🔢Arithmetic Geometry Unit 7 – Abelian varieties

Definition and Basic Properties

• Abelian varieties defined as complete algebraic varieties equipped with a group law that is compatible with the algebraic structure
• Projective varieties lack a natural group structure, so the group law on an Abelian variety must be explicitly defined
• Simplest examples of Abelian varieties include elliptic curves and Jacobians of algebraic curves
  • Elliptic curves are one-dimensional Abelian varieties defined by a cubic equation in the projective plane
  • Jacobians generalize elliptic curves to higher dimensions and encode important geometric information about algebraic curves
• Abelian varieties are always commutative groups, meaning the group operation satisfies the commutative property $a + b = b + a$ for all points $a$ and $b$
• Fundamental theorem states that every Abelian variety is isogenous to a product of simple Abelian varieties
  • Simple Abelian varieties cannot be decomposed further into non-trivial Abelian subvarieties
• Abelian varieties are non-singular, meaning they do not contain any singular points where the tangent space has higher dimension than expected
• Dimension of an Abelian variety refers to its dimension as a complex manifold, which is always a non-negative integer

Algebraic Structure

• Group law on an Abelian variety is defined by regular functions, making it an algebraic group
• Rational points of an Abelian variety form a finitely generated Abelian group, known as the Mordell-Weil group
  • Rank of the Mordell-Weil group measures the number of independent rational points of infinite order
• Endomorphism ring of an Abelian variety consists of all morphisms from the variety to itself that preserve the group structure
  • Endomorphism ring is a finitely generated $\mathbb{Z}$-module and carries important arithmetic information
• Abelian varieties can be classified up to isogeny by their endomorphism algebras, which are obtained by tensoring the endomorphism ring with $\mathbb{Q}$
• Rosati involution is a special involution on the endomorphism algebra that generalizes complex conjugation
  • Rosati involution plays a crucial role in the study of polarizations and dual varieties
• Tate modules of an Abelian variety encode the $\ell$-adic Galois representation attached to the variety, where $\ell$ is a prime number
  • Tate modules are important tools for studying the Galois action on the torsion points of the variety

Geometric Interpretation

• Abelian varieties can be viewed as higher-dimensional generalizations of elliptic curves
• Complex points of an Abelian variety form a complex torus $\mathbb{C}^g/\Lambda$, where $\Lambda$ is a lattice of rank $2g$
  • Lattice $\Lambda$ determines the shape and geometry of the Abelian variety
• Riemann bilinear relations provide a characterization of the lattices that give rise to Abelian varieties
  • Riemann bilinear relations involve the periods of the variety and ensure the existence of a compatible polarization
• Abelian varieties can be embedded into projective space using theta functions, which are special functions that generalize the Weierstrass $\wp$-function for elliptic curves
• Jacobians of algebraic curves are examples of principally polarized Abelian varieties
  • Principal polarization corresponds to an embedding of the Jacobian into projective space that is induced by the canonical divisor of the curve
• Néron-Severi group of an Abelian variety classifies the algebraic equivalence classes of divisors on the variety
  • Elements of the Néron-Severi group can be represented by hermitian forms on the complex vector space $\mathbb{C}^g$

Morphisms and Isogenies

• Morphisms between Abelian varieties are regular maps that preserve the group structure
• Isogenies are surjective morphisms with finite kernel, which can be viewed as analogues of isomorphisms in the category of Abelian varieties
  • Degree of an isogeny is the cardinality of its kernel, which is always a finite group
• Isogenies form a partially ordered set, with the ordering given by the division of degrees
  • Isogeny class of an Abelian variety consists of all varieties that are isogenous to it
• Isogeny theorem states that two Abelian varieties are isogenous if and only if their Tate modules are isomorphic as Galois modules
• Frobenius morphism is a special endomorphism of an Abelian variety defined over a finite field
  • Characteristic polynomial of the Frobenius morphism encodes important arithmetic information about the variety
• Isogeny-based cryptography utilizes the difficulty of computing isogenies between Abelian varieties for constructing secure cryptographic protocols
  • Isogeny-based cryptosystems are believed to be resistant to attacks by quantum computers

Torsion Points and Group Law

• Torsion points of an Abelian variety are the points of finite order with respect to the group law
• $n$-torsion subgroup $A[n]$ consists of all points that are annihilated by multiplication by $n$, where $n$ is a positive integer
  • $A[n]$ is a finite group of order $n^{2g}$, where $g$ is the dimension of the Abelian variety
• Torsion points are the key to understanding the group structure of an Abelian variety
  • Group law can be described explicitly in terms of the addition of torsion points
• Weil pairing is a bilinear pairing on the $n$-torsion subgroup that takes values in the group of $n$-th roots of unity
  • Weil pairing is alternating and non-degenerate, making it a powerful tool for studying the arithmetic of Abelian varieties
• Tate-Shafarevich group measures the failure of the local-global principle for Abelian varieties
  • Elements of the Tate-Shafarevich group are torsion points that are locally trivial but globally non-trivial
• Torsion conjecture states that for any fixed positive integer $n$, there are only finitely many isomorphism classes of Abelian varieties over a given number field with a point of order $n$
• Nagell-Lutz theorem provides a criterion for determining the torsion points on an elliptic curve defined over the rational numbers

Polarizations and Dual Varieties

• Polarization on an Abelian variety is an isogeny from the variety to its dual variety that satisfies certain symmetry conditions
• Dual variety of an Abelian variety $A$ is another Abelian variety $A^{\vee}$ that parametrizes the line bundles on $A$
  • Dual variety is functorially associated to the original variety and has the same dimension
• Principal polarizations are polarizations that are also isomorphisms between the variety and its dual
  • Jacobians of algebraic curves are examples of principally polarized Abelian varieties
• Polarizations can be described analytically in terms of positive definite hermitian forms on the complex vector space $\mathbb{C}^g$
• Polarized Abelian varieties form a moduli space, which is a fundamental object in algebraic geometry
  • Moduli space of principally polarized Abelian varieties has a natural compactification called the Satake compactification
• Polarizations are closely related to the intersection theory on the variety
  • Néron-Tate height is a quadratic form on the Mordell-Weil group that is induced by a polarization
• Kernel of a polarization is a finite subgroup of the variety that is isotropic with respect to the Weil pairing
  • Isotropic subgroups play a crucial role in the construction of isogenies and the study of the isogeny graph

Complex Analytic Theory

• Complex analytic theory provides a powerful set of tools for studying Abelian varieties over the complex numbers
• Uniformization theorem states that every complex Abelian variety is analytically isomorphic to a complex torus $\mathbb{C}^g/\Lambda$
  • Lattice $\Lambda$ is determined uniquely up to homothety by the Abelian variety
• Period matrix of an Abelian variety is a $g \times 2g$ matrix that encodes the lattice $\Lambda$ in terms of a basis of holomorphic differentials
  • Riemann bilinear relations impose conditions on the period matrix that ensure the existence of a compatible polarization
• Theta functions are holomorphic functions on the complex vector space $\mathbb{C}^g$ that are quasi-periodic with respect to the lattice $\Lambda$
  • Theta functions provide a way to embed Abelian varieties into projective space and study their geometry
• Siegel upper half-space is the moduli space of principally polarized complex Abelian varieties
  • Points in the Siegel upper half-space correspond to period matrices that satisfy the Riemann bilinear relations
• Schottky problem asks for a characterization of the Jacobians of algebraic curves among all principally polarized Abelian varieties
  • Schottky problem is a central question in the study of the geometry of Abelian varieties
• Intermediate Jacobians are a generalization of Jacobians that are associated to higher-dimensional algebraic varieties
  • Intermediate Jacobians provide a way to study the geometry of varieties that are not necessarily curves

Applications in Number Theory

• Abelian varieties play a central role in modern number theory and have numerous applications to classical problems
• Birch and Swinnerton-Dyer conjecture relates the rank of the Mordell-Weil group of an elliptic curve to the behavior of its $L$-function
  • Birch and Swinnerton-Dyer conjecture is one of the most important open problems in number theory and has far-reaching consequences
• Modular curves are algebraic curves that parametrize elliptic curves with additional structure
  • Modular curves are fundamental objects in the study of elliptic curves and their arithmetic
• Fermat's Last Theorem was famously proved by Andrew Wiles using techniques from the theory of elliptic curves and modular forms
  • Proof of Fermat's Last Theorem relies on the modularity theorem, which relates elliptic curves to modular forms
• Elliptic curve cryptography is a widely used cryptographic system that is based on the arithmetic of elliptic curves
  • Security of elliptic curve cryptography relies on the difficulty of the discrete logarithm problem in the group of rational points
• Serre's open image theorem describes the structure of the Galois representations attached to elliptic curves
  • Serre's theorem has important applications to the study of the torsion points and the isogeny graph
• Faltings' theorem states that there are only finitely many rational points on a curve of genus greater than one
  • Faltings' theorem is a deep result in arithmetic geometry that generalizes the Mordell conjecture for curves
• Langlands program is a vast generalization of the modularity theorem that relates Galois representations to automorphic forms
  • Abelian varieties and their $L$-functions are central objects in the Langlands program and its geometric counterpart