Modular Arithmetic Geometry
Modular Curves
Modular curve creation and attributes.
Small Modular Curves
Explicit models, projections, and automorphisms of small modular curves.
Congruence Subgroups of PSL2(R)
Congruence subgroups, their structure, and group elements.
Arithmetic Fuchsian Groups and Shimura Curves
Fuchsian groups, fundamental domains, and triangle groups.
Modular Forms
Spaces of modular forms, bases, and q-expansions.
Modular Symbols
Modular symbols spaces, creation, and bases.
Brandt Modules
Brandt modules, subspaces, and Hecke operators.
Supersingular Divisors on Modular Curves
Supersingular divisors: creation, basis, and properties.
Modular Abelian Varieties
Modular abelian varieties, homology, and morphisms.
Hilbert Modular Forms
Hilbert modular forms: creation and basic properties.
Modular Forms over Imaginary Quadratic Fields
Bianchi modular forms, basic properties, and Hecke operators.
Algebraic Modular Forms
Algebraic modular forms: creation, properties, and elements.
Representations of Local Fields
Admissible representations of GL\ :sub:2\ (Q\ :sub:p\ ) and local Langlands.