1.6 Elliptic curves over various fields

Elliptic curves are fascinating mathematical objects studied across various fields. They have a rich structure that connects complex analysis, number theory, and cryptography. Understanding their behavior over different fields is key to grasping their full potential.

This topic explores elliptic curves defined over real numbers, complex numbers, finite fields, local fields, and global fields. We'll examine their unique properties, group structures, and applications in each setting, building a comprehensive view of these powerful curves.

Definition of elliptic curves

• Elliptic curves are a fundamental object of study in algebraic geometry and number theory
• They have a rich structure and connections to various areas of mathematics, including complex analysis, arithmetic geometry, and cryptography
• Understanding the basic definitions and properties of elliptic curves is essential for studying their behavior over different fields

Weierstrass equation

• An elliptic curve can be defined by a Weierstrass equation of the form $y^2 = x^3 + ax + b$, where $a$ and $b$ are constants
• The Weierstrass equation provides a canonical way to represent elliptic curves and study their properties
• The coefficients $a$ and $b$ determine the shape and features of the elliptic curve
• The Weierstrass equation can be generalized to other fields, such as finite fields or p-adic numbers

Projective closure

• The projective closure of an elliptic curve is obtained by adding a point at infinity, denoted by $\mathcal{O}$
• The point at infinity serves as the identity element for the group law on the elliptic curve
• The projective closure allows for a more uniform treatment of the curve and its points
• In projective coordinates, the Weierstrass equation becomes $Y^2Z = X^3 + aXZ^2 + bZ^3$

Discriminant and singularities

• The discriminant of an elliptic curve, denoted by $\Delta$, is a quantity that determines whether the curve is singular or nonsingular
• For the Weierstrass equation $y^2 = x^3 + ax + b$, the discriminant is given by $\Delta = -16(4a^3 + 27b^2)$
• If $\Delta \neq 0$, the elliptic curve is nonsingular and has distinct roots
• If $\Delta = 0$, the elliptic curve is singular and has a node or a cusp singularity
• Singular elliptic curves have different properties and are not suitable for many applications

Elliptic curves over real numbers

• Elliptic curves over the real numbers have a rich geometric structure and can be visualized as graphs in the Cartesian plane
• Studying elliptic curves over the reals provides intuition and insights into their behavior and properties
• Many concepts and results for elliptic curves over the reals can be generalized to other fields

Graph and basic properties

• The graph of an elliptic curve over the reals is a smooth curve that is symmetric about the x-axis
• The curve intersects the x-axis at three points (counting multiplicity) or a single point if the curve is singular
• The shape of the curve depends on the coefficients $a$ and $b$ in the Weierstrass equation
• The curve has two components: an infinite component and a finite component (if the curve is nonsingular)

Group law and geometric interpretation

• Elliptic curves over the reals have a group law that allows for the addition of points on the curve
• The group law has a geometric interpretation based on the chord-and-tangent method
  • To add two points $P$ and $Q$, draw a line through $P$ and $Q$ (or a tangent line if $P = Q$) and find the third point of intersection with the curve, denoted by $R$
  • The sum $P + Q$ is defined as the reflection of $R$ about the x-axis
• The point at infinity $\mathcal{O}$ serves as the identity element for the group law
• The group law satisfies the associative, inverse, and identity properties

Torsion points and generators

• Torsion points on an elliptic curve over the reals are points of finite order under the group law
• The order of a torsion point $P$ is the smallest positive integer $n$ such that $nP = \mathcal{O}$
• The set of torsion points forms a subgroup of the elliptic curve group
• The structure of the torsion subgroup is described by Mazur's theorem, which states that it can be isomorphic to $\mathbb{Z}/n\mathbb{Z}$ for $n = 1, 2, \ldots, 10, 12$ or $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2n\mathbb{Z}$ for $n = 1, 2, 3, 4$
• Generators of the elliptic curve group are points that generate the entire group under the group law
• The rank of an elliptic curve over the reals is the number of independent generators (modulo torsion)

Elliptic curves over complex numbers

• Elliptic curves over the complex numbers have a rich structure and connections to complex analysis and algebraic geometry
• The study of elliptic curves over the complex numbers reveals deep properties and leads to important results in number theory
• Many concepts and techniques from complex analysis, such as lattices and complex tori, play a crucial role in understanding elliptic curves over $\mathbb{C}$

Lattices and complex tori

• A lattice in the complex plane is a discrete subgroup of $\mathbb{C}$ of the form $\Lambda = \{m\omega_1 + n\omega_2 : m, n \in \mathbb{Z}\}$, where $\omega_1$ and $\omega_2$ are linearly independent complex numbers
• The quotient space $\mathbb{C}/\Lambda$ is called a complex torus and has the structure of a compact Riemann surface
• Every complex torus is isomorphic to an elliptic curve over $\mathbb{C}$
• The periods $\omega_1$ and $\omega_2$ determine the shape and properties of the corresponding elliptic curve

Isomorphism with complex tori

• There is a canonical isomorphism between elliptic curves over $\mathbb{C}$ and complex tori
• The isomorphism is given by the Weierstrass $\wp$-function and its derivative
• The Weierstrass $\wp$-function is a doubly periodic meromorphic function that satisfies the differential equation $(\wp')^2 = 4\wp^3 - g_2\wp - g_3$, where $g_2$ and $g_3$ are constants depending on the lattice
• The isomorphism allows for the study of elliptic curves using the tools and techniques of complex analysis

Endomorphism ring and complex multiplication

• An endomorphism of an elliptic curve over $\mathbb{C}$ is a complex-analytic map from the curve to itself that preserves the group law
• The set of endomorphisms of an elliptic curve forms a ring, called the endomorphism ring
• The endomorphism ring is either $\mathbb{Z}$ or an order in an imaginary quadratic field
• Elliptic curves with complex multiplication are those whose endomorphism ring is larger than $\mathbb{Z}$
• Complex multiplication plays a crucial role in the study of elliptic curves and their arithmetic properties
• Elliptic curves with complex multiplication have special properties and are important in cryptography and the construction of abelian extensions of number fields

Elliptic curves over finite fields

• Elliptic curves over finite fields have important applications in cryptography and coding theory
• The study of elliptic curves over finite fields reveals interesting arithmetic and algebraic properties
• Many concepts and results from the theory of elliptic curves over $\mathbb{C}$ and $\mathbb{R}$ can be adapted to the finite field setting

Definition and basic properties

• An elliptic curve over a finite field $\mathbb{F}_q$ is defined by a Weierstrass equation $y^2 = x^3 + ax + b$, where $a, b \in \mathbb{F}_q$ and the discriminant $\Delta \neq 0$
• The set of points on an elliptic curve over $\mathbb{F}_q$, denoted by $E(\mathbb{F}_q)$, consists of the solutions to the Weierstrass equation together with the point at infinity $\mathcal{O}$
• The set $E(\mathbb{F}_q)$ forms a finite abelian group under the chord-and-tangent group law
• The group law on $E(\mathbb{F}_q)$ can be computed efficiently using arithmetic in the finite field

Group structure and order of points

• The group structure of $E(\mathbb{F}_q)$ is determined by the number of points on the curve, denoted by $\#E(\mathbb{F}_q)$
• The order of a point $P \in E(\mathbb{F}_q)$ is the smallest positive integer $n$ such that $nP = \mathcal{O}$
• The group $E(\mathbb{F}_q)$ is either cyclic or isomorphic to a product of two cyclic groups
• The possible group structures of $E(\mathbb{F}_q)$ are classified by the Rück-Voloch theorem
• The number of points on an elliptic curve over $\mathbb{F}_q$ satisfies the Hasse-Weil bound: $|\#E(\mathbb{F}_q) - (q+1)| \leq 2\sqrt{q}$

Hasse's theorem and bounds

• Hasse's theorem provides a tight bound on the number of points on an elliptic curve over a finite field
• The theorem states that $|\#E(\mathbb{F}_q) - (q+1)| \leq 2\sqrt{q}$
• The quantity $a_q = q+1 - \#E(\mathbb{F}_q)$ is called the trace of Frobenius and satisfies $|a_q| \leq 2\sqrt{q}$
• The Hasse bound is a consequence of the Riemann hypothesis for the zeta function of the elliptic curve
• The Hasse bound is used in the construction of elliptic curve cryptographic schemes and in the analysis of their security

Supersingular vs ordinary curves

• Elliptic curves over finite fields can be classified as either supersingular or ordinary
• An elliptic curve $E$ over $\mathbb{F}_q$ is supersingular if the trace of Frobenius $a_q$ satisfies $a_q \equiv 0 \pmod{p}$, where $q = p^n$ and $p$ is the characteristic of the field
• Equivalently, $E$ is supersingular if the endomorphism ring of $E$ over the algebraic closure of $\mathbb{F}_q$ is an order in a quaternion algebra
• Supersingular curves have special properties and are important in cryptography and the construction of abelian varieties
• Ordinary curves are those that are not supersingular
• Most elliptic curves over finite fields are ordinary, and they have different properties compared to supersingular curves

Elliptic curves over local fields

• Local fields are an important tool in number theory and arithmetic geometry
• The study of elliptic curves over local fields provides insights into their local behavior and leads to global results
• Many concepts and techniques from the theory of elliptic curves over $\mathbb{C}$ and $\mathbb{R}$ can be adapted to the local field setting

p-adic numbers and local fields

• The p-adic numbers $\mathbb{Q}_p$ are a completion of the rational numbers with respect to the p-adic absolute value
• A local field is a field that is complete with respect to a discrete valuation and has a finite residue field
• Examples of local fields include the p-adic numbers $\mathbb{Q}_p$, the field of formal Laurent series $\mathbb{F}_q((t))$, and the completion of a number field at a prime ideal
• Local fields have a rich arithmetic structure and are used in the study of Diophantine equations and arithmetic geometry

Reduction modulo p and Néron models

• The reduction of an elliptic curve $E$ over a local field $K$ modulo the maximal ideal of the valuation ring of $K$ is an important tool in the study of elliptic curves
• The reduction of $E$ can be either good, multiplicative, or additive, depending on the valuation of the discriminant of $E$
• The Néron model of an elliptic curve $E$ over a local field $K$ is a smooth group scheme over the valuation ring of $K$ that extends $E$
• The Néron model captures the local behavior of the elliptic curve and its points
• The special fiber of the Néron model is related to the reduction type of the elliptic curve

Kodaira-Néron classification of special fibers

• The Kodaira-Néron classification describes the possible special fibers of the Néron model of an elliptic curve over a local field
• The classification is based on the reduction type of the elliptic curve and the structure of the special fiber
• The possible reduction types are: $\mathrm{I}_n$ (good reduction), $\mathrm{I}_n^*$ (multiplicative reduction), $\mathrm{II}$, $\mathrm{III}$, $\mathrm{IV}$, $\mathrm{II}^*$, $\mathrm{III}^*$, $\mathrm{IV}^*$ (additive reduction)
• The special fiber can have several components and can be described using the concept of intersection graphs
• The Kodaira-Néron classification is used in the study of the arithmetic and geometric properties of elliptic curves over local fields

Tate's algorithm and local zeta functions

• Tate's algorithm is a method for computing the reduction type and the local zeta function of an elliptic curve over a local field
• The algorithm is based on the properties of the Weierstrass equation of the elliptic curve and the valuation of its coefficients
• The local zeta function of an elliptic curve $E$ over a local field $K$ is a generating function that encodes information about the number of points on the reductions of $E$ modulo powers of the maximal ideal of $K$
• The local zeta function is a rational function and satisfies a functional equation
• Tate's algorithm and local zeta functions are important tools in the study of the arithmetic of elliptic curves and their L-functions

Elliptic curves over global fields

• Global fields are an important class of fields in number theory and arithmetic geometry
• The study of elliptic curves over global fields combines local and global techniques and leads to deep results in Diophantine geometry
• Many concepts and results from the theory of elliptic curves over local fields and finite fields can be combined to study elliptic curves over global fields

Mordell-Weil theorem and rank

• The Mordell-Weil theorem states that the group of rational points $E(K)$ on an elliptic curve $E$ over a number field $K$ is finitely generated
• The group $E(K)$ is isomorphic to $E(K)_{\text{tors}} \oplus \mathbb{Z}^r$, where $E(K)_{\text{tors}}$ is the torsion subgroup and $r$ is the rank of $E$ over $K$
• The rank of an elliptic curve measures the size of the free part of the group of rational points
• Computing the rank of an elliptic curve is a difficult problem, and there are no known algorithms for determining the rank in general
• The Birch and Swinnerton-Dyer conjecture relates the rank of an elliptic curve to the behavior of its L-function at $s=1$

Torsion subgroup and Nagell-Lutz theorem

• The torsion subgroup $E(K)_{\text{tors}}$ of an elliptic curve $E$ over a number field $K$ consists of the points of finite order
• The structure of the torsion subgroup is described by the Mazur-Kenku theorem, which states that $E(K)_{\text{tors}}$ is isomorphic to one of 15 possible groups
• The Nagell-Lutz theorem provides a criterion for determining whether a point on an elliptic curve over $\mathbb{Q}$ is a torsion point
• The theorem states that if $P = (x, y)$ is a torsion point on $E: y^2 = x^3 + ax + b$ with $a, b \in \mathbb{Z}$, then $x, y \in \mathbb{Z}$ and either $y = 0$ or $y^2$ divides the discriminant of $E$
• The Nagell-Lutz theorem is a useful tool for computing the torsion subgroup of an elliptic curve over $\mathbb{Q}$

Height functions and canonical height

• Height functions are an important tool in Diophantine geometry and the study of rational points on algebraic varieties
• The height of a point measures its arithmetic complexity and provides a way to quantify the size of rational points
• The naive height of a point $P = (x, y)$ on