Contents

• Characteristic Polynomials of Hecke Operators
  • Level $1$ Weight $24$
  • Higher Level
  • High Weight
  
  
• Modular Forms Modulo $p$
  • Big images
  • Non-liftable weight one modular forms over  $\mathbb{F}_p$
  
  
• Hilbert Modular Forms
  • Database
  • Investigate the Birch and Swinnteron-Dyer conjecture
  
  
• Computing with Classical Modular Forms
  • Compute Every Elliptic Curve of Conductor $234446$
  • Find all elliptic curves of conductor $\leq 234446$
  • Weight $144169$
  • A Problem About Bernoulli Numbers
  • Half Integral Weight Modular Forms
  
  
• Non-classical Modular Forms
  • Hilbert-Siegel Modular Forms
  • Modular forms on higher rank groups
  
  
• $p$ -adic Heights
  • Background
  • An Example: Computing $\mathcal{E}_2((37a,5))$
    • Introduction
    • Example: Computing the Matrix of Frobenius for $37A$ at $p = 5$
    • Source Code
    • Future Problems
  
  
• Level Raising and Lowering Modulo $p^n$
  • Problems
  
  
• Invariants of Modular Abelian Varieties
  • Endomorphism ring of $\bar{A}/\mathbb{F}_p$
  • Endomorphism Rings over Number Fields
  • Component Groups
  • Mordell-Weil Groups and Torsion subgroups
  • Analytic invariants
  • The Manin constant
  • Birch and Swinnerton-Dyer Conjectures
    • Refined Conjectures
    • Intrinsic Characterization of Optimality
  • The Shafarevich-Tate Group
    • Verifying the Full Conjecture for Elliptic Curves
    • Finiteness of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}$
  
  
• Fun with Visualizing Modular Forms (in SAGE)
• Other Projects
• Bibliography