Elliptic curves over rational numbers are a fascinating area of study in number theory. They combine algebraic geometry with arithmetic, offering insights into fundamental mathematical problems.
These curves have a rich structure, forming groups under addition. Understanding their rational points, integral points, and isogenies is crucial for applications in cryptography and solving Diophantine equations.
Definition of elliptic curves
Elliptic curves are a fundamental object of study in number theory and algebraic geometry
They have a rich structure and connections to various branches of mathematics
Understanding their definition is crucial for studying their properties and applications
Weierstrass equation
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An elliptic curve over a field K can be defined by a Weierstrass equation of the form y2=x3+Ax+B, where A,B∈K
The coefficients A and B must satisfy the condition that the Δ=−16(4A3+27B2) is nonzero to ensure the curve is smooth
The Weierstrass equation can be transformed into a shorter form y2=x3−27c4x−54c6 using a change of variables
Smooth projective curves
Elliptic curves are smooth projective curves of genus one with a specified base point
They can be viewed as the set of solutions to a cubic equation in the projective plane P2
The smoothness condition means that the curve has no cusps or self-intersections
The genus one condition implies that the curve has a unique , which serves as the identity element for the
Discriminant and j-invariant
The discriminant Δ of an elliptic curve is a quantity that measures how singular the curve is
If Δ=0, the curve is smooth and has no singular points
The j(E)=c43/Δ is an important invariant of an elliptic curve that characterizes its isomorphism class
Two elliptic curves over a field K are isomorphic if and only if they have the same j-invariant
Rational points on elliptic curves
Rational points on an elliptic curve are points whose coordinates are rational numbers
They form a group under a natural group law, which gives elliptic curves a rich algebraic structure
Studying rational points is a central problem in arithmetic geometry and has applications in cryptography
Definition and examples
A rational point on an elliptic curve E defined over Q is a point (x,y)∈E such that x,y∈Q
The set of rational points on E is denoted by E(Q)
Examples of rational points on the elliptic curve y2=x3−x include (0,0), (1,0), and (−1,0)
Group law
The set of rational points E(Q) forms an abelian group under a natural group law
The group law is defined geometrically by the chord-and-tangent process
Given two points P and Q on E, the sum P+Q is defined as the reflection of the third intersection point of the line through P and Q with E
The identity element is the point at infinity, denoted by O
Geometric interpretation
The group law on an elliptic curve has a beautiful geometric interpretation
Adding two points P and Q can be visualized by drawing a line through P and Q, finding the third intersection point with the curve, and reflecting it across the x-axis
If P=Q, the line is taken to be the tangent line at P, and the reflection of the double intersection point is the result
This geometric description allows for a visual understanding of the group structure
Algebraic formulas
The group law on an elliptic curve can also be described algebraically using explicit formulas
Let P=(x1,y1) and Q=(x2,y2) be two points on an elliptic curve E given by y2=x3+Ax+B
If P=Q, then the sum P+Q=(x3,y3) is given by:
x3=λ2−x1−x2
y3=λ(x1−x3)−y1
where λ=(y2−y1)/(x2−x1)
If P=Q, then the double 2P=(x3,y3) is given by:
x3=λ2−2x1
y3=λ(x1−x3)−y1
where λ=(3x12+A)/(2y1)
Mordell-Weil theorem
The is a fundamental result in the theory of elliptic curves
It describes the structure of the group of rational points on an elliptic curve over a number field
The theorem has important consequences for understanding the arithmetic of elliptic curves
Statement and consequences
The Mordell-Weil theorem states that for an elliptic curve E over a number field K, the group E(K) of K-rational points is finitely generated
This means that E(K) is isomorphic to Zr⊕E(K)tors, where r is a non-negative integer called the of E over K, and E(K)tors is the torsion subgroup of E(K)
The theorem implies that there are only finitely many and that the rank is a measure of the "size" of the group of rational points
It also provides a basis for studying the arithmetic of elliptic curves and their rational points
Rank and torsion
The rank r of an elliptic curve E over a number field K is the number of independent points of infinite order in E(K)
Determining the rank is a difficult problem, and there is no general algorithm known for computing it
The torsion subgroup E(K)tors consists of the points of finite order in E(K)
The possible torsion subgroups over Q are known and classified by Mazur's theorem
Understanding the rank and torsion of an elliptic curve provides insights into its structure and properties
Examples and computations
Consider the elliptic curve E:y2=x3−x over Q
The torsion subgroup E(Q)tors consists of the points O, (0,0), (1,0), and (−1,0), forming a group isomorphic to Z/2Z×Z/2Z
The rank of E over Q is 0, so E(Q)≅Z/2Z×Z/2Z
Another example is the elliptic curve E:y2=x3−4x, which has rank 1 over Q and torsion subgroup isomorphic to Z/2Z
Computing the rank and torsion of an elliptic curve often involves a combination of algebraic and analytic techniques, such as methods and L-functions
Integral points on elliptic curves
Integral points on an elliptic curve are points whose coordinates are integers
Studying integral points is a natural question in Diophantine geometry and has connections to other problems in number theory
Several important theorems and results are known about the structure and finiteness of integral points
Nagell-Lutz theorem
The Nagell-Lutz theorem provides a criterion for determining the torsion points on an elliptic curve over Q with integral coefficients
It states that if E:y2=x3+Ax+B with A,B∈Z, then any torsion point (x,y)∈E(Q)tors satisfies either y=0 or y2 divides the discriminant Δ
This theorem gives a practical way to find all the torsion points on an elliptic curve and helps in understanding the torsion subgroup
Siegel's theorem
Siegel's theorem is a fundamental result about the finiteness of integral points on curves of genus at least one
For an elliptic curve E over Q, Siegel's theorem implies that the set of integral points E(Z) is finite
The proof of Siegel's theorem is non-effective, meaning it does not provide an explicit bound on the size of the integral points
Effective versions of Siegel's theorem have been proved for specific classes of elliptic curves, such as those with complex multiplication
Elliptic logarithms
Elliptic logarithms are a tool for studying integral points on elliptic curves
The elliptic logarithm is a function that maps points on an elliptic curve to a complex number, analogous to the natural logarithm for real numbers
It satisfies a group homomorphism property and can be used to derive bounds on the size of integral points
Elliptic logarithms play a role in the proof of Siegel's theorem and in the study of linear forms in elliptic logarithms, which has applications to Diophantine equations
Elliptic curves over finite fields
Elliptic curves can also be studied over finite fields, where they exhibit interesting properties and have important applications
The theory of elliptic curves over finite fields is a rich area of research with connections to number theory, algebraic geometry, and cryptography
Several key results and algorithms are known for elliptic curves over finite fields
Hasse's theorem
Hasse's theorem, also known as the Hasse-Weil bound, gives an estimate for the number of points on an elliptic curve over a finite field
It states that for an elliptic curve E over a finite field Fq of characteristic p, the number of Fq-rational points #E(Fq) satisfies the inequality ∣#E(Fq)−(q+1)∣≤2q
This theorem provides a tight bound on the number of points and has implications for the structure and properties of elliptic curves over finite fields
Supersingular vs ordinary curves
Elliptic curves over finite fields can be classified into two types: supersingular and ordinary curves
An elliptic curve E over a finite field of characteristic p is called supersingular if p divides the trace of Frobenius ap=p+1−#E(Fp), and ordinary otherwise
Supersingular curves have special properties and are of interest in cryptography and the theory of modular forms
Ordinary curves are more common and have a simpler structure, making them suitable for cryptographic applications
Schoof's algorithm
Schoof's algorithm is a polynomial-time algorithm for counting the number of points on an elliptic curve over a finite field
It uses the action of the Frobenius endomorphism on the ℓ-torsion points of the curve for various small primes ℓ to determine the trace of Frobenius modulo ℓ
By combining the information modulo several primes, Schoof's algorithm computes the exact number of points on the curve
The algorithm has a running time of O(log8q) for a curve over Fq and has been further improved by various optimizations and variants (SEA, AGM, etc.)
Rational isogenies
Isogenies are a fundamental concept in the study of elliptic curves and their relationships
A rational isogeny between two elliptic curves is a non-constant morphism that preserves the group structure
Isogenies provide a way to relate different elliptic curves and have applications in cryptography and the theory of modular curves
Definition and examples
An isogeny between two elliptic curves E1 and E2 over a field K is a non-constant rational map ϕ:E1→E2 that is also a group homomorphism
The degree of an isogeny is the degree of the corresponding rational map
Examples of isogenies include multiplication-by-n maps, which are isogenies from an elliptic curve to itself, and the Frobenius endomorphism over finite fields
Isogenies can be classified into separable and inseparable isogenies based on the separability of the corresponding function field extension
Vélu's formulas
Vélu's formulas provide explicit equations for computing isogenies between elliptic curves
Given an elliptic curve E and a finite subgroup G of E, Vélu's formulas describe the equation of the quotient curve E/G and the isogeny ϕ:E→E/G
The formulas involve the coordinates of the points in G and the coefficients of the curve E
Vélu's formulas are used in the computation of isogenies and the construction of isogeny graphs
Isogeny graphs
Isogeny graphs are a way to visualize the relationships between elliptic curves through isogenies
The vertices of an isogeny graph represent elliptic curves (up to isomorphism), and the edges represent isogenies between them
The degree of an isogeny is often attached as a label to the corresponding edge
Isogeny graphs have a rich structure and are studied in the context of modular curves and the moduli space of elliptic curves
They also have applications in cryptography, such as in the construction of hash functions and the analysis of isogeny-based cryptographic protocols
Isogeny-based cryptography
Isogeny-based cryptography is a relatively new area that uses isogenies between elliptic curves for constructing cryptographic protocols
The security of these protocols relies on the difficulty of computing isogenies between elliptic curves and the hardness of the isogeny problem
Examples of isogeny-based cryptographic protocols include the supersingular isogeny Diffie-Hellman (SIDH) key exchange and the supersingular isogeny hash function
Isogeny-based cryptography is believed to be resistant to attacks by quantum computers and is a candidate for post-quantum cryptography
Elliptic curve cryptography
(ECC) is a modern public-key cryptography approach that uses the algebraic structure of elliptic curves over finite fields
ECC provides similar security levels to traditional cryptosystems (like RSA) with smaller key sizes, making it efficient for use in constrained environments
The security of ECC relies on the difficulty of the elliptic curve discrete logarithm problem (ECDLP)
Diffie-Hellman key exchange
The Diffie-Hellman key exchange protocol can be adapted to use elliptic curves, resulting in the elliptic curve Diffie-Hellman (ECDH) key exchange
In ECDH, two parties agree on an elliptic curve E over a finite field and a base point P∈E of large order
Each party chooses a secret integer a and b, respectively, and computes the public points aP and bP
The shared secret is then computed as abP, which can be used to derive a symmetric encryption key
The security of ECDH relies on the difficulty of computing the secret integers a and b given the public points aP and bP (the ECDLP)
Elliptic curve digital signature algorithm (ECDSA)
The elliptic curve digital signature algorithm (ECDSA) is a variant of the digital signature algorithm (DSA) that uses elliptic curves
In ECDSA, the signer has a private key d and a corresponding public key Q=dP, where P is a base point on an agreed-upon elliptic curve
To sign a message m, the signer chooses a random integer k and computes the point kP=(x,y)
The signature consists of two components: $r = x \bmo
Key Terms to Review (17)
Andrew Wiles: Andrew Wiles is a British mathematician best known for proving Fermat's Last Theorem, a problem that remained unsolved for over 350 years. His groundbreaking work not only established the truth of this theorem but also had profound implications for elliptic curves, modular forms, and number theory.
Birch and Swinnerton-Dyer Conjecture: The Birch and Swinnerton-Dyer Conjecture is a significant unsolved problem in number theory that relates the number of rational points on an elliptic curve to the behavior of its L-function at a specific point. This conjecture connects the fields of elliptic curves, L-functions, and algebraic number theory, suggesting that the rank of an elliptic curve, which measures the number of independent rational points, can be determined by analyzing the order of the zero of its associated L-function at s=1.
David Hilbert: David Hilbert was a prominent German mathematician known for his foundational contributions to various areas of mathematics, including algebra, number theory, and geometry. His work laid the groundwork for the modern understanding of elliptic curves and their properties, influencing numerous aspects of mathematics and theoretical physics.
Descent: Descent refers to a method in algebraic geometry that allows us to relate properties of a given algebraic variety to its subvarieties, especially in the context of elliptic curves. This process is crucial for understanding the rational points on elliptic curves and provides a way to study their behavior over different fields, particularly rational numbers and their extensions. By employing descent, mathematicians can establish connections between seemingly unrelated curves and use this knowledge to prove significant results, such as those pertaining to the Modularity theorem.
Discriminant: The discriminant is a mathematical expression that provides important information about the roots of a polynomial, particularly in the context of elliptic curves. In relation to elliptic curves defined by Weierstrass equations, the discriminant helps to determine the singularity of the curve; if the discriminant is zero, the curve has singular points and is not considered an elliptic curve. Understanding the discriminant is crucial for studying properties of elliptic curves over different fields, analyzing their rational points, and exploring their applications in number theory and cryptography.
Elliptic Curve Cryptography: Elliptic Curve Cryptography (ECC) is a form of public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC allows for smaller keys compared to traditional methods while maintaining high levels of security, making it efficient for use in digital communication and data protection.
Elliptic Curve Equation: An elliptic curve equation is a mathematical equation of the form $$y^2 = x^3 + ax + b$$ where the coefficients a and b are constants that satisfy a specific condition to ensure that the curve has no singular points. These equations define elliptic curves, which are essential in number theory and cryptography, providing a framework for operations like point doubling and exploring their properties over various fields, such as rational numbers.
Group law: In the context of elliptic curves, group law refers to the set of rules that define how to add points on an elliptic curve, forming a mathematical group. This concept is crucial as it provides a structured way to perform point addition and ensures that the operation adheres to properties like associativity, commutativity, and the existence of an identity element, which are fundamental in various applications including cryptography and number theory.
J-invariant: The j-invariant is a complex analytic invariant associated with an elliptic curve, which classifies the curve up to isomorphism over the complex numbers. It plays a crucial role in understanding the properties of elliptic curves, allowing for distinctions between different curves that may look similar algebraically but differ in their complex structure.
Mordell-Weil Theorem: The Mordell-Weil Theorem states that the group of rational points on an elliptic curve over the rational numbers is finitely generated. This theorem highlights a deep connection between algebraic geometry and number theory, establishing that the set of rational points can be expressed as a finite direct sum of a torsion subgroup and a free abelian group. It plays a crucial role in understanding the structure of elliptic curves and their rational solutions.
Mordell's Theorem: Mordell's Theorem states that the group of rational points on an elliptic curve defined over the rational numbers is finitely generated. This means that the set of rational solutions to the equation describing the elliptic curve can be expressed as a finite combination of a finite number of generators and a torsion subgroup. This theorem connects the structure of elliptic curves to the nature of rational numbers, illustrating how solutions behave over various fields.
Non-singular elliptic curves: Non-singular elliptic curves are a special type of algebraic curve defined over a field, characterized by their smoothness and having a specific structure. These curves can be represented by a Weierstrass equation of the form $$y^2 = x^3 + ax + b$$, where the discriminant is non-zero, ensuring there are no singular points. This smoothness is crucial because it allows for the rich theory of elliptic curves to develop, linking number theory and algebraic geometry.
Point at Infinity: The point at infinity is a unique point that serves as the identity element in the context of elliptic curves, representing a limit point that is added to the elliptic curve. This concept is crucial for defining the group law on elliptic curves, where it plays a central role in operations involving other points on the curve. Additionally, it connects with projective geometry, where it helps manage the behavior of lines and curves at infinity.
Rank: In the context of elliptic curves, the rank refers to the number of independent rational points that can be generated on an elliptic curve over a given field, particularly over the rational numbers. This concept is crucial as it helps in understanding the structure of the group of rational points, leading to insights about the solutions to equations defined by the curve and their distributions over various fields.
Supersingular Elliptic Curves: Supersingular elliptic curves are a special class of elliptic curves that exhibit unique properties, particularly over finite fields. These curves have distinct behavior in terms of their endomorphism rings and lack a point of order equal to the characteristic of the field, which means they are not ordinary. Supersingular elliptic curves play an important role in various areas such as number theory, cryptography, and coding theory, impacting the study of elliptic curves over rational numbers and their applications in linear codes.
Torsion Points: Torsion points on an elliptic curve are points that have finite order with respect to the group structure of the curve. This means that if you repeatedly add a torsion point to itself a certain number of times, you will eventually return to the identity element (the point at infinity). Torsion points are essential for understanding the structure of elliptic curves and are linked to many important concepts, such as the group law, rational points, and their applications in number theory and cryptography.
Weierstrass form: Weierstrass form is a specific way of representing elliptic curves using a cubic equation in two variables, typically expressed as $$y^2 = x^3 + ax + b$$, where $$a$$ and $$b$$ are constants. This representation is fundamental because it simplifies the study of elliptic curves, enabling clear definitions of point addition and doubling, and serving as a basis for various applications in number theory and cryptography.