\contentsline {chapter}{\hbox to\@tempdima {\hfil }Preface}{5}{chapter*.2}
\contentsline {chapter}{\numberline {1}Modular Forms}{7}{chapter.1}
\contentsline {section}{\numberline {1.1}Basic Definitions}{7}{section.1.1}
\contentsline {section}{\numberline {1.2}Modular Forms of Level 1}{8}{section.1.2}
\contentsline {section}{\numberline {1.3}Modular Forms of Any Level}{10}{section.1.3}
\contentsline {subsection}{\numberline {1.3.1}Computing Widths of Cusps}{12}{subsection.1.3.1}
\contentsline {section}{\numberline {1.4}Examples of Modular Forms of Level $1$}{13}{section.1.4}
\contentsline {subsection}{\numberline {1.4.1}The Cusp Form $\Delta $}{14}{subsection.1.4.1}
\contentsline {subsection}{\numberline {1.4.2}Fourier Expansions of Eisenstein Series}{14}{subsection.1.4.2}
\contentsline {section}{\numberline {1.5}Structure Theorem}{16}{section.1.5}
\contentsline {section}{\numberline {1.6}The Victor Miller Basis}{18}{section.1.6}
\contentsline {section}{\numberline {1.7}Hecke Operators}{20}{section.1.7}
\contentsline {section}{\numberline {1.8}Computing Hecke Operators}{23}{section.1.8}
\contentsline {subsection}{\numberline {1.8.1}A Conjecture about Complexity}{24}{subsection.1.8.1}
\contentsline {section}{\numberline {1.9}Exercises}{25}{section.1.9}
\contentsline {chapter}{\numberline {2}Dirichlet Characters}{27}{chapter.2}
\contentsline {section}{\numberline {2.1}Decomposing Modular Forms Using Dirichlet Characters}{28}{section.2.1}
\contentsline {section}{\numberline {2.2}Representation and Arithmetic}{29}{section.2.2}
\contentsline {section}{\numberline {2.3}Algorithms}{34}{section.2.3}
\contentsline {section}{\numberline {2.4}Alternative Representations of Characters}{38}{section.2.4}
\contentsline {section}{\numberline {2.5}Exercises}{39}{section.2.5}
\contentsline {chapter}{\numberline {3}Eisenstein Series}{41}{chapter.3}
\contentsline {section}{\numberline {3.1}Generalized Bernoulli Numbers}{41}{section.3.1}
\contentsline {section}{\numberline {3.2}Explicit Basis for the Eisenstein Subspace}{43}{section.3.2}
\contentsline {section}{\numberline {3.3}Exercises}{45}{section.3.3}
\contentsline {chapter}{\numberline {4}Dimensions Formulas}{47}{chapter.4}
\contentsline {section}{\numberline {4.1}Modular Forms for $\Gamma _0(N)$}{48}{section.4.1}
\contentsline {subsection}{\numberline {4.1.1}New and Old Subspaces}{49}{subsection.4.1.1}
\contentsline {section}{\numberline {4.2}Modular Forms for $\Gamma _1(N)$}{52}{section.4.2}
\contentsline {section}{\numberline {4.3}Modular Forms with Character}{53}{section.4.3}
\contentsline {section}{\numberline {4.4}Exercises}{56}{section.4.4}
\contentsline {chapter}{\numberline {5}Linear Algebra}{57}{chapter.5}
\contentsline {section}{\numberline {5.1}Echelon Forms of Matrices}{57}{section.5.1}
\contentsline {section}{\numberline {5.2}Echelon Forms over $\@mathbb {Q}$}{60}{section.5.2}
\contentsline {section}{\numberline {5.3}Polynomials}{65}{section.5.3}
\contentsline {section}{\numberline {5.4}Decomposing Spaces}{65}{section.5.4}
\contentsline {subsection}{\numberline {5.4.1}Wiedemann's Minimal Polynomial Algorithm}{66}{subsection.5.4.1}
\contentsline {subsection}{\numberline {5.4.2}Polynomial Factorization}{70}{subsection.5.4.2}
\contentsline {subsection}{\numberline {5.4.3}Decomposition Using Kernels}{70}{subsection.5.4.3}
\contentsline {subsection}{\numberline {5.4.4}Multi-Modular Decomposition Algorithm}{70}{subsection.5.4.4}
\contentsline {chapter}{\numberline {6}Modular Symbols}{73}{chapter.6}
\contentsline {section}{\numberline {6.1}Modular Symbols}{74}{section.6.1}
\contentsline {section}{\numberline {6.2}Manin Symbols}{75}{section.6.2}
\contentsline {subsection}{\numberline {6.2.1}Coset Representatives and Manin Symbols}{79}{subsection.6.2.1}
\contentsline {subsection}{\numberline {6.2.2}Modular Symbols With Character}{80}{subsection.6.2.2}
\contentsline {section}{\numberline {6.3}Hecke Operators}{80}{section.6.3}
\contentsline {subsection}{\numberline {6.3.1}General Definition of Hecke Operators}{81}{subsection.6.3.1}
\contentsline {subsection}{\numberline {6.3.2}Hecke Operators on Manin Symbols}{83}{subsection.6.3.2}
\contentsline {subsection}{\numberline {6.3.3}Remarks on Complexity}{84}{subsection.6.3.3}
\contentsline {section}{\numberline {6.4}Cuspidal Modular Symbols}{85}{section.6.4}
\contentsline {section}{\numberline {6.5}The Pairing Between Modular Symbols and Modular Forms}{86}{section.6.5}
\contentsline {section}{\numberline {6.6}Explicitly Computing $\@mathbb {M}_k(\Gamma _0(N)$}{90}{section.6.6}
\contentsline {subsection}{\numberline {6.6.1}Computing $\@mathbb {P}^1(\@mathbb {Z}/N\@mathbb {Z})$}{91}{subsection.6.6.1}
\contentsline {subsection}{\numberline {6.6.2}Examples of Computation of $\@mathbb {M}_k(\Gamma _0(N))$}{94}{subsection.6.6.2}
\contentsline {subsection}{\numberline {6.6.3}Refined Algorithm For Computing Presentation}{102}{subsection.6.6.3}
\contentsline {section}{\numberline {6.7}Applications}{105}{section.6.7}
\contentsline {subsection}{\numberline {6.7.1}Later in this Book}{105}{subsection.6.7.1}
\contentsline {subsection}{\numberline {6.7.2}Discussion of the Literature and Research}{105}{subsection.6.7.2}
\contentsline {section}{\numberline {6.8}Exercises}{106}{section.6.8}
\contentsline {chapter}{\numberline {7}Computing Spaces of Modular Forms}{107}{chapter.7}
\contentsline {section}{\numberline {7.1}Atkin-Lehner-Li Theory}{107}{section.7.1}
\contentsline {section}{\numberline {7.2}Computing Cuspforms Using\\Modular Symbols}{109}{section.7.2}
\contentsline {section}{\numberline {7.3}Computing Systems of Eigenvalues}{110}{section.7.3}
\contentsline {subsection}{\numberline {7.3.1}Computing Projection Onto a Subspace}{110}{subsection.7.3.1}
\contentsline {subsection}{\numberline {7.3.2}Systems of Eigenvalues}{111}{subsection.7.3.2}
\contentsline {chapter}{\numberline {8}Periods and Special Values of $L$-functions}{115}{chapter.8}
\contentsline {section}{\numberline {8.1}The Period Mapping and Complex Torus Attached to a Newform}{115}{section.8.1}
\contentsline {section}{\numberline {8.2}Extended Modular Symbols}{117}{section.8.2}
\contentsline {section}{\numberline {8.3}Numerically Approximating Period Integrals}{117}{section.8.3}
\contentsline {section}{\numberline {8.4}Speeding Convergence Using the Atkin-Lehner Operator}{122}{section.8.4}
\contentsline {subsection}{\numberline {8.4.1}Another Atkin-Lehner Trick}{123}{subsection.8.4.1}
\contentsline {section}{\numberline {8.5}Computing the Period Mapping}{124}{section.8.5}
\contentsline {section}{\numberline {8.6}Computing Elliptic Curves of Given Conductor}{125}{section.8.6}
\contentsline {subsection}{\numberline {8.6.1}Using Modular Symbols}{125}{subsection.8.6.1}
\contentsline {subsection}{\numberline {8.6.2}Finding Curves by Finding $S$-Integral Points}{127}{subsection.8.6.2}
\contentsline {section}{\numberline {8.7}Examples}{128}{section.8.7}
\contentsline {subsection}{\numberline {8.7.1}Jacobians of genus-two curves}{128}{subsection.8.7.1}
\contentsline {subsection}{\numberline {8.7.2}Level one cusp forms}{129}{subsection.8.7.2}
\contentsline {subsection}{\numberline {8.7.3}CM elliptic curves of weight greater than two}{130}{subsection.8.7.3}
\contentsline {section}{\numberline {8.8}Exercises}{130}{section.8.8}
\contentsline {chapter}{\numberline {9}Congruences}{133}{chapter.9}
\contentsline {section}{\numberline {9.1}Congruences Between Modular Forms}{133}{section.9.1}
\contentsline {subsection}{\numberline {9.1.1}The $j$-invariant}{133}{subsection.9.1.1}
\contentsline {subsection}{\numberline {9.1.2}Congruences for Modular Forms}{134}{subsection.9.1.2}
\contentsline {subsection}{\numberline {9.1.3}Congruence for Newforms}{137}{subsection.9.1.3}
\contentsline {section}{\numberline {9.2}Generating the Hecke Algebra as a $\@mathbb {Z}$-module}{138}{section.9.2}