Elliptic curves are smooth, projective algebraic curves of genus one with a specified base point. They have a rich geometric structure, including a group law that allows for the addition of points on the curve. This group law enables fascinating applications in cryptography and number theory.

Abelian varieties generalize elliptic curves to higher dimensions, maintaining the algebraic group structure. They share many properties with elliptic curves, including the Mordell-Weil theorem. Studying abelian varieties deepens our understanding of algebraic geometry and number theory connections.

Elliptic curve structure and properties

Defining elliptic curves and their canonical form

• An elliptic curve is a smooth, projective algebraic curve of genus one with a specified base point
• Elliptic curves can be defined over any field, but they are most commonly studied over the complex numbers, the rational numbers, and finite fields
• The Weierstrass equation $y^2 = x^3 + ax + b$ is a canonical form for elliptic curves
  • $a$ and $b$ are constants satisfying certain conditions to ensure smoothness
  • The Weierstrass equation provides a standard way to represent elliptic curves algebraically

Geometric structure and the group law

• Elliptic curves have a rich geometric structure, including a group law that allows for the addition of points on the curve
• The group law on an elliptic curve is defined by the chord-and-tangent process
  • Drawing a line through two points on the curve and finding the third point of intersection
  • The group law enables the construction of an abelian group structure on the points of the elliptic curve
• The group law on an elliptic curve satisfies the axioms of an abelian group, with the specified base point serving as the identity element
• The torsion subgroup of an elliptic curve consists of the points of finite order under the group law
  • The structure of the torsion subgroup is described by the Nagell-Lutz theorem
  • Understanding the torsion subgroup provides insights into the arithmetic properties of the elliptic curve

Group law on elliptic curves

Applications in cryptography

• The group law on elliptic curves has numerous applications in cryptography
  • Elliptic Curve Diffie-Hellman (ECDH) key exchange protocol
  • Elliptic Curve Digital Signature Algorithm (ECDSA)
• Elliptic curve cryptography (ECC) is based on the difficulty of the Elliptic Curve Discrete Logarithm Problem (ECDLP)
  • ECDLP involves finding an integer $n$ such that $nP = Q$ for given points $P$ and $Q$ on an elliptic curve
  • The hardness of ECDLP provides the security foundation for ECC

Applications in number theory and arithmetic

• The group law on elliptic curves can be used to construct elliptic curve factorization methods
  • Lenstra's Elliptic Curve Method (ECM) for integer factorization
  • ECM utilizes the group structure of elliptic curves to find factors of large integers efficiently
• The group structure of elliptic curves over finite fields is used in the construction of elliptic curve primality proving (ECPP) algorithms
  • Goldwasser-Kilian algorithm is an example of an ECPP algorithm
  • ECPP algorithms leverage the properties of elliptic curves to deterministically prove the primality of integers
• The Mordell-Weil theorem states that the group of rational points on an elliptic curve over a number field is finitely generated
  • This theorem has implications for the arithmetic of elliptic curves
  • Understanding the structure of the group of rational points provides insights into the Diophantine properties of elliptic curves
• The Birch and Swinnerton-Dyer conjecture relates the rank of the group of rational points on an elliptic curve to the behavior of its L-function
  • The conjecture provides a deep connection between elliptic curves and analytic number theory
  • If proven, the conjecture would have significant implications for the arithmetic of elliptic curves

Elliptic curves vs abelian varieties

Generalizing elliptic curves to abelian varieties

• An abelian variety is a complete algebraic variety that is also an algebraic group
  • Abelian varieties generalize the concept of elliptic curves to higher dimensions
  • Elliptic curves are the one-dimensional case of abelian varieties
• Abelian varieties can be defined over any field and are classified by their dimension, which is always a positive integer
• Every abelian variety is isomorphic to a projective variety embedded in projective space
  • Abelian varieties can be described by a set of homogeneous polynomial equations
  • The projective embedding provides a geometric realization of abelian varieties

Properties and structure of abelian varieties

• The group law on an abelian variety is given by regular maps, making it an algebraic group
• The Mordell-Weil theorem generalizes to abelian varieties
  • The group of rational points on an abelian variety over a number field is finitely generated
  • This generalization extends the arithmetic properties of elliptic curves to higher-dimensional abelian varieties
• The dual abelian variety of an abelian variety $A$ is another abelian variety $A^∨$
  • $A^∨$ parametrizes the line bundles on $A$
  • The dual of $A^∨$ is isomorphic to $A$, exhibiting a duality between abelian varieties

Arithmetic and geometry of abelian varieties

Algebraic cycles and line bundles

• The Picard group of an abelian variety $A$, denoted $Pic(A)$, is the group of isomorphism classes of line bundles on $A$
  • $Pic(A)$ plays a crucial role in the study of abelian varieties
  • Understanding the structure of $Pic(A)$ provides insights into the geometry of the abelian variety
• The Néron-Severi group of an abelian variety is a finitely generated abelian group
  • It describes the algebraic cycles on the variety modulo numerical equivalence
  • The Néron-Severi group captures important geometric information about the abelian variety

Endomorphisms and Galois representations

• The Tate module of an abelian variety $A$ over a field $k$ is the inverse limit of the n-torsion subgroups of $A$ over the algebraic closure of $k$
  • The Tate module carries a natural action of the Galois group of $k$
  • Studying the Galois action on the Tate module provides insights into the arithmetic of the abelian variety
• The endomorphism ring of an abelian variety $A$, denoted $End(A)$, is the ring of regular maps from $A$ to itself that preserve the group structure
  • $End(A)$ reflects the symmetries of the abelian variety
  • The structure of $End(A)$ is closely related to the arithmetic properties of the abelian variety
• The Rosati involution is a positive involution on the endomorphism ring of an abelian variety
  • It is used to define important subgroups, such as the Lefschetz group and the Hodge group
  • The Rosati involution plays a key role in the study of the endomorphism algebra of an abelian variety

Complex multiplication and the Mumford-Tate conjecture

• Abelian varieties with complex multiplication are those whose endomorphism ring contains an order in a number field of degree equal to twice the dimension of the variety
  • Abelian varieties with complex multiplication have special arithmetic properties
  • The theory of complex multiplication provides a rich interplay between abelian varieties and number theory
• The Mumford-Tate conjecture relates the Hodge group of an abelian variety to its Galois representations
  • The conjecture provides a deep connection between the arithmetic and geometry of abelian varieties
  • If proven, the Mumford-Tate conjecture would have significant implications for the study of abelian varieties and their associated Galois representations