Artin L-function

Definition

An Artin L-function is a complex function associated with a Dirichlet character and a Galois representation that arises in number theory, particularly in the study of field extensions and the distribution of prime numbers. These functions are designed to generalize the Riemann zeta function and have deep connections to algebraic number theory, particularly in understanding the properties of abelian extensions of number fields.

5 Must Know Facts For Your Next Test

1. Artin L-functions can be expressed as an Euler product over primes, which captures important information about the distribution of primes in number fields.
2. These functions are used to prove deep results like the Artin reciprocity law, which relates L-functions to the Galois groups of field extensions.
3. The analytic properties of Artin L-functions can provide insights into their zeros, which is related to important conjectures in number theory such as the Generalized Riemann Hypothesis.
4. Artin L-functions are connected to various conjectures about the non-vanishing of L-functions at specific points, like the critical line in complex analysis.
5. They play a central role in the Langlands program, which seeks to relate number theory and representation theory through a series of conjectured connections.

Review Questions

• How do Artin L-functions relate to Dirichlet characters and Galois representations?
  • Artin L-functions arise from associating a Dirichlet character to a Galois representation of a number field. The Dirichlet character captures information about how primes behave in relation to the Galois group, while the Galois representation allows for deeper insights into the symmetry and structure of field extensions. Together, they create a framework for understanding the arithmetic properties of numbers through complex functions.
• Discuss the significance of Artin L-functions in proving results like the Artin reciprocity law.
  • The Artin reciprocity law establishes a profound link between Artin L-functions and Galois groups, showing how these functions can be used to understand the solvability of polynomial equations over different fields. This law articulates how L-functions encapsulate information about abelian extensions of number fields, providing tools to analyze their behavior under various arithmetic operations. The results derived from this law have far-reaching implications across number theory.
• Evaluate how Artin L-functions contribute to the Langlands program and their role in modern number theory.
  • Artin L-functions are pivotal in the Langlands program as they serve as key components in establishing relationships between number theory and representation theory. This program proposes conjectures that connect Galois representations with automorphic forms, where Artin L-functions act as crucial examples illustrating these connections. Their properties help mathematicians to advance toward unifying various branches of mathematics and tackling significant open questions related to prime distribution and L-function behavior.

Related terms

Dirichlet Character: A Dirichlet character is a completely multiplicative arithmetic function that is periodic and takes values in the complex numbers, which plays a crucial role in the study of Dirichlet L-functions.

Galois Representation: A Galois representation is a homomorphism from the Galois group of a field extension into a linear group, allowing one to study field extensions through linear algebraic methods.

Riemann Zeta Function: The Riemann zeta function is a special function of great significance in number theory and mathematical analysis, defined as the infinite series $\\sum_{n=1}^{\infty} \frac{1}{n^s}$ for complex numbers s with real part greater than 1.