Mathematics / research
Modular forms and automorphic symmetry
Study holomorphic functions with arithmetic symmetry through fundamental domains, Fourier coefficients, and Hecke operators.
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Module 01 research
Modular forms and automorphic symmetry
Study holomorphic functions with arithmetic symmetry through fundamental domains, Fourier coefficients, and Hecke operators.
- 01
The modular group and fundamental domains
Track fractional-linear actions, boundary identifications, and stabilizers before treating the quotient as a parameter space.
31 min - 02
Modular forms, cusps, and transformation laws
Combine automorphy with interior and cusp holomorphy, keeping weight, local width, and coefficient conventions explicit.
32 min - 03
Eisenstein series and lattice sums
Derive modular symmetry from absolutely convergent lattice reindexing and isolate the exceptional behavior of weight two.
34 min - 04
The discriminant function and q-expansions
Fix the normalized discriminant, compute product coefficients, and separate finite Fourier evidence from a proved modular identity.
33 min - 05
The valence formula and dimension calculations
Count weighted zeros on the full modular quotient and turn the count into bases and finite Fourier certificates.
35 min - 06
Poincaré series and Petersson inner products
Average a Fourier seed over a coset space and unfold an invariant integral with every measure and conjugation convention fixed.
35 min - 07
Hecke operators and eigenforms
Fix the arithmetic Hecke normalization and derive prime-power coefficient relations from simultaneous eigenvectors.
35 min - 08
L-series and Mellin transforms
Derive the archimedean factor and functional equation from a cusp form while separating analytic continuation from Euler-product structure.
35 min - 09
Modular curves and congruence subgroups
Compactify finite-index quotients, compute cusp widths and genus, and distinguish complex quotient geometry from arithmetic models.
35 min - 10
Elliptic curves and modularity
Match rational elliptic-curve Euler factors with normalized weight-two newforms and separate arithmetic modularity from complex uniformization.
36 min