Artin L-functions

Definition

Artin L-functions are complex functions associated with Artin representations, which arise from Galois representations of number fields. They generalize Dirichlet L-functions and play a crucial role in number theory, particularly in understanding the behavior of primes and solving various problems related to Galois cohomology and class field theory. These functions encode important arithmetic information and have connections to the conjectural properties of automorphic forms and the Langlands program.

5 Must Know Facts For Your Next Test

1. Artin L-functions can be expressed as a product over primes, reflecting how primes split in the number field associated with the Artin representation.
2. They satisfy functional equations that relate values at $s$ and $1-s$, mirroring properties found in other types of L-functions.
3. The order of the zeroes and poles of Artin L-functions is closely related to the degree of the associated Artin representation.
4. There exists a conjecture relating Artin L-functions to the Riemann Hypothesis, suggesting that their nontrivial zeroes lie on a critical line.
5. Artin L-functions play a significant role in the Langlands program, connecting number theory and representation theory through their properties.

Review Questions

• How do Artin L-functions relate to Artin representations and what significance do they have in number theory?
  • Artin L-functions are constructed from Artin representations, which capture how Galois groups act on algebraic objects. These functions provide deep insights into the distribution of primes and serve as crucial tools in studying Galois cohomology and class field theory. By encoding arithmetic data about number fields, Artin L-functions help to bridge connections between different areas in mathematics.
• Discuss the similarities and differences between Artin L-functions and Dirichlet L-functions.
  • Both Artin and Dirichlet L-functions are types of L-functions used in number theory, but they arise in different contexts. Dirichlet L-functions are associated specifically with characters of the integers, particularly useful for understanding primes in arithmetic progressions. In contrast, Artin L-functions generalize this concept to encompass representations of Galois groups over number fields, allowing for a richer framework that incorporates more complex algebraic structures and relationships.
• Evaluate the implications of the conjectural relationships between Artin L-functions and the Riemann Hypothesis on modern number theory.
  • The conjectured connections between Artin L-functions and the Riemann Hypothesis suggest that if proven true, these relationships could lead to a deeper understanding of the distribution of prime numbers across various number fields. Such insights could unify many branches of number theory under a common framework, impacting not only theoretical research but also practical applications in cryptography and coding theory. Addressing these conjectures may open new pathways for exploration within the Langlands program, potentially revolutionizing how we view connections between arithmetic and geometry.

Related terms

Artin Representations: These are representations of Galois groups of number fields into general linear groups, capturing the action of Galois on the roots of polynomials.

L-function: A class of complex functions that arise in number theory, used to study various properties of numbers, including the distribution of prime numbers.

Dirichlet L-functions: These are specific examples of L-functions associated with Dirichlet characters, crucial for studying primes in arithmetic progressions.