Modular form
Introduction
A modular form is a complex analytic function that exhibits a high degree of symmetry and structure. These functions play a significant role in various areas of mathematics, including number theory, algebraic geometry, and mathematical physics. Modular forms are defined on the upper half-plane and are invariant under the action of the modular group, which consists of transformations of the form \(\frac{az + b}{cz + d}\) where \(a, b, c,\) and \(d\) are integers satisfying \(ad - bc = 1\).
Basic Definitions and Properties
Modular Group
The modular group \( \text{SL}(2, \mathbb{Z}) \) consists of 2x2 matrices with integer entries and determinant 1. The action of this group on the upper half-plane \( \mathbb{H} \) is given by: \[ \gamma z = \frac{az + b}{cz + d} \] for \( \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \text{SL}(2, \mathbb{Z}) \).
Modular Forms
A modular form of weight \(k\) is a holomorphic function \(f: \mathbb{H} \to \mathbb{C}\) that satisfies: \[ f\left( \frac{az + b}{cz + d} \right) = (cz + d)^k f(z) \] for all \( \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \text{SL}(2, \mathbb{Z}) \) and all \( z \in \mathbb{H} \). Additionally, \(f\) must be holomorphic at the cusps, including \( \infty \).
Eisenstein Series
One of the simplest examples of a modular form is the Eisenstein series. For a positive even integer \(k\), the Eisenstein series \(E_k(z)\) is given by: \[ E_k(z) = 1 - \frac{2k}{B_k} \sum_{n=1}^{\infty} \sigma_{k-1}(n) q^n \] where \( q = e^{2\pi i z} \), \( B_k \) are the Bernoulli numbers, and \( \sigma_{k-1}(n) \) is the sum of the \((k-1)\)-th powers of the divisors of \(n\).
Fourier Expansion
Modular forms can be expressed in terms of their Fourier series. For a modular form \(f\) of weight \(k\), the Fourier expansion is: \[ f(z) = \sum_{n=0}^{\infty} a_n q^n \] where \( q = e^{2\pi i z} \). The coefficients \(a_n\) often encode significant arithmetic information.
Hecke Operators
Hecke operators are an important tool in the study of modular forms. For a modular form \(f\) of weight \(k\), the Hecke operator \(T_n\) acts on \(f\) to produce another modular form of the same weight. The action is defined by: \[ (T_n f)(z) = n^{k-1} \sum_{ad=n, a>0} \sum_{b \mod d} f\left( \frac{az + b}{d} \right) \]
Modular Forms and L-functions
Modular forms are closely related to L-functions. Given a modular form \(f(z) = \sum_{n=1}^{\infty} a_n q^n\), one can associate an L-function \(L(f, s)\) defined by: \[ L(f, s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s} \] This series converges for \(\text{Re}(s) > k/2 + 1\) and can often be analytically continued to the entire complex plane.
Applications in Number Theory
Modular forms have profound applications in number theory. One of the most famous results is the proof of Fermat's Last Theorem by Andrew Wiles, which relied on the modularity theorem. This theorem states that every rational elliptic curve is associated with a modular form.
Modular Forms and Elliptic Curves
There is a deep connection between modular forms and elliptic curves. For an elliptic curve \(E\) defined over \(\mathbb{Q}\), there exists a modular form \(f\) such that the L-function of \(E\) matches the L-function of \(f\). This correspondence is a cornerstone of the modularity theorem.