6.4 Artin L-functions

Artin L-functions connect Galois representations to complex analysis, providing a powerful tool for studying algebraic number fields. They encode arithmetic information about field extensions and reveal deep connections between various areas of mathematics.

These functions exhibit remarkable properties, including functional equations and analytic continuation. Understanding Artin L-functions is crucial for exploring deeper connections in algebraic number theory and applying them to solve problems in arithmetic geometry.

Definition of Artin L-functions

• Artin L-functions play a crucial role in arithmetic geometry by connecting Galois representations to complex analysis
• These functions provide a powerful tool for studying algebraic number fields and their extensions
• Understanding Artin L-functions forms a foundation for exploring deeper connections in algebraic number theory and representation theory

Galois representations

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• Linear representations of Galois groups over number fields
• Map Galois groups to general linear groups GL(n,C)
• Encode arithmetic information about field extensions
• Provide a framework for studying symmetries in algebraic structures

Artin characters

• Character functions associated with Galois representations
• Defined as traces of representation matrices
• Determine the representation up to isomorphism
• Play a key role in defining Artin L-functions
• Used to classify representations and study their properties

Meromorphic continuation

• Artin L-functions initially defined for Re(s) > 1
• Extended to meromorphic functions on the entire complex plane
• Analytic continuation achieved through complex analysis techniques
• Reveals deep connections between arithmetic and analysis
• Allows for the study of L-functions' behavior at critical points

Properties of Artin L-functions

• Artin L-functions exhibit remarkable analytical and arithmetic properties
• These properties connect various areas of mathematics, including number theory, complex analysis, and representation theory
• Understanding these properties is essential for applying Artin L-functions to solve problems in arithmetic geometry

Functional equation

• Relates values of L(s,χ) to L(1-s,χ̄)
• Involves gamma factors and a root number
• Symmetry around the critical line Re(s) = 1/2
• Provides information about zeros and poles of the L-function
• Crucial for understanding the behavior of L-functions in the critical strip

Analytic continuation

• Extends the domain of L-functions beyond their initial region of convergence
• Achieved through various methods (Hecke's method, Tate's thesis)
• Reveals the true nature of L-functions as global objects
• Allows for the study of special values and zeros
• Connects to the Riemann Hypothesis for Artin L-functions

Special values

• L-function values at specific points carry arithmetic significance
• Include values at non-positive integers and s = 1
• Related to class numbers, regulators, and other number-theoretic invariants
• Studied through p-adic L-functions and Iwasawa theory
• Provide insights into the structure of algebraic number fields

Artin's conjecture

• Artin's conjecture represents a fundamental problem in the theory of L-functions
• This conjecture connects representation theory, complex analysis, and number theory
• Resolving Artin's conjecture would have far-reaching implications for arithmetic geometry

Statement of conjecture

• Asserts holomorphy of Artin L-functions for non-trivial irreducible representations
• Excludes the trivial representation, which has a pole at s = 1
• Implies that L(s,χ) is entire for χ ≠ 1
• Generalizes known results for abelian representations
• Remains one of the most important open problems in number theory

Known cases

• Proved for abelian representations (Artin reciprocity law)
• Established for certain low-dimensional representations
• Verified for specific families of non-abelian extensions
• Partial results obtained through modular forms and automorphic representations
• Recent advancements using potential automorphy techniques

Implications for number theory

• Would establish a deep connection between Galois representations and automorphic forms
• Provides evidence for the Langlands program
• Impacts the study of zeros of L-functions and the Riemann Hypothesis
• Influences research on algebraic varieties and their zeta functions
• Has applications to elliptic curves and modular forms

Relationship to other L-functions

• Artin L-functions form part of a larger family of L-functions in number theory
• Understanding these relationships provides a unified view of various L-functions
• Connections between different L-functions reveal deep structures in arithmetic geometry

Dedekind zeta functions

• Generalize the Riemann zeta function to number fields
• Express as products of Artin L-functions (Artin factorization)
• Encode information about prime ideals and class numbers
• Related to the analytic class number formula
• Study of zeros connects to generalized Riemann Hypothesis

Hecke L-functions

• Generalize Dirichlet L-functions to number fields
• Can be expressed as Artin L-functions for certain representations
• Associated with Hecke characters and ideal class groups
• Play a role in the theory of complex multiplication
• Connect to special values of modular forms

Automorphic L-functions

• Arise from automorphic representations of reductive groups
• Conjecturally include all Artin L-functions (Langlands functoriality)
• Satisfy similar analytic properties (functional equation, meromorphic continuation)
• Central to the Langlands program and modern number theory
• Provide a framework for studying symmetries in arithmetic geometry

Applications in arithmetic geometry

• Artin L-functions serve as powerful tools in various areas of arithmetic geometry
• These applications demonstrate the deep connections between L-functions and geometric objects
• Understanding these applications is crucial for advancing research in algebraic number theory and arithmetic geometry

Modularity theorem

• Relates elliptic curves over Q to modular forms
• Proved using Artin representations and compatible systems
• Key ingredient in the proof of Fermat's Last Theorem
• Generalizes to higher-dimensional varieties (Serre's modularity conjecture)
• Connects geometric objects to automorphic forms

Sato-Tate conjecture

• Describes the distribution of Frobenius eigenvalues for elliptic curves
• Proved for elliptic curves over totally real fields with non-integral j-invariant
• Utilizes potential automorphy of symmetric powers of Galois representations
• Relates to the analytic properties of symmetric power L-functions
• Generalizes to higher-dimensional varieties and abelian varieties

Langlands program

• Unifies various areas of mathematics through automorphic representations
• Proposes a correspondence between Galois representations and automorphic forms
• Artin L-functions play a central role in formulating and testing conjectures
• Influences research in algebraic geometry, representation theory, and number theory
• Provides a framework for understanding symmetries in arithmetic objects

Computational aspects

• Computational methods play a crucial role in studying Artin L-functions
• These techniques allow for numerical verification of conjectures and exploration of L-function properties
• Computational approaches bridge the gap between theoretical results and concrete examples in arithmetic geometry

Algorithms for Artin L-functions

• Compute coefficients of L-series expansions
• Utilize techniques from computational algebraic number theory
• Involve calculations in Galois theory and representation theory
• Implement efficient methods for handling large degree extensions
• Develop algorithms for computing Artin conductors and root numbers

Numerical evaluation

• Compute special values of Artin L-functions to high precision
• Utilize techniques like the Euler-Maclaurin summation formula
• Implement methods for locating zeros on the critical line
• Develop algorithms for computing functional equation parameters
• Explore connections between L-function values and arithmetic invariants

Software implementations

• Develop specialized software packages for L-function computations (PARI/GP, Magma)
• Implement algorithms in computer algebra systems (SageMath, GAP)
• Create databases of L-functions and their properties (LMFDB)
• Provide tools for visualizing L-function behavior and zero distributions
• Facilitate collaborative research through open-source software development

Generalizations and extensions

• The theory of Artin L-functions has inspired various generalizations and extensions
• These developments expand the applicability of L-functions to broader contexts in arithmetic geometry
• Studying these generalizations provides insights into the fundamental nature of L-functions

Non-abelian L-functions

• Extend Artin L-functions to more general Galois representations
• Study L-functions associated with representations of profinite groups
• Investigate analytic properties and functional equations in non-abelian settings
• Explore connections to Langlands functoriality and base change
• Develop techniques for studying L-functions of higher-dimensional varieties

Artin-Mazur L-functions

• Generalize Artin L-functions to étale cohomology of schemes
• Encode information about arithmetic and geometric properties of varieties
• Satisfy conjectural functional equations and meromorphic continuation
• Connect to the Weil conjectures and the Hasse-Weil zeta function
• Provide a framework for studying L-functions of motives

p-adic Artin L-functions

• Construct p-adic analogues of complex Artin L-functions
• Utilize p-adic measures and Iwasawa theory
• Study p-adic interpolation of special values
• Investigate connections to p-adic Hodge theory and Galois cohomology
• Explore applications to Iwasawa main conjectures and p-adic BSD conjecture

Historical development

• The theory of Artin L-functions has evolved significantly since its inception
• Tracing this development provides context for understanding modern research in arithmetic geometry
• Key historical contributions have shaped our current understanding of L-functions and their applications

Artin's original work

• Introduced Artin L-functions in the 1920s
• Generalized Dirichlet L-functions to non-abelian Galois extensions
• Formulated the Artin reciprocity law, generalizing quadratic reciprocity
• Proposed the Artin conjecture on the holomorphy of L-functions
• Laid the foundation for the modern theory of L-functions

Brauer's theorem

• Proved by Richard Brauer in the 1940s
• Showed that every Artin L-function is a ratio of Hecke L-functions
• Established meromorphic continuation and functional equation for Artin L-functions
• Provided a crucial tool for studying non-abelian L-functions
• Influenced later developments in representation theory and class field theory

Modern advancements

• Langlands program formulated in the 1960s, providing a broader context for Artin L-functions
• Development of ℓ-adic cohomology and étale cohomology in the 1960s and 1970s
• Advances in modularity and potential automorphy in the late 20th and early 21st centuries
• Computational techniques and algorithms developed for studying L-functions
• Recent progress on cases of Artin's conjecture and Sato-Tate conjecture

Open problems and conjectures

• The study of Artin L-functions continues to generate important open problems and conjectures
• These unresolved questions drive current research in arithmetic geometry and related fields
• Understanding these open problems provides insight into the frontiers of modern number theory

Artin's holomorphy conjecture

• Remains unproven for general non-abelian representations
• Partial results obtained for specific families of representations
• Connected to the theory of automorphic representations and functoriality
• Influences research on Galois representations and their L-functions
• Resolution would have significant implications for the Langlands program

Dedekind conjecture

• Asserts non-vanishing of Dedekind zeta functions at s = 1
• Equivalent to the non-vanishing of Artin L-functions at s = 1 for the regular representation
• Proved for abelian extensions (Dirichlet's theorem)
• Connected to class field theory and the structure of ideal class groups
• Resolution would provide insights into the arithmetic of number fields

Langlands reciprocity conjecture

• Proposes a correspondence between Galois representations and automorphic representations
• Generalizes class field theory to non-abelian extensions
• Implies Artin's conjecture and the Sato-Tate conjecture
• Connects various areas of mathematics (number theory, algebraic geometry, representation theory)
• Resolution would unify large parts of modern number theory and representation theory