6.1 Elliptic curves as algebraic varieties

Elliptic curves are fascinating algebraic varieties with deep connections to number theory and cryptography. As cubic curves, they possess a unique group structure that enables powerful applications in various fields of mathematics and computer science.

This section explores the defining equations of elliptic curves, their geometric properties as cubic curves, and the group law that governs their points. We'll also examine isomorphisms between curves, their behavior over different fields, and the intriguing properties of torsion points.

Defining equations of elliptic curves

• Elliptic curves are algebraic curves defined by specific types of equations called Weierstrass equations
• Understanding the various forms of equations is crucial for studying the properties and applications of elliptic curves
• Different forms of equations offer advantages in specific contexts, such as simplifying computations or highlighting certain geometric features

Weierstrass equations

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• General form: $y^2 + a_1xy + a_3y = x^3 + a_2x^2 + a_4x + a_6$ where $a_1, a_2, a_3, a_4, a_6$ are constants
• Simplified Weierstrass equation: $y^2 = x^3 + ax + b$ where $a, b$ are constants
• Most commonly used form due to its generality and simplicity
• Coefficients determine the shape and properties of the elliptic curve

Legendre normal form

• Equation: $y^2 = x(x-1)(x-\lambda)$ where $\lambda$ is a constant not equal to 0 or 1
• Obtained from the simplified Weierstrass equation by a change of variables
• Useful for studying elliptic curves over finite fields and in cryptography (e.g., Diffie-Hellman key exchange)

Hesse normal form

• Equation: $x^3 + y^3 + z^3 = 3\mu xyz$ where $\mu$ is a constant not equal to 1
• Projective form of an elliptic curve, using homogeneous coordinates $[x:y:z]$
• Symmetric form that highlights the cubic nature of the curve
• Facilitates the study of elliptic curves over various fields (e.g., complex numbers)

Montgomery form

• Equation: $By^2 = x^3 + Ax^2 + x$ where $A, B$ are constants satisfying certain conditions
• Designed to optimize computations in elliptic curve cryptography
• Enables faster point multiplication and resistance to side-channel attacks (e.g., timing attacks)

Edwards form

• Equation: $x^2 + y^2 = 1 + dx^2y^2$ where $d$ is a constant not equal to 0 or 1
• Alternative form that provides efficient arithmetic and symmetry
• Useful for implementing elliptic curve cryptography (e.g., Edwards25519 curve)
• Allows for unified addition formulas and resistance to exceptional points

Elliptic curves as cubic curves

• Elliptic curves are a special class of algebraic curves defined by cubic equations
• Understanding the geometric properties of cubic curves is essential for studying elliptic curves
• Cubic curves have nine points of inflection, which play a crucial role in the group law on elliptic curves

Nonsingularity condition

• For a cubic curve to be an elliptic curve, it must be nonsingular (i.e., have no cusps or self-intersections)
• Nonsingularity ensures that the curve is smooth and has well-defined tangent lines at every point
• The nonsingularity condition is expressed in terms of the discriminant of the cubic equation

Discriminant of cubic polynomials

• The discriminant $\Delta$ of a cubic polynomial $f(x) = ax^3 + bx^2 + cx + d$ is given by $\Delta = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2$
• For an elliptic curve in simplified Weierstrass form $y^2 = x^3 + ax + b$, the discriminant is $\Delta = -16(4a^3 + 27b^2)$
• The curve is nonsingular if and only if $\Delta \neq 0$
• The sign of the discriminant determines the nature of the roots (e.g., three real roots, one real and two complex roots)

Projective closure

• Elliptic curves can be extended to the projective plane by adding a point at infinity
• The projective closure of an elliptic curve is obtained by homogenizing the equation (e.g., $y^2z = x^3 + axz^2 + bz^3$)
• The point at infinity, denoted by $\mathcal{O}$, is the unique point where the projective curve intersects the line at infinity ($z = 0$)
• The projective closure is essential for defining the group law on elliptic curves

Points at infinity

• In the projective closure of an elliptic curve, there is a single point at infinity $\mathcal{O}$
• The point at infinity serves as the identity element for the group law on the elliptic curve
• Geometrically, the point at infinity can be thought of as the point where all vertical lines intersect the curve
• The existence of the point at infinity is crucial for the completeness and consistency of the group law

Group law on elliptic curves

• Elliptic curves have a natural group structure, which is the foundation for their applications in cryptography and number theory
• The group law defines an addition operation on the points of an elliptic curve, resulting in an abelian group
• The group law has a geometric interpretation that involves drawing lines and finding their intersections with the curve

Chord-tangent construction

• To add two points $P$ and $Q$ on an elliptic curve, draw a line through $P$ and $Q$ (if $P \neq Q$) or the tangent line at $P$ (if $P = Q$)
• The line intersects the curve at a third point $R'$
• The sum $P + Q$ is defined as the reflection of $R'$ across the x-axis, denoted by $-R'$
• If the line is vertical, the sum is defined as the point at infinity $\mathcal{O}$

Associativity of group law

• The group law on elliptic curves is associative: $(P + Q) + R = P + (Q + R)$ for any points $P, Q, R$ on the curve
• Associativity allows for efficient computation of point multiples (e.g., $nP = P + P + \cdots + P$ $n$ times)
• Associativity is a consequence of the geometric properties of the chord-tangent construction

Identity element

• The point at infinity $\mathcal{O}$ serves as the identity element for the group law on elliptic curves
• For any point $P$ on the curve, $P + \mathcal{O} = P$ and $\mathcal{O} + P = P$
• The identity element is the neutral element under the addition operation

Inverse elements

• For any point $P$ on an elliptic curve, there exists an inverse element $-P$ such that $P + (-P) = \mathcal{O}$
• The inverse of a point $P = (x, y)$ is given by $-P = (x, -y)$, the reflection of $P$ across the x-axis
• The existence of inverse elements makes the group law on elliptic curves an abelian group

Geometric interpretation of group law

• The group law on elliptic curves has a beautiful geometric interpretation
• Adding points corresponds to drawing lines and finding their intersections with the curve
• The chord-tangent construction provides a visual way to understand the addition operation
• The geometric interpretation is useful for proving properties of the group law and for visualizing cryptographic protocols

Isomorphisms of elliptic curves

• Isomorphisms are mappings between elliptic curves that preserve the group structure
• Studying isomorphisms is essential for understanding the equivalence classes of elliptic curves and their moduli spaces
• Isomorphisms are used to classify elliptic curves and to simplify their equations

Changes of variables

• Elliptic curves can be transformed into different forms by applying changes of variables
• Linear changes of variables (e.g., $x \mapsto u^2x + r, y \mapsto u^3y + su^2x + t$) preserve the Weierstrass form of the equation
• Quadratic changes of variables (e.g., $x \mapsto \frac{x}{z}, y \mapsto \frac{y}{z}$) lead to projective forms of the equation (e.g., Hesse normal form)

Admissible changes of coordinates

• Admissible changes of coordinates are changes of variables that preserve the group structure of an elliptic curve
• For Weierstrass equations, admissible changes are linear changes with $u \neq 0$
• Admissible changes form a group called the group of admissible changes of coordinates
• The group of admissible changes is isomorphic to the projective linear group $\text{PGL}(2, K)$ over the base field $K$

Isomorphism classes

• Two elliptic curves are isomorphic if there exists an admissible change of coordinates that transforms one curve into the other
• Isomorphic curves have the same group structure and share many properties (e.g., the same number of points over finite fields)
• Isomorphism classes partition the set of all elliptic curves into equivalence classes
• The moduli space of elliptic curves parametrizes the isomorphism classes

j-invariant

• The j-invariant is a numerical invariant that characterizes the isomorphism class of an elliptic curve
• For a Weierstrass equation $y^2 = x^3 + ax + b$, the j-invariant is given by $j = 1728 \frac{4a^3}{4a^3 + 27b^2}$
• Two elliptic curves are isomorphic if and only if they have the same j-invariant
• The j-invariant is a crucial tool for classifying elliptic curves and studying their moduli spaces

Automorphisms of elliptic curves

• An automorphism of an elliptic curve is an isomorphism from the curve to itself
• The set of automorphisms of an elliptic curve forms a group under composition
• For elliptic curves over fields of characteristic not equal to 2 or 3, the automorphism group is either cyclic of order 2, 4, or 6
• The order of the automorphism group is related to the j-invariant and the presence of complex multiplication

Elliptic curves over various fields

• Elliptic curves can be defined over any field, and their properties depend on the characteristics of the underlying field
• Studying elliptic curves over different fields reveals rich connections to various branches of mathematics
• The choice of the field affects the group structure, torsion subgroups, and arithmetic properties of elliptic curves

Elliptic curves over real numbers

• Over the field of real numbers $\mathbb{R}$, elliptic curves have a geometric interpretation as curves in the real plane
• The real points of an elliptic curve form either one or two connected components (e.g., a single infinite component or a bounded component and an infinite component)
• The group law on real elliptic curves can be visualized using the chord-tangent construction
• Real elliptic curves are used in some cryptographic protocols and have applications in physics and engineering

Elliptic curves over complex numbers

• Over the field of complex numbers $\mathbb{C}$, elliptic curves have a rich geometric and analytic structure
• Complex elliptic curves are isomorphic to complex tori $\mathbb{C}/\Lambda$, where $\Lambda$ is a lattice in the complex plane
• The group law on complex elliptic curves corresponds to the addition of complex numbers modulo the lattice
• Complex elliptic curves are central objects in complex analysis, algebraic geometry, and number theory (e.g., the Modularity Theorem)

Elliptic curves over finite fields

• Elliptic curves over finite fields $\mathbb{F}_q$ (where $q$ is a prime power) have a finite number of points
• The group of points on an elliptic curve over a finite field is a finite abelian group
• The order of the group, denoted by $\#E(\mathbb{F}_q)$, satisfies the Hasse-Weil bound: $|q + 1 - \#E(\mathbb{F}_q)| \leq 2\sqrt{q}$
• Elliptic curves over finite fields are the foundation for elliptic curve cryptography (e.g., ECDSA, ECDH)

Elliptic curves over p-adic fields

• p-adic fields $\mathbb{Q}_p$ are completions of the rational numbers with respect to the p-adic norm
• Elliptic curves over p-adic fields have a rich arithmetic structure and are studied in p-adic analysis and p-adic Hodge theory
• The group of points on an elliptic curve over a p-adic field is an infinite abelian group
• p-adic elliptic curves are used in cryptography (e.g., p-adic point counting algorithms) and in the study of rational points on elliptic curves

Reduction mod p

• For an elliptic curve defined over the rational numbers $\mathbb{Q}$, the reduction mod $p$ is the elliptic curve obtained by reducing the coefficients modulo a prime $p$
• The reduction mod $p$ is not always an elliptic curve; it may be singular (i.e., have a cusp or a node)
• Primes for which the reduction is singular are called bad primes; others are called good primes
• The reduction type (good, multiplicative, or additive) provides information about the arithmetic properties of the elliptic curve (e.g., the Néron-Ogg-Shafarevich criterion)

Torsion points on elliptic curves

• Torsion points are points of finite order on an elliptic curve, i.e., points $P$ such that $nP = \mathcal{O}$ for some positive integer $n$
• The set of torsion points forms a subgroup of the elliptic curve group, called the torsion subgroup
• Studying torsion points and torsion subgroups is crucial for understanding the arithmetic structure of elliptic curves

Torsion subgroups

• The torsion subgroup of an elliptic curve $E$ over a field $K$, denoted by $E(K)_{\text{tors}}$, is the subgroup of points of finite order
• Over the complex numbers, the torsion subgroup is isomorphic to a product of two cyclic groups: $E(\mathbb{C})_{\text{tors}} \cong \mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$, where $m$ divides $n$
• Over the rational numbers, the torsion subgroup is isomorphic to one of 15 possible groups (by Mazur's theorem)
• The structure of the torsion subgroup provides information about the arithmetic and geometric properties of the elliptic curve

Nagell-Lutz theorem

• The Nagell-Lutz theorem gives a criterion for a point on an elliptic curve over the rational numbers to be a torsion point
• For an elliptic curve $y^2 = x^3 + ax + b$ with $a, b \in \mathbb{Z}$, a rational point $(x, y)$ is a torsion point if and only if:
  1. $x, y \in \mathbb{Z}$, and
  2. either $y = 0$ or $y^2$ divides the discriminant $\Delta = -16(4a^3 + 27b^2)$
• The Nagell-Lutz theorem is a useful tool for finding torsion points and determining the torsion subgroup of an elliptic curve over $\mathbb{Q}$

Mazur's theorem on torsion

• Mazur's theorem classifies the possible torsion subgroups of elliptic curves over the rational numbers
• The torsion subgroup of an elliptic curve over $\mathbb{Q}$ is isomorphic to one of the following 15 groups:
  • $\mathbb{Z}/n\mathbb{Z}$ for $n = 1, 2, ..., 10, 12$
  • $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2n\mathbb{Z}$ for $n = 1, 2, 3, 4$
• Mazur's theorem is a deep result in arithmetic geometry and has important consequences for the study of rational points on elliptic curves

Rational torsion points

• Rational torsion points are torsion points whose coordinates are rational numbers
• The set of rational torsion points forms a subgroup of the torsion subgroup, denoted by $E(\mathbb{Q})_{\text{tors}}$
• Finding rational torsion points is a fundamental problem in the arithmetic of elliptic curves
• Rational torsion points