🔢Arithmetic Geometry Unit 3 – Modular forms

Definition and Basic Properties

• Modular forms are complex analytic functions defined on the upper half-plane $\mathbb{H} = \{z \in \mathbb{C} : \text{Im}(z) > 0\}$
• Satisfy certain transformation properties under the action of discrete subgroups of $\text{SL}_2(\mathbb{R})$ (modular groups)
• Can be viewed as sections of line bundles on modular curves, which are quotients of $\mathbb{H}$ by modular groups
• Have Fourier expansions with integer coefficients, allowing for arithmetic study
• Play a central role in connecting complex analysis, algebraic geometry, and number theory
• Classified by their weight $k$, a positive integer that determines the transformation behavior under the modular group
  • For a modular form $f$ and a matrix $\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \text{SL}_2(\mathbb{Z})$, $f(\gamma z) = (cz+d)^k f(z)$
• Can be holomorphic (analytic everywhere) or meromorphic (allowing poles)

Historical Context and Motivation

• Modular forms emerged in the 19th century through the work of mathematicians such as Eisenstein, Dedekind, and Klein
• Initially studied in the context of elliptic functions and the modular group $\text{SL}_2(\mathbb{Z})$
• Gained prominence in the 20th century with the development of the theory of automorphic forms and the Langlands program
• Motivated by the study of L-functions, which encode arithmetic information and are closely related to modular forms
• Played a crucial role in Andrew Wiles' proof of Fermat's Last Theorem, demonstrating their deep connections to elliptic curves and Galois representations
• Have applications in various areas of mathematics, including number theory, algebraic geometry, and mathematical physics
• Serve as a bridge between the classical theory of elliptic functions and the modern theory of automorphic forms and representation theory

Modular Groups and Congruence Subgroups

• Modular groups are discrete subgroups of $\text{SL}_2(\mathbb{R})$ that act on the upper half-plane $\mathbb{H}$ by fractional linear transformations
• The most important modular group is the full modular group $\text{SL}_2(\mathbb{Z})$, consisting of 2x2 integer matrices with determinant 1
• Congruence subgroups are subgroups of $\text{SL}_2(\mathbb{Z})$ defined by congruence conditions on the matrix entries modulo some integer $N$
  • Examples include the principal congruence subgroup $\Gamma(N)$ and the Hecke congruence subgroups $\Gamma_0(N)$ and $\Gamma_1(N)$
• Modular curves are quotients of $\mathbb{H}$ by congruence subgroups, and they parametrize elliptic curves with additional structure (level structure)
• The study of modular forms for congruence subgroups leads to the theory of newforms and oldforms, which is crucial for understanding the structure of spaces of modular forms
• Congruence subgroups and their associated modular curves play a central role in the study of modular forms and their arithmetic properties

Fourier Expansions and q-series

• Modular forms have Fourier expansions in terms of the variable $q = e^{2\pi i z}$, where $z \in \mathbb{H}$
• The Fourier coefficients of a modular form encode important arithmetic information and are often the main object of study
• For a modular form $f(z) = \sum_{n=0}^{\infty} a_n q^n$, the coefficients $a_n$ are complex numbers that satisfy certain growth conditions
• The Fourier expansion of a modular form is a q-series, a formal power series in the variable $q$
• Many important q-series arise as the Fourier expansions of modular forms, such as the Eisenstein series and the Dedekind eta function
• The study of q-series identities and their connections to modular forms is an active area of research in number theory and combinatorics
• The Fourier coefficients of modular forms often satisfy interesting congruence relations and multiplicative properties, which are studied using the theory of Hecke operators

Eisenstein Series and Cusp Forms

• Modular forms can be classified into two main types: Eisenstein series and cusp forms
• Eisenstein series are modular forms that are constructed using infinite sums over lattice points in the plane
  • For even integers $k \geq 4$, the Eisenstein series $E_k(z) = \sum_{(m,n) \neq (0,0)} \frac{1}{(mz+n)^k}$ is a modular form of weight $k$ for $\text{SL}_2(\mathbb{Z})$
• Cusp forms are modular forms that vanish at the cusps (points at infinity) of the modular curve
  • They form a subspace of the space of modular forms and are often the most interesting from an arithmetic perspective
• The space of modular forms can be decomposed as a direct sum of the space of Eisenstein series and the space of cusp forms
• Cusp forms have vanishing constant term in their Fourier expansion, while Eisenstein series have non-zero constant term
• The study of Eisenstein series and cusp forms is central to understanding the structure of spaces of modular forms and their associated L-functions
• Many important examples of modular forms, such as the Ramanujan delta function, are cusp forms

Hecke Operators and Eigenforms

• Hecke operators are linear operators that act on spaces of modular forms and preserve important arithmetic properties
• For each prime $p$, there is a Hecke operator $T_p$ that acts on modular forms by modifying their Fourier coefficients
  • For a modular form $f(z) = \sum_{n=0}^{\infty} a_n q^n$, $(T_p f)(z) = \sum_{n=0}^{\infty} (a_{pn} + p^{k-1} a_{n/p}) q^n$, where $k$ is the weight of $f$
• Hecke operators commute with each other and satisfy certain multiplicative relations, forming a commutative algebra (the Hecke algebra)
• Eigenforms are modular forms that are simultaneous eigenvectors for all Hecke operators
  • They play a crucial role in the theory of modular forms and their associated L-functions
• The Fourier coefficients of eigenforms satisfy multiplicative relations determined by their eigenvalues under the Hecke operators
• The study of Hecke operators and eigenforms is essential for understanding the arithmetic properties of modular forms and their connections to Galois representations
• Hecke operators can be used to construct bases of spaces of modular forms consisting of eigenforms, which is important for computational and theoretical purposes

L-functions and Modularity Theorem

• L-functions are complex analytic functions that encode arithmetic information about mathematical objects, such as modular forms and elliptic curves
• The L-function of a modular form $f(z) = \sum_{n=0}^{\infty} a_n q^n$ is defined as the Dirichlet series $L(f, s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$, where $s$ is a complex variable
• L-functions of modular forms have an Euler product representation, reflecting the multiplicative properties of the Fourier coefficients
• The modularity theorem, proved by Wiles and others, establishes a deep connection between elliptic curves over $\mathbb{Q}$ and modular forms
  • It states that every elliptic curve over $\mathbb{Q}$ is modular, meaning that its L-function coincides with the L-function of a modular form
• The modularity theorem was a key ingredient in the proof of Fermat's Last Theorem, as it allowed Wiles to relate the Galois representations attached to elliptic curves to those attached to modular forms
• The study of L-functions and the modularity theorem has led to significant advances in number theory, including the Sato-Tate conjecture and the Langlands program
• Generalized notions of modularity, such as automorphic forms and Galois representations, are active areas of research in modern number theory

Applications in Number Theory and Cryptography

• Modular forms have numerous applications in various branches of number theory, including the study of elliptic curves, Galois representations, and Diophantine equations
• The Fourier coefficients of modular forms often encode arithmetic information, such as the number of points on elliptic curves over finite fields
• The study of congruences between modular forms and their Fourier coefficients has led to important results in number theory, such as the proof of the Ramanujan-Petersson conjecture
• Modular forms play a crucial role in the Langlands program, which seeks to unify various branches of mathematics through the study of automorphic forms and Galois representations
• In cryptography, modular forms have been used to construct error-correcting codes and to study the security of certain cryptographic protocols
• The theory of modular forms has been applied to the study of sphere packings and other problems in discrete geometry
• Modular forms have also found applications in mathematical physics, such as in the study of conformal field theories and string theory
• The computational aspects of modular forms, such as the efficient computation of Fourier coefficients and the construction of bases of spaces of modular forms, are important in both theoretical and applied settings