\contentsline {chapter}{\numberline {1}Characteristic Polynomials of Hecke Operators}{3}{chapter.1}
\contentsline {section}{\numberline {1.1}Level $1$ Weight $24$}{3}{section.1.1}
\contentsline {subsection}{\numberline {1.1.1}Preliminary theorems}{4}{subsection.1.1.1}
\contentsline {subsection}{\numberline {1.1.2}Main Result and its Proof}{4}{subsection.1.1.2}
\contentsline {section}{\numberline {1.2}Higher Level}{6}{section.1.2}
\contentsline {section}{\numberline {1.3}High Weight}{6}{section.1.3}
\contentsline {chapter}{\numberline {2}Modular Forms Modulo $p$}{7}{chapter.2}
\contentsline {section}{\numberline {2.1}Big images}{7}{section.2.1}
\contentsline {section}{\numberline {2.2}Non-liftable weight one modular forms over\nobreakspace {}$\@mathbb {F}_p$}{8}{section.2.2}
\contentsline {chapter}{\numberline {3}Hilbert Modular Forms}{9}{chapter.3}
\contentsline {section}{\numberline {3.1}Database}{9}{section.3.1}
\contentsline {section}{\numberline {3.2}Investigate the Birch and Swinnteron-Dyer conjecture}{9}{section.3.2}
\contentsline {chapter}{\numberline {4}Computing with Classical Modular Forms}{10}{chapter.4}
\contentsline {section}{\numberline {4.1}Compute Every Elliptic Curve of Conductor $234446$}{10}{section.4.1}
\contentsline {section}{\numberline {4.2}Find all elliptic curves of conductor $\leq 234446$}{11}{section.4.2}
\contentsline {section}{\numberline {4.3}Weight $144169$}{12}{section.4.3}
\contentsline {section}{\numberline {4.4}A Problem About Bernoulli Numbers}{12}{section.4.4}
\contentsline {section}{\numberline {4.5}Half Integral Weight Modular Forms}{13}{section.4.5}
\contentsline {chapter}{\numberline {5}Non-classical Modular Forms}{14}{chapter.5}
\contentsline {section}{\numberline {5.1}Hilbert-Siegel Modular Forms}{14}{section.5.1}
\contentsline {section}{\numberline {5.2}Modular forms on higher rank groups}{15}{section.5.2}
\contentsline {chapter}{\numberline {6}$p$-adic Heights}{18}{chapter.6}
\contentsline {section}{\numberline {6.1}Background}{18}{section.6.1}
\contentsline {section}{\numberline {6.2}An Example: Computing $\@mathcal {E}_2((37a,5))$}{21}{section.6.2}
\contentsline {subsection}{\numberline {6.2.1}Introduction}{21}{subsection.6.2.1}
\contentsline {subsection}{\numberline {6.2.2}Example: Computing the Matrix of Frobenius for $37A$ at $p=5$}{24}{subsection.6.2.2}
\contentsline {subsection}{\numberline {6.2.3}Source Code}{26}{subsection.6.2.3}
\contentsline {subsection}{\numberline {6.2.4}Future Problems}{28}{subsection.6.2.4}
\contentsline {chapter}{\numberline {7}Level Raising and Lowering Modulo $p^n$}{30}{chapter.7}
\contentsline {section}{\numberline {7.1}Problems}{31}{section.7.1}
\contentsline {chapter}{\numberline {8}Invariants of Modular Abelian Varieties}{32}{chapter.8}
\contentsline {section}{\numberline {8.1}Endomorphism ring of $\mathaccentV {bar}016{A}/\@mathbb {F}_p$}{32}{section.8.1}
\contentsline {section}{\numberline {8.2}Endomorphism Rings over Number Fields}{34}{section.8.2}
\contentsline {section}{\numberline {8.3}Component Groups}{35}{section.8.3}
\contentsline {section}{\numberline {8.4}Mordell-Weil Groups and Torsion subgroups}{35}{section.8.4}
\contentsline {section}{\numberline {8.5}Analytic invariants}{35}{section.8.5}
\contentsline {section}{\numberline {8.6}The Manin constant}{36}{section.8.6}
\contentsline {section}{\numberline {8.7}Birch and Swinnerton-Dyer Conjectures}{37}{section.8.7}
\contentsline {subsection}{\numberline {8.7.1}Refined Conjectures}{37}{subsection.8.7.1}
\contentsline {subsection}{\numberline {8.7.2}Intrinsic Characterization of Optimality}{37}{subsection.8.7.2}
\contentsline {section}{\numberline {8.8}The Shafarevich-Tate Group}{37}{section.8.8}
\contentsline {subsection}{\numberline {8.8.1}Verifying the Full Conjecture for Elliptic Curves}{37}{subsection.8.8.1}
\contentsline {subsection}{\numberline {8.8.2}Finiteness of ${\unhbox \voidb@x \hbox {{\fontencoding {OT2}\fontfamily {wncyr}\fontseries {m}\fontshape {n}\selectfont Sh}}}$}{38}{subsection.8.8.2}
\contentsline {chapter}{\numberline {9}Fun with Visualizing Modular Forms (in SAGE)}{40}{chapter.9}
\contentsline {chapter}{\numberline {10}Other Projects}{42}{chapter.10}