A holomorphic function f on the upper half-plane satisfying f((az+b)/(cz+d)) = (cz+d)^k f(z) for all (a,b;c,d) ∈ SL₂(ℤ) (or a congruence subgroup), plus growth at cusps. Fourier coefficients encode deep arithmetic.
Modular form
Related concepts
- Holomorphic function
- Elliptic curve
- Langlands program
- Modularity theorem (Taniyama-Shimura-Weil)
- Modular curves X_0(N)
- Hecke operators T_p
- Modular forms / L-functions (Langlands)
- Monstrous moonshine: j-function coefficients as Monster irreducible dimensions (string-theory)
- Modular invariance (CFT partition function)