Problem 2 - Complex Analysis

Question

The natural group homomorphism $$ \mathrm{SL}(2, \mathbb{Z}) \longrightarrow \mathrm{SL}(2, \mathbb{Z} / q \mathbb{Z}) $$ is surjective. In particular, $$ [\Gamma: \Gamma[q]]=\\# \operatorname{SL}(2, \mathbb{Z} / q \mathbb{Z}) $$

Step-by-Step Solution

Answer

The homomorphism is surjective and \([\Gamma: \Gamma[q]] = \# \operatorname{SL}(2, \mathbb{Z}/q\mathbb{Z})\).

Step 1: Understanding the Problem

We need to show that the natural homomorphism from \( \mathrm{SL}(2, \mathbb{Z}) \) to \( \mathrm{SL}(2, \mathbb{Z}/q\mathbb{Z}) \) is surjective. This means every element in \( \mathrm{SL}(2, \mathbb{Z}/q\mathbb{Z}) \) has a preimage in \( \mathrm{SL}(2, \mathbb{Z}) \). Additionally, we need to verify the index equivalence \( [\Gamma: \Gamma[q]] = \# \operatorname{SL}(2, \mathbb{Z}/q\mathbb{Z}) \), where \( \Gamma = \mathrm{SL}(2, \mathbb{Z}) \) and \( \Gamma[q] \) is the kernel of this homomorphism.

Step 2: Set Up the Homomorphism

Consider the homomorphism \( \phi: \mathrm{SL}(2, \mathbb{Z}) \rightarrow \mathrm{SL}(2, \mathbb{Z}/q\mathbb{Z}) \) defined by reducing the entries of a matrix modulo \( q \). For a matrix \( A \in \mathrm{SL}(2, \mathbb{Z}) \), \( \phi(A) \in \mathrm{SL}(2, \mathbb{Z}/q\mathbb{Z}) \) because it retains the property that \( \det(A) = 1 \), now considered in \( \mathbb{Z}/q\mathbb{Z} \).

Step 3: Show Surjectivity

We need to demonstrate that every matrix in \( \mathrm{SL}(2, \mathbb{Z}/q\mathbb{Z}) \) is an image under \( \phi \) of some matrix in \( \mathrm{SL}(2, \mathbb{Z}) \). This proof involves showing that for any matrix \( B \in \mathrm{SL}(2, \mathbb{Z}/q\mathbb{Z}) \), we can find a matrix in \( \mathrm{SL}(2, \mathbb{Z}) \) which reduces to \( B \). This is achievable as according to standard results in arithmetic group theory, the reduction map is always surjective for modular arithmetic over integers.

Step 4: Verify the Index

The index \( [\Gamma: \Gamma[q]] \) represents the number of distinct cosets of \( \Gamma[q] \) in \( \Gamma \). Since \( \phi \) is surjective, \( \Gamma / \Gamma[q] \cong \operatorname{SL}(2, \mathbb{Z}/q\mathbb{Z}) \) as groups. Thus, the index is equal to the order of \( \operatorname{SL}(2, \mathbb{Z}/q\mathbb{Z}) \), proving that \( [\Gamma: \Gamma[q]] = \# \operatorname{SL}(2, \mathbb{Z}/q\mathbb{Z}) \).

Step 5: Conclusion

Hence, we have shown that the natural group homomorphism \( \mathrm{SL}(2, \mathbb{Z}) \longrightarrow \mathrm{SL}(2, \mathbb{Z}/q\mathbb{Z}) \) is surjective. Consequently, the index \( [\Gamma: \Gamma[q]] \) is equal to the cardinality of \( \mathrm{SL}(2, \mathbb{Z}/q\mathbb{Z}) \), confirming the given exercise statement.

Key Concepts

Modular Arithmetic

Modular arithmetic is like clock arithmetic. It's a system where numbers wrap around after reaching a certain value, called the modulus. For instance, imagine a clock where after 12, it resets to 1. Similarly, in modular arithmetic, numbers wrap around after a specified modulus.

• This type of arithmetic is essential for computations involving a 'moduli', which could be any positive integer.
• When we say a number is equivalent "modulo some number", we mean after division by that number (the modulus), the remainder is the same.
• For example, in modulo 5 arithmetic: 7, 12, and 17 are equivalent because they leave a remainder of 2 when divided by 5.

Using modular arithmetic is helpful in group theory, which focuses on structures like matrices and operations within them.

SL(2, Z)

The term \( \mathrm{SL}(2, \mathbb{Z}) \) represents the special linear group of 2x2 integer matrices with determinant 1.

• This group plays a crucial role in various areas of mathematics like geometry and number theory.
• Matrices in \( \mathrm{SL}(2, \mathbb{Z}) \) transform a two-dimensional integer grid but preserve area because their determinant is 1.
• To be in \( \mathrm{SL}(2, \mathbb{Z}) \), a matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \) must satisfy \( ad - bc = 1 \).

Knowing this helps in understanding how such matrices operate and impact group structures when used in conditions involving modular arithmetic.

Surjective Mapping

A map or function is deemed surjective (or onto) if every element in the target set is mapped by at least one element from the domain.

• Surjectivity assures that the mapping is covering the entire target set, leaving no element unmapped.
• In our context, it indicates that every matrix in \( \mathrm{SL}(2, \mathbb{Z}/q\mathbb{Z}) \) comes from some matrix in \( \mathrm{SL}(2, \mathbb{Z}) \).
• This property is pivotal when proving that two groups are essentially structured in a certain way under a given mapping, revealing relationships between their elements.

Understanding surjectivity is crucial in algebra as it provides insights into potential equivalences and isomorphisms between group structures.

Kernel of a Homomorphism

The kernel of a homomorphism is a subset of elements in the domain that map to the identity element in the codomain.

• It's a measure of how far the homomorphism is from being an isomorphism.
• For a homomorphism \( \phi: G \to H \), the kernel is defined as \( \text{ker}(\phi) = \{ g \in G \mid \phi(g) = e_H \} \), where \( e_H \) is the identity element in \( H \).
• The kernel's size or structure often provides information about the nature of the surjective mapping, helping identify whether or not the mapping is injective.

This concept is key in determining the quotient groups and understanding the fundamental theorem of homomorphisms, providing a direct insight into the structure of mathematical objects.