Galois representations and are powerful tools in modern number theory. They connect abstract algebra with complex analysis, providing insights into fundamental mathematical structures and relationships.
These concepts have led to groundbreaking results, including the proof of . The , which links Galois representations and automorphic forms, continues to drive research and shape our understanding of arithmetic geometry.
Galois representations and modular forms
Definition and connection
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: continuous homomorphism from the absolute Galois group of a field to the general linear group of a vector space over a field
Modular forms: complex analytic functions on the upper half-plane that satisfy certain transformation properties under the action of the modular group
Transformation properties include invariance under specific subgroups of the modular group and holomorphicity at cusps
Galois representations associated with modular forms obtained by studying the action of the absolute Galois group on the étale cohomology of modular curves
Étale cohomology is a cohomology theory for algebraic varieties that takes into account the arithmetic properties of the variety
Modular curves are algebraic curves that parametrize with additional structure (level structure)
Key results and conjectures
Langlands correspondence predicts a bijection between certain Galois representations and of the adelic points of a reductive group
Automorphic representations are representations of reductive groups over that satisfy certain analytic and algebraic properties
Adeles are a ring of "local" fields that contain information about all completions of a global field
(now a theorem) states that every elliptic curve over the rational numbers is modular, i.e., its L-function agrees with the L-function of a modular form
are complex analytic functions associated with arithmetic objects that encode important information about their properties
Proof of Fermat's Last Theorem by Andrew Wiles relied on establishing the modularity of , connecting Galois representations and modular forms
Semistable elliptic curves have a specific type of reduction at primes of bad reduction (multiplicative reduction)
Significance of the Langlands program
Overview and scope
Langlands program is a series of far-reaching conjectures that relate Galois representations and automorphic forms
Seeks to establish a correspondence between of the absolute Galois group of a number field and automorphic representations of the adelic points of a reductive group
l-adic Galois representations are Galois representations on vector spaces over the field of l-adic numbers (completions of the algebraic closure of the rational numbers with respect to a prime l)
Langlands correspondence proven in special cases (GL(1), GL(2)) but remains open in general
GL(n) denotes the general linear group of degree n, consisting of invertible n×n matrices
Global and local Langlands correspondence
relates global Galois representations to automorphic representations
Global Galois representations are representations of the absolute Galois group of a number field
deals with local Galois representations and local automorphic representations
Local Galois representations are representations of the absolute Galois group of a local field (completion of a number field with respect to a prime)
Local automorphic representations are representations of the points of a reductive group over a local field
Langlands program has deep connections with number theory, representation theory, and harmonic analysis
Proof of the Taniyama-Shimura conjecture can be viewed as a special case of the Langlands correspondence for GL(2) over the rational numbers
Galois representations in elliptic curves
Galois representations and L-functions
Elliptic curves are algebraic curves defined by a cubic equation in two variables, and their L-functions encode important arithmetic information
Galois representation associated with an elliptic curve obtained by studying the action of the absolute Galois group on the of the curve
Tate module is the inverse limit of the of an elliptic curve
Taniyama-Shimura conjecture (now a theorem) states that the L-function of an elliptic curve over the rational numbers agrees with the L-function of a modular form
Proof of the Taniyama-Shimura conjecture by Andrew Wiles and Richard Taylor relied on establishing the modularity of semistable elliptic curves using Galois representations
Applications and related conjectures
relates the rank of an elliptic curve to the order of vanishing of its L-function at s=1, and Galois representations play a crucial role in its study
Rank of an elliptic curve is the number of independent rational points of infinite order
Galois representations have been used to prove important results about the torsion points and of elliptic curves
Torsion points are points of finite order on an elliptic curve
Isogenies are morphisms between elliptic curves that preserve the group structure
Impact on modern number theory
Breakthroughs and new directions
Galois representations have revolutionized number theory by providing a powerful tool for studying arithmetic objects (elliptic curves, modular forms)
Proof of Fermat's Last Theorem by Andrew Wiles relied heavily on Galois representations to establish the modularity of semistable elliptic curves
Langlands program, relating Galois representations and automorphic forms, has become a central focus of research in modern number theory
Galois representations used to prove important results in Iwasawa theory, which studies the behavior of arithmetic objects in towers of number fields
Iwasawa theory investigates the growth of arithmetic invariants (class groups, unit groups) in infinite extensions of number fields
Connections with other fields
Study of has led to significant advances in understanding and
p-adic L-functions are analogues of complex L-functions that take values in p-adic fields
p-adic Hodge theory studies the relationship between p-adic Galois representations and p-adic differential equations
Galois representations have played a key role in the development of the theory of motives, which aims to provide a unified framework for studying arithmetic and geometric objects
Motives are abstract objects that capture the essential properties of algebraic varieties and their cohomology
Use of Galois representations has opened up new avenues for collaboration between number theory and other areas of mathematics (representation theory, algebraic geometry)