Modular forms
Introduction
Modular forms are a fundamental concept in the field of number theory, a branch of pure mathematics. They are complex analytic functions on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore involves complex analysis, algebra, and the theory of automorphic forms.
Definition
A modular form is a complex analytic function on the upper half-plane that transforms in a specific way under the action of the group of Möbius transformations. It also satisfies a growth condition at the cusps. The precise definition of a modular form is as follows:
Let Γ be a subgroup of SL(2,ℤ), the group of 2×2 matrices with integer coefficients and determinant 1. A function f: H → ℂ is called a modular form of weight k for Γ if it satisfies the following conditions:
1. For all γ = (abcd) in Γ and all z in H, we have f(γz) = (cz+d)kf(z). 2. f is holomorphic on H. 3. f has at most a pole at the cusps of Γ.
Properties
Modular forms have a number of important properties. For example, the space of modular forms of a given weight for a given group is a finite-dimensional complex vector space. This allows us to talk about the dimension of the space of modular forms, which is a central concept in the theory of modular forms.
Another important property of modular forms is their Fourier expansion. Every modular form has a Fourier expansion, which is a way of representing the modular form as an infinite series. This Fourier expansion is of fundamental importance in the study of modular forms, as it allows us to study the coefficients of the modular form, which often have interesting arithmetic properties.
Examples
There are many examples of modular forms. The simplest example is the constant function 1, which is a modular form of weight 0 for any group. Another simple example is the function Δ(z), which is a modular form of weight 12 for the group SL(2,ℤ). This function is known as the discriminant function, and its coefficients have deep connections to the theory of partitions.
Applications
Modular forms have many applications in number theory and related fields. One of the most famous applications is in the proof of Fermat's Last Theorem by Andrew Wiles. In his proof, Wiles used the theory of modular forms to construct a certain elliptic curve with the same L-series as a given modular form, which allowed him to prove the modularity theorem and hence Fermat's Last Theorem.
Modular forms also have applications in the theory of elliptic curves, in the theory of L-functions, and in the theory of automorphic representations. They are also used in physics, in particular in string theory and conformal field theory.