre's Conjecture}{115}{chapter.10}
\contentsline {section}{\numberline {10.1}Congruences and Reduction Modulo a Prime}{115}{section.10.1}
\contentsline {section}{\numberline {10.2}Statement of the Conjecture}{115}{section.10.2}
\contentsline {section}{\numberline {10.3}Determining Irreducibility}{115}{section.10.3}
\contentsline {section}{\numberline {10.4}Computing the Serre Invariants}{115}{section.10.4}
\contentsline {section}{\numberline {10.5}Finding the Newforms that Give Rise to a Representation}{116}{section.10.5}
\contentsline {chapter}{\numberline {11}Software for Computing With Modular Forms}{117}{chapter.11}
\contentsline {section}{\numberline {11.1}MAGMA}{117}{section.11.1}
\contentsline {section}{\numberline {11.2}Python / MANIN}{117}{section.11.2}
\contentsline {section}{\numberline {11.3}Cremona's mwrank}{118}{section.11.3}
\contentsline {section}{\numberline {11.4}HECKE C++ Library}{118}{section.11.4}
\contentsline {section}{\numberline {11.5}PARI/GP Package}{118}{section.11.5}
\contentsline {chapter}{\hbox to\@tempdima {\hfil }Appendix: GNU Free Documentation License}{119}{chapter*.3}