12.3 Galois representations and modular forms

Galois representations and modular forms 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 Fermat's Last Theorem. The Langlands program, 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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• Galois representation: 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 elliptic curves with additional structure (level structure)

Key results and conjectures

• Langlands correspondence predicts a bijection between certain Galois representations and automorphic representations of the adelic points of a reductive group
  • Automorphic representations are representations of reductive groups over adeles that satisfy certain analytic and algebraic properties
  • Adeles are a ring of "local" fields that contain information about all completions of a global field
• Taniyama-Shimura conjecture (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
  • L-functions 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 semistable elliptic curves, 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 l-adic Galois representations 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

• Global Langlands correspondence relates global Galois representations to automorphic representations
  • Global Galois representations are representations of the absolute Galois group of a number field
• Local Langlands correspondence 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 Tate module of the curve
  • Tate module is the inverse limit of the torsion points 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

• Birch and Swinnerton-Dyer conjecture 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 isogenies 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 p-adic Galois representations has led to significant advances in understanding p-adic L-functions and p-adic Hodge theory
  • 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)