🔢Arithmetic Geometry Unit 5 – Galois representations

Key Concepts and Definitions

• Galois group $\text{Gal}(L/K)$ consists of all field automorphisms of $L$ that fix $K$ pointwise, where $L/K$ is a Galois extension
• Field extension $L/K$ implies $K$ is a subfield of $L$ and $L$ is a vector space over $K$
  • Degree of extension $[L:K]$ equals the dimension of $L$ as a vector space over $K$
• Galois extension $L/K$ has $\text{Aut}(L/K) = \text{Gal}(L/K)$ acting transitively on the roots of irreducible polynomials over $K$ that split in $L$
• Representation of a group $G$ on a vector space $V$ is a group homomorphism $\rho: G \to \text{GL}(V)$
  • Linear action of $G$ on $V$ preserves the vector space structure
• Character of a representation $\rho$ is the trace function $\chi_\rho(g) = \text{tr}(\rho(g))$, encoding essential information about $\rho$
• Arithmetic geometry studies algebraic varieties over number fields, combining tools from algebraic geometry and number theory

Historical Context and Motivation

• Galois theory originated in the work of Évariste Galois (1811-1832) on solving polynomial equations by radicals
• Galois introduced permutation groups and their connection to field extensions, laying the foundation for modern Galois theory
• Representation theory emerged in the late 19th century, with contributions from Frobenius, Schur, and Burnside
  • Motivated by the study of permutation groups and their actions on vector spaces
• Galois representations combine Galois theory and representation theory, providing a powerful tool for studying arithmetic properties of algebraic varieties
• Grothendieck's theory of étale cohomology in the 1960s revolutionized arithmetic geometry and highlighted the importance of Galois representations
• Langlands program, formulated in the late 1960s, conjectures deep connections between Galois representations and automorphic forms

Galois Groups and Field Extensions

• Galois group $\text{Gal}(L/K)$ measures the symmetries of the field extension $L/K$
• Fundamental Theorem of Galois Theory establishes a correspondence between subgroups of $\text{Gal}(L/K)$ and intermediate fields $K \subseteq M \subseteq L$
  • Normal subgroups correspond to Galois extensions
• Splitting field of a polynomial $f(x) \in K[x]$ is the smallest field extension $L/K$ where $f(x)$ factors into linear factors
  • $\text{Gal}(L/K)$ permutes the roots of $f(x)$, giving a faithful action on the roots
• Galois correspondence allows the study of field extensions through their Galois groups
• Inverse Galois problem asks which groups occur as Galois groups of extensions of a given field (e.g., $\mathbb{Q}$)
• Kummer theory describes abelian extensions of a field containing roots of unity, using the multiplicative group structure

Representation Theory Basics

• Representation $\rho: G \to \text{GL}(V)$ encodes the action of a group $G$ on a vector space $V$
• Irreducible representation cannot be decomposed into smaller subrepresentations
  • Every representation is a direct sum of irreducible representations
• Character $\chi_\rho$ determines the representation $\rho$ up to isomorphism (by the Artin-Wedderburn theorem)
• Schur's lemma states that morphisms between irreducible representations are either zero or isomorphisms
• Tensor products and dual representations allow the construction of new representations from existing ones
• Induced representations, such as the permutation representation, are built from representations of subgroups
• Characters form a basis for the space of class functions on $G$, enabling the decomposition of representations

Galois Representations: Construction and Properties

• Galois representation is a continuous homomorphism $\rho: \text{Gal}(\overline{K}/K) \to \text{GL}_n(\overline{\mathbb{Q}}_\ell)$ for a prime $\ell$ and a field $K$
  • Encodes the action of the absolute Galois group on an $\ell$-adic vector space
• Galois representations arise naturally from the action of $\text{Gal}(\overline{K}/K)$ on the étale cohomology of algebraic varieties
  • Étale cohomology groups $H^i_{\text{ét}}(X_{\overline{K}}, \mathbb{Q}_\ell)$ are $\ell$-adic vector spaces with a continuous $\text{Gal}(\overline{K}/K)$-action
• Tate module $T_\ell(A)$ of an abelian variety $A$ over $K$ gives a Galois representation, capturing arithmetic information about $A$
• Galois representations are unramified at all but finitely many primes, allowing the definition of local $L$-factors
• Modularity of Galois representations relates them to automorphic forms and has deep arithmetic consequences (e.g., Fermat's Last Theorem)

Applications in Arithmetic Geometry

• Galois representations provide a powerful tool for studying the arithmetic of algebraic varieties over number fields
• Tate conjecture relates the Galois action on étale cohomology to the existence of algebraic cycles
  • Proven in some cases, such as for abelian varieties over finite fields (Tate-Honda theory)
• Serre's modularity conjecture (now a theorem) characterizes odd, irreducible, 2-dimensional mod $p$ Galois representations as arising from modular forms
  • Crucial ingredient in the proof of Fermat's Last Theorem
• Galois representations are central to the Langlands program, which predicts a correspondence between Galois representations and automorphic representations
• Fontaine-Mazur conjecture describes the geometric Galois representations that should arise from étale cohomology of algebraic varieties
• Galois representations are used to study the Birch and Swinnerton-Dyer conjecture, relating the rank of an elliptic curve to its $L$-function

Computational Techniques and Examples

• Explicit computation of Galois groups is possible for polynomials of low degree using resolvent polynomials and discriminants
  • Example: The polynomial $x^3 - 2$ has Galois group $S_3$ over $\mathbb{Q}$
• Magma, PARI/GP, and SageMath are computer algebra systems with built-in functions for working with Galois groups and representations
• Modular forms can be computed using modular symbols, allowing the explicit construction of Galois representations
  • Example: The modular form $\Delta(z)$ corresponds to a Galois representation $\rho_\Delta: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{Z}_\ell)$
• Elliptic curves over finite fields provide a rich source of examples for studying Galois representations
  • Example: The Frobenius endomorphism of an elliptic curve over $\mathbb{F}_q$ generates a Galois representation
• Computational methods, such as point counting and $L$-function computations, are used to investigate the arithmetic of Galois representations

Advanced Topics and Current Research

• $p$-adic Hodge theory, developed by Fontaine and others, studies $p$-adic Galois representations and their relation to $p$-adic analysis
  • Crystalline, semi-stable, and de Rham representations are important classes of $p$-adic representations
• Geometric Langlands program, initiated by Beilinson and Drinfeld, is an analog of the Langlands program for curves over complex numbers
  • Relates Galois representations to $D$-modules and connections on vector bundles
• Motive is a conjectural generalization of the notion of a cohomology group, aiming to unify various cohomology theories
  • Motivic Galois groups are expected to govern the Galois representations arising from motives
• Shimura varieties are higher-dimensional analogs of modular curves, providing a geometric framework for studying Galois representations and automorphic forms
• Perfectoid spaces, introduced by Scholze, have revolutionized $p$-adic geometry and have applications to Galois representations and the Langlands program
  • Example: The construction of Galois representations attached to torsion classes in the cohomology of locally symmetric spaces