4.5 Elliptic curves and Diophantine equations

Elliptic curves are smooth, projective algebraic curves with a rich structure. They're key in number theory and cryptography, defined by equations like y^2 = x^3 + ax + b. Their points form an abelian group under a geometric addition operation.

Diophantine equations are polynomial equations with integer coefficients, seeking integer solutions. They connect to elliptic curves, as many Diophantine problems can be reframed using elliptic curves. This link has led to breakthroughs in number theory and cryptography.

Elliptic curves

• Elliptic curves are smooth, projective algebraic curves of genus one with a specified basepoint
• They have a rich algebraic structure and are important objects of study in number theory and cryptography
• Elliptic curves can be defined over any field, including the complex numbers, the rational numbers, and finite fields

Weierstrass form

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• The Weierstrass form of an elliptic curve is the equation $y^2 = x^3 + ax + b$, where $a$ and $b$ are constants
• Every elliptic curve can be written in Weierstrass form by a suitable change of variables
• The discriminant of the Weierstrass equation, given by $\Delta = -16(4a^3 + 27b^2)$, must be nonzero for the curve to be smooth

Elliptic curve group law

• The set of points on an elliptic curve forms an abelian group under a geometric addition operation called the group law
• The group law is defined by the following rule: three points on the curve add up to the identity element (the point at infinity) if and only if they are collinear
• The group law can be expressed algebraically using the Weierstrass equation, making it efficient to compute

Points of finite order

• A point $P$ on an elliptic curve is said to have finite order if there exists a positive integer $n$ such that $nP = O$, where $O$ is the identity element (point at infinity)
• The order of a point $P$ is the smallest positive integer $n$ such that $nP = O$
• Points of finite order are also called torsion points

Torsion subgroup

• The set of all points of finite order on an elliptic curve forms a subgroup called the torsion subgroup
• The torsion subgroup is always finite and can be classified completely over the complex numbers (Mazur's theorem) and the rational numbers (Nagell-Lutz theorem)
• The torsion subgroup is an important invariant of an elliptic curve and can provide information about its structure and properties

Rank of elliptic curves

• The rank of an elliptic curve is the number of independent points of infinite order in its group of rational points
• The rank is a measure of the "size" of the group of rational points and is an important invariant of the curve
• Determining the rank of an elliptic curve is a difficult problem, and there is no known algorithm for computing it in general

Diophantine equations

• Diophantine equations are polynomial equations with integer coefficients for which integer solutions are sought
• They are named after the ancient Greek mathematician Diophantus of Alexandria, who studied them extensively
• Diophantine equations have a rich history and have been the subject of much research in number theory

Linear Diophantine equations

• A linear Diophantine equation is an equation of the form $ax + by = c$, where $a$, $b$, and $c$ are integers
• The existence of solutions to a linear Diophantine equation can be determined using the Euclidean algorithm
• If solutions exist, they can be parametrized using the extended Euclidean algorithm

Pythagorean triples

• A Pythagorean triple is a set of three positive integers $(a, b, c)$ satisfying the equation $a^2 + b^2 = c^2$
• Pythagorean triples correspond to right triangles with integer side lengths
• All Pythagorean triples can be generated from the primitive ones using scaling and permutation

Fermat's Last Theorem

• Fermat's Last Theorem states that the equation $x^n + y^n = z^n$ has no non-trivial integer solutions for $n > 2$
• The theorem was conjectured by Pierre de Fermat in 1637 but was not proved until 1995 by Andrew Wiles
• The proof of Fermat's Last Theorem relies on deep connections between elliptic curves and modular forms

Elliptic curves as Diophantine equations

• Elliptic curves can be viewed as Diophantine equations since they are defined by polynomial equations with integer coefficients
• The group of rational points on an elliptic curve corresponds to the integer solutions of its defining equation
• Many Diophantine problems, such as finding perfect powers or solving certain types of equations, can be reduced to questions about elliptic curves

Mordell-Weil Theorem

• The Mordell-Weil Theorem states that the group of rational points on an elliptic curve is finitely generated
• It was first proved by Louis Mordell in 1922 for elliptic curves over $\mathbb{Q}$ and later generalized by André Weil to elliptic curves over any number field
• The Mordell-Weil Theorem is a fundamental result in the study of elliptic curves and has many important consequences

Finitely generated abelian groups

• An abelian group is finitely generated if it can be generated by a finite set of elements
• Every finitely generated abelian group is isomorphic to a direct sum of cyclic groups
• The structure theorem for finitely generated abelian groups allows us to classify them up to isomorphism

Proof of Mordell-Weil Theorem

• The proof of the Mordell-Weil Theorem relies on the theory of heights on elliptic curves
• The height of a rational point measures its "complexity" and satisfies certain properties that allow for a descent argument
• The proof proceeds by showing that there are only finitely many rational points of bounded height, and then using the group law to generate the entire group from these points

Consequences of Mordell-Weil Theorem

• The Mordell-Weil Theorem implies that the group of rational points on an elliptic curve has a finite torsion subgroup and a free abelian part of finite rank
• The theorem allows us to study the structure of the group of rational points and to compute it in specific cases
• The Mordell-Weil Theorem has applications to many Diophantine problems, such as the congruent number problem and the study of integral points on elliptic curves

Elliptic curve cryptography

• Elliptic curve cryptography (ECC) is a public-key cryptography approach based on the algebraic structure of elliptic curves over finite fields
• ECC requires smaller key sizes than other public-key cryptography systems (such as RSA) for equivalent security, making it more efficient
• Elliptic curve cryptography is widely used in practice, including in the Bitcoin protocol and in the TLS standard

Elliptic Curve Diffie-Hellman (ECDH)

• Elliptic Curve Diffie-Hellman is a key agreement protocol that allows two parties to establish a shared secret over an insecure channel
• ECDH is based on the difficulty of the elliptic curve discrete logarithm problem (ECDLP)
• In ECDH, the parties agree on an elliptic curve and a base point, then each party generates a private-public key pair and exchanges the public keys to compute the shared secret

Elliptic Curve Digital Signature Algorithm (ECDSA)

• The Elliptic Curve Digital Signature Algorithm is a digital signature scheme based on elliptic curve cryptography
• ECDSA is used to verify the authenticity of a message and the identity of the sender
• In ECDSA, the signer generates a private-public key pair and uses the private key to sign the message, while the verifier uses the public key to check the signature

Security of elliptic curve cryptography

• The security of elliptic curve cryptography relies on the difficulty of solving the elliptic curve discrete logarithm problem (ECDLP)
• The ECDLP is believed to be harder than the discrete logarithm problem in finite fields (for comparable key sizes), which is the basis for the security of other cryptographic systems like DSA
• The most efficient known algorithms for solving the ECDLP have exponential running time, making ECC secure for appropriately chosen parameters

Elliptic curves over finite fields

• Elliptic curves can be defined over finite fields $\mathbb{F}_q$, where $q$ is a prime power
• The group of $\mathbb{F}_q$-rational points on an elliptic curve, denoted by $E(\mathbb{F}_q)$, is a finite group
• Elliptic curves over finite fields have applications in cryptography and coding theory

Hasse's Theorem

• Hasse's Theorem gives a bound on the number of $\mathbb{F}_q$-rational points on an elliptic curve $E$
• The theorem states that $|E(\mathbb{F}_q) - (q+1)| \leq 2\sqrt{q}$
• Hasse's Theorem allows us to estimate the size of the group $E(\mathbb{F}_q)$ and to study its properties

Supersingular vs ordinary curves

• An elliptic curve over a finite field is called supersingular if its endomorphism ring is an order in a quaternion algebra, and ordinary otherwise
• Supersingular curves have special properties that make them useful in certain cryptographic applications (such as pairing-based cryptography)
• Most elliptic curves over finite fields are ordinary, and they are used in standard elliptic curve cryptography

Pairing-based cryptography

• Pairing-based cryptography is a subfield of elliptic curve cryptography that uses bilinear pairings on elliptic curves
• A bilinear pairing is a map that takes two points on an elliptic curve and returns an element of a finite field, satisfying certain properties
• Pairings can be used to construct advanced cryptographic protocols, such as identity-based encryption and short signatures

Elliptic curves in number theory

• Elliptic curves are central objects in modern number theory and have connections to many other areas of mathematics
• The study of elliptic curves over various fields (such as the rational numbers, number fields, and finite fields) leads to deep and fascinating questions
• Elliptic curves have been used to solve many long-standing problems in number theory, such as Fermat's Last Theorem and the congruent number problem

Congruent number problem

• A congruent number is a positive integer that is the area of a right triangle with rational side lengths
• The congruent number problem asks which integers are congruent numbers
• The problem can be reformulated in terms of the existence of rational points on certain elliptic curves, providing a link between geometry and arithmetic

Birch and Swinnerton-Dyer conjecture

• The Birch and Swinnerton-Dyer conjecture is one of the most important open problems in number theory and is one of the Millennium Prize Problems
• The conjecture relates the rank of an elliptic curve (the number of independent infinite-order rational points) to the behavior of its L-function at $s=1$
• The conjecture has been proved in some special cases but remains open in general

Elliptic curves and modular forms

• There is a deep connection between elliptic curves and modular forms, which are certain analytic functions on the upper half-plane satisfying transformation properties
• The Modularity Theorem, proved by Wiles and others, states that every elliptic curve over $\mathbb{Q}$ is modular, meaning that it corresponds to a modular form
• This connection between elliptic curves and modular forms was a crucial ingredient in the proof of Fermat's Last Theorem and has led to many other important results in number theory