A modular form is a complex analytic function that transforms in a specific way under the action of a discrete group.A modular form is a holomorphic function on the upper complex half-plane satisfying the functional equation f((az+b)/(cz+d)) = (cz+d)^k f(z) for all matrices in the modular group SL(2,Z) and possessing certain growth conditions. The weight k is an integer. Modular forms constitute finite-dimensional vector spaces, and their Fourier series contain deep arithmetic information.
The origins of the theory of modular forms go back to the works of Gauss and Eisenstein, but systematic study began in the late 19th century with Poincaré, who introduced the general concept of automorphic forms. Felix Klein studied modular functions in connection with the icosahedron. In the 20th century, Hecke developed the theory of operators, Ramanujan discovered deep properties of the τ-function, and later the Taniyama–Shimura conjecture about the modularity of elliptic curves led to Andrew Wiles' proof of Fermat's Last Theorem. Modern development includes the Langlands program and the modularity of all elliptic curves over ℚ.
A modular form is a section of a certain line bundle over the modular curve. The functional equation ensures that the function transforms well under the action of the modular group, while holomorphicity and the growth condition mean smoothness and finiteness at the cusps. Fourier series allow expressing the form through coefficients a_n, which often carry arithmetic meaning (e.g., the number of points on an elliptic curve over a finite field). The connection to elliptic curves is through modularity: every elliptic curve corresponds to a modular form whose L-series coincide. This is used in the proof of Fermat's theorem. Modular forms also generate representations of the group GL(2) over the adeles.
Proof of Fermat's Last Theorem, error-correcting codes, foundation for string theory and quantum invariants.
The Riemann zeta function, a mysterious function about the distribution of prime numbers, is closely related to modular forms through a modular form called the theta function.
| x | variable | Sequence in normed space — sequence |
| \lambda | variable | Scaling parameter — scaling parameter |
| n | variable | Index of sequence element — index |
| \inf | operator | Takes the greatest lower bound of a set — infimum |
| \sum_{n=1}^\infty | operator | Adds terms of a sequence — series summation |
| f | variable | scalar function value — function value |
| x | variable | input vector — vector argument |
| Q | variable | quadratic coefficient matrix — quadratic coefficient matrix |
| b | variable | linear coefficient vector — linear coefficient vector |
| c | variable | constant term — constant term |
| T | operator | flips a matrix or vector over its diagonal — transpose operator |
| s | variable | Complex argument of the L-function |
| n | variable | Summation index (positive integer) |
| χ | variable | Dirichlet character modulo some integer |
| ∑ | operator | Sum over positive integers — summation |
| Y | variable | observed response — response variable |
| X | variable | input features — feature vector |
| f | variable | true underlying function — function |
| ε | variable | noise term — noise |
| D | variable | monotonic transformation function — monotonic function |
