3.1 Modular groups

Modular groups connect number theory, complex analysis, and algebraic geometry. They arise in studying elliptic curves and modular forms, bridging discrete and continuous math. Understanding them lays the groundwork for exploring deeper concepts in arithmetic geometry.

These groups act on the upper half-plane through fractional linear transformations. This action preserves hyperbolic geometry and helps visualize quotient spaces. Modular groups exhibit both algebraic and geometric properties, making them powerful tools in arithmetic geometry.

Definition of modular groups

• Modular groups play a crucial role in arithmetic geometry by connecting number theory, complex analysis, and algebraic geometry
• These groups arise naturally in the study of elliptic curves and modular forms, providing a bridge between discrete and continuous mathematics
• Understanding modular groups lays the foundation for exploring deeper concepts in arithmetic geometry, such as modular curves and their arithmetic properties

SL(2,Z) and PSL(2,Z)

• SL(2,Z) denotes the special linear group of 2x2 matrices with integer entries and determinant 1
• Elements of SL(2,Z) take the form $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ where $ad - bc = 1$ and $a, b, c, d \in \mathbb{Z}$
• PSL(2,Z) represents the projective special linear group, obtained by quotienting SL(2,Z) by its center $\{\pm I\}$
• PSL(2,Z) consists of equivalence classes of matrices in SL(2,Z) under the relation $A \sim -A$
• These groups have important applications in the theory of modular forms and the arithmetic of elliptic curves

Action on upper half-plane

• Modular groups act on the upper half-plane $\mathbb{H} = \{z \in \mathbb{C} : \text{Im}(z) > 0\}$ via fractional linear transformations
• The action of a matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ on $z \in \mathbb{H}$ is given by $z \mapsto \frac{az + b}{cz + d}$
• This action preserves the hyperbolic geometry of the upper half-plane
• Orbits of points under this action correspond to equivalence classes of lattices in $\mathbb{C}$
• Understanding this action helps visualize the quotient space $\mathbb{H}/\text{PSL}(2,\mathbb{Z})$, which has a rich geometric structure

Properties of modular groups

• Modular groups exhibit both algebraic and geometric properties, making them powerful tools in arithmetic geometry
• These groups bridge discrete and continuous aspects of mathematics, connecting number-theoretic problems to geometric and analytic methods
• Studying the properties of modular groups provides insights into the structure of modular forms and the arithmetic of elliptic curves

Discrete subgroups

• Modular groups are discrete subgroups of SL(2,R), the group of 2x2 real matrices with determinant 1
• Discreteness implies that the group elements are isolated points in the topology of SL(2,R)
• This property allows for the construction of fundamental domains and quotient spaces with rich geometric structure
• Discreteness ensures that the action on the upper half-plane has no accumulation points
• Important examples of discrete subgroups include congruence subgroups (Γ(N), Γ₀(N), Γ₁(N))

Generators and relations

• PSL(2,Z) can be generated by two elements: S and T, where S(z) = -1/z and T(z) = z + 1
• These generators satisfy the relations S² = (ST)³ = I, where I is the identity element
• The presentation of PSL(2,Z) in terms of generators and relations is $\langle S, T | S² = (ST)³ = I \rangle$
• Understanding the generators and relations helps in studying the group structure and its action on the upper half-plane
• This presentation connects modular groups to the theory of Fuchsian groups and hyperbolic geometry

Fundamental domain

• The concept of fundamental domains is crucial in arithmetic geometry for understanding quotient spaces and modular forms
• Fundamental domains provide a geometric realization of the action of modular groups on the upper half-plane
• Studying fundamental domains helps visualize the structure of modular curves and their compactifications

Construction and visualization

• A fundamental domain for PSL(2,Z) is the region $\{z \in \mathbb{H} : |z| \geq 1, |\text{Re}(z)| \leq \frac{1}{2}\}$
• This domain can be visualized as a hyperbolic triangle in the upper half-plane
• The boundary of the fundamental domain consists of three geodesic segments
• Points on the boundary are identified under the action of the generators S and T
• Constructing fundamental domains for other modular groups involves similar principles but may result in more complex shapes

Tessellation of upper half-plane

• The images of the fundamental domain under the action of PSL(2,Z) form a tessellation of the upper half-plane
• This tessellation consists of congruent hyperbolic triangles covering the entire upper half-plane without overlaps
• Each triangle in the tessellation corresponds to a coset of PSL(2,Z)
• The tessellation provides a visual representation of the group action and the quotient space
• Understanding the tessellation helps in studying modular forms and their behavior under the group action

Modular forms

• Modular forms are complex-analytic functions on the upper half-plane with specific transformation properties under modular groups
• These functions play a central role in arithmetic geometry, connecting number theory, complex analysis, and algebraic geometry
• Modular forms encode deep arithmetic information and have applications in various areas of mathematics and physics

Definition and examples

• A modular form of weight k for PSL(2,Z) is a holomorphic function f on $\mathbb{H}$ satisfying $f(\frac{az+b}{cz+d}) = (cz+d)^k f(z)$ for all $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \text{PSL}(2,\mathbb{Z})$
• Modular forms must also be holomorphic at the cusp $i\infty$
• The simplest non-trivial example is the Eisenstein series $G_k(z) = \sum_{(m,n) \neq (0,0)} \frac{1}{(mz+n)^k}$ for even k ≥ 4
• The discriminant function $\Delta(z) = (2\pi)^{12} \eta(z)^{24}$, where η is the Dedekind eta function, is a weight 12 cusp form
• Theta functions associated with lattices provide another important class of modular forms

Weight and level

• The weight k of a modular form determines its transformation behavior under the group action
• Integer weights correspond to scalar-valued modular forms, while half-integer weights relate to metaplectic forms
• The level N of a modular form specifies the congruence subgroup for which it is invariant
• Forms of higher level often have richer arithmetic properties and are associated with modular curves of higher genus
• The space of modular forms of a given weight and level has a finite dimension, which can be calculated using the Riemann-Roch theorem