modular form

The section of line bundles over the moduli stack of elliptic curves ${\mathcal{M}}_{ell}$.
$M_k\simeq S_{k+12}$
Modularity theorem: the map ${\mathrm{\Gamma }}_0(N)\mathrm{\setminus }(\mathbb{H}\cup \{cusps\})\to E$ from the modular curves to elliptic curves (through cusp forms) is surjective. It's part of the Langlands program: Galois representation$\leftrightarrow$automorphic form
Hecke operators

topological modular forms

2d CFT on torus

VOA