\contentsline {section}{\numberline {1}Introduction}{1}{section.1}
\contentsline {subsection}{\numberline {1.1}$\GL _2$-type and Modular Abelian Varieties}{2}{subsection.1.1}
\contentsline {subsection}{\numberline {1.2}Explicit Defining Data for Modular Abelian Varieties}{3}{subsection.1.2}
\contentsline {subsection}{\numberline {1.3}Contents}{3}{subsection.1.3}
\contentsline {section}{\numberline {2}Computing with Modular Abelian Varieties}{5}{section.2}
\contentsline {subsection}{\numberline {2.1}Ambient Modular Abelian Varieties}{5}{subsection.2.1}
\contentsline {subsection}{\numberline {2.2}Enumerating New Simple Modular Abelian Varieties}{6}{subsection.2.2}
\contentsline {subsection}{\numberline {2.3}Decomposition}{6}{subsection.2.3}
\contentsline {subsubsection}{\numberline {2.3.1}New and Old Subvarieties and Quotients}{6}{subsubsection.2.3.1}
\contentsline {subsubsection}{\numberline {2.3.2}Decomposition as a Product of Simples}{6}{subsubsection.2.3.2}
\contentsline {subsubsection}{\numberline {2.3.3}Verifying Defining Data of a Modular Abelian Variety}{6}{subsubsection.2.3.3}
\contentsline {subsection}{\numberline {2.4}Arithmetic with Modular Abelian Varieties}{6}{subsection.2.4}
\contentsline {subsubsection}{\numberline {2.4.1}Sums and Products}{6}{subsubsection.2.4.1}
\contentsline {subsubsection}{\numberline {2.4.2}Intersection}{6}{subsubsection.2.4.2}
\contentsline {subsubsection}{\numberline {2.4.3}Complements (Poincare Reducibility)}{7}{subsubsection.2.4.3}
\contentsline {subsubsection}{\numberline {2.4.4}Quotients by Subgroups and Subvarieties}{7}{subsubsection.2.4.4}
\contentsline {subsection}{\numberline {2.5}Finite Subgroups}{7}{subsection.2.5}
\contentsline {subsubsection}{\numberline {2.5.1}Defining Data}{7}{subsubsection.2.5.1}
\contentsline {subsubsection}{\numberline {2.5.2}The $n$-Torsion Subgroup}{7}{subsubsection.2.5.2}
\contentsline {subsubsection}{\numberline {2.5.3}Intersection of Finite Subgroups}{7}{subsubsection.2.5.3}
\contentsline {subsubsection}{\numberline {2.5.4}The Cuspidal Subgroup}{7}{subsubsection.2.5.4}
\contentsline {subsubsection}{\numberline {2.5.5}The Rational Cuspidal Subgroup}{7}{subsubsection.2.5.5}
\contentsline {subsubsection}{\numberline {2.5.6}The Torsion Subgroup}{7}{subsubsection.2.5.6}
\contentsline {subsubsection}{\numberline {2.5.7}The Shimura Subgroup}{8}{subsubsection.2.5.7}
\contentsline {subsection}{\numberline {2.6}Morphisms Between Modular Abelian Varieties}{8}{subsection.2.6}
\contentsline {subsubsection}{\numberline {2.6.1}Defining Data for Morphisms}{8}{subsubsection.2.6.1}
\contentsline {subsubsection}{\numberline {2.6.2}Natural Maps}{8}{subsubsection.2.6.2}
\contentsline {subsubsection}{\numberline {2.6.3}Morphisms Defined by Finite Subgroups}{8}{subsubsection.2.6.3}
\contentsline {subsubsection}{\numberline {2.6.4}Kernels of Morphisms}{8}{subsubsection.2.6.4}
\contentsline {subsubsection}{\numberline {2.6.5}Images of Morphisms}{8}{subsubsection.2.6.5}
\contentsline {subsubsection}{\numberline {2.6.6}The Universal Property of the Cokernel}{8}{subsubsection.2.6.6}
\contentsline {subsubsection}{\numberline {2.6.7}The Projection Morphism}{8}{subsubsection.2.6.7}
\contentsline {subsubsection}{\numberline {2.6.8}Left and Right Inverses}{8}{subsubsection.2.6.8}
\contentsline {subsection}{\numberline {2.7}Endomorphism Rings and Hom Spaces}{8}{subsection.2.7}
\contentsline {subsubsection}{\numberline {2.7.1}Computing End and Hom}{8}{subsubsection.2.7.1}
\contentsline {subsubsection}{\numberline {2.7.2}Computing Discriminants of Endomorphism Rings}{11}{subsubsection.2.7.2}
\contentsline {subsubsection}{\numberline {2.7.3}The Hecke Subring}{11}{subsubsection.2.7.3}
\contentsline {subsubsection}{\numberline {2.7.4}Atkin-Lehner Operators}{11}{subsubsection.2.7.4}
\contentsline {subsubsection}{\numberline {2.7.5}The $I$-torsion Subgroup for any Ideal $I$}{11}{subsubsection.2.7.5}
\contentsline {subsection}{\numberline {2.8}Isogenies and Isomorphisms of Modular Abelian Varieties}{11}{subsection.2.8}
\contentsline {subsubsection}{\numberline {2.8.1}Isogenies From $A$ to $B$}{11}{subsubsection.2.8.1}
\contentsline {subsubsection}{\numberline {2.8.2}Isomorphisms from $A$ to $B$}{12}{subsubsection.2.8.2}
\contentsline {subsubsection}{\numberline {2.8.3}The Minimal Isogeny}{14}{subsubsection.2.8.3}
\contentsline {subsection}{\numberline {2.9}Complex Periods}{15}{subsection.2.9}
\contentsline {subsubsection}{\numberline {2.9.1}The Period Lattice}{15}{subsubsection.2.9.1}
\contentsline {subsubsection}{\numberline {2.9.2}The BSD Real Volume}{15}{subsubsection.2.9.2}
\contentsline {subsection}{\numberline {2.10}Component Groups}{15}{subsection.2.10}
\contentsline {subsubsection}{\numberline {2.10.1}Supersingular Curves}{15}{subsubsection.2.10.1}
\contentsline {subsubsection}{\numberline {2.10.2}Definite Quaternion Algebras}{15}{subsubsection.2.10.2}
\contentsline {subsubsection}{\numberline {2.10.3}The Component Group}{15}{subsubsection.2.10.3}
\contentsline {subsubsection}{\numberline {2.10.4}Tamagawa Numbers}{15}{subsubsection.2.10.4}
\contentsline {subsubsection}{\numberline {2.10.5}$J_1(N)$}{15}{subsubsection.2.10.5}
\contentsline {subsection}{\numberline {2.11}Complex $L$-Series}{15}{subsection.2.11}
\contentsline {subsubsection}{\numberline {2.11.1}Local $L$-factors}{15}{subsubsection.2.11.1}
\contentsline {subsubsection}{\numberline {2.11.2}Numerical Evaluation at any Point}{15}{subsubsection.2.11.2}
\contentsline {subsubsection}{\numberline {2.11.3}The Rational Part of the Special Value}{16}{subsubsection.2.11.3}
\contentsline {subsubsection}{\numberline {2.11.4}Order of Vanishing (Analytic Rank)}{16}{subsubsection.2.11.4}
\contentsline {subsubsection}{\numberline {2.11.5}Zeros in the Critical Strip}{16}{subsubsection.2.11.5}
\contentsline {subsection}{\numberline {2.12}$p$-adic $L$-Series}{16}{subsection.2.12}
\contentsline {subsubsection}{\numberline {2.12.1}The Definition}{16}{subsubsection.2.12.1}
\contentsline {subsubsection}{\numberline {2.12.2}Computing to Given Precision}{16}{subsubsection.2.12.2}
\contentsline {subsubsection}{\numberline {2.12.3}Computing the Leading Coefficient and Order of Vanishing}{16}{subsubsection.2.12.3}
\contentsline {section}{\numberline {3}Computing the Isogeny Class}{16}{section.3}
\contentsline {subsection}{\numberline {3.1}The Class Group -- Noneisenstein Isogenies}{18}{subsection.3.1}
\contentsline {subsection}{\numberline {3.2}Eisenstein Isogenies}{18}{subsection.3.2}
\contentsline {subsection}{\numberline {3.3}Enumerating the Isogeny Class}{19}{subsection.3.3}
\contentsline {section}{\numberline {4}Tables of Modular Abelian Varieties}{19}{section.4}
\contentsline {subsection}{\numberline {4.1}Contents}{19}{subsection.4.1}
\contentsline {subsection}{\numberline {4.2}Factors of $J_0(N)$ for $N\leq 125$}{20}{subsection.4.2}
\contentsline {subsection}{\numberline {4.3}Factors of $J_H(N)$ for $N\leq 49$}{20}{subsection.4.3}
\contentsline {subsection}{\numberline {4.4}Minimal Isogenies}{20}{subsection.4.4}
\contentsline {subsection}{\numberline {4.5}Birch and Swinnerton-Dyer}{20}{subsection.4.5}
\contentsline {subsection}{\numberline {4.6}Other Examples}{20}{subsection.4.6}
\contentsline {subsection}{\numberline {4.7}Level $35$}{20}{subsection.4.7}
\contentsline {subsection}{\numberline {4.8}Level $69$: The first $A_f$ that is not isomorphic to $A_f^{\vee }$}{20}{subsection.4.8}
\contentsline {subsection}{\numberline {4.9}Level $195$: An $A_f$ not isomorphic to its dual, though there are solutions to the norm equation}{21}{subsection.4.9}