6.6 Elliptic curves and the Mordell-Weil theorem

Elliptic curves are fascinating mathematical objects with deep connections to number theory and algebraic geometry. They're defined by special equations and have a unique group structure that allows for addition of points on the curve.

The Mordell-Weil theorem is a key result about elliptic curves. It states that the group of rational points on an elliptic curve over a number field is finitely generated, revealing important insights about the curve's structure.

Definition of elliptic curves

• Elliptic curves are a fundamental object of study in number theory and algebraic geometry
• They have a rich structure and deep connections to various areas of mathematics
• Understanding the basic definitions and properties of elliptic curves is essential for studying the Mordell-Weil theorem

Weierstrass equations

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• Elliptic curves can be defined by Weierstrass equations of the form $y^2 = x^3 + ax + b$, where $a$ and $b$ are constants satisfying certain conditions
• The Weierstrass equation provides a standard way to represent elliptic curves algebraically
• The coefficients $a$ and $b$ determine the shape and properties of the curve (e.g., $y^2 = x^3 - x$ defines a curve with two components)

Smooth projective curves

• Elliptic curves are smooth projective curves of genus one with a specified base point
• Smoothness means the curve has no singularities or self-intersections
• Projective curves are defined in projective space, which allows for points at infinity (e.g., the point $[0:1:0]$ on the projective closure of the Weierstrass equation)
• The genus of a curve is a measure of its complexity, and elliptic curves have genus one

Rational points

• Elliptic curves are studied over various fields, but the most interesting case is often over the rational numbers $\mathbb{Q}$
• A rational point on an elliptic curve is a point whose coordinates are rational numbers
• The set of rational points on an elliptic curve forms a group under a certain operation called the group law (e.g., the points $(0,0)$ and $(1,0)$ on the curve $y^2 = x^3 - x$)

Group law on elliptic curves

• The set of points on an elliptic curve, together with a special point called the point at infinity, forms an abelian group under the group law
• The group law allows us to add points on the curve and obtain another point on the curve
• Understanding the group structure of elliptic curves is crucial for studying their arithmetic properties

Geometric description

• The group law on elliptic curves can be described geometrically using the chord-and-tangent process
• To add two points $P$ and $Q$, draw a line through them and find the third point of intersection with the curve, then reflect that point across the $x$-axis to obtain $P+Q$
• If $P=Q$, draw the tangent line at $P$ and follow the same process (e.g., adding the points $(0,0)$ and $(1,0)$ on the curve $y^2 = x^3 - x$)

Algebraic formulas

• The group law can also be described algebraically using explicit formulas in terms of the coordinates of the points
• These formulas involve rational functions and can be derived from the geometric description
• The algebraic formulas are useful for computations and implementing the group law in practice (e.g., the formula for the $x$-coordinate of $P+Q$ in terms of the coordinates of $P$ and $Q$)

Associativity and identity element

• The group law on elliptic curves is associative, meaning that $(P+Q)+R = P+(Q+R)$ for any points $P$, $Q$, and $R$
• The point at infinity serves as the identity element of the group, denoted by $\mathcal{O}$
• For any point $P$ on the curve, $P+\mathcal{O} = P$ and $P+(-P) = \mathcal{O}$, where $-P$ is the reflection of $P$ across the $x$-axis

Mordell-Weil theorem

• The Mordell-Weil theorem is a fundamental result in the theory of elliptic curves, describing the structure of the group of rational points
• It states that the group of rational points on an elliptic curve over a number field is finitely generated
• The theorem has important consequences for the arithmetic and Diophantine properties of elliptic curves

Statement of theorem

• Let $E$ be an elliptic curve defined over a number field $K$. Then the group $E(K)$ of $K$-rational points on $E$ is a finitely generated abelian group
• This means that $E(K)$ is isomorphic to a direct product of a finite torsion subgroup and a free abelian group of finite rank
• The theorem was first proved by Louis Mordell for $K=\mathbb{Q}$ and later generalized by André Weil to arbitrary number fields

Finitely generated abelian groups

• A finitely generated abelian group is an abelian group that can be generated by a finite set of elements
• The structure theorem for finitely generated abelian groups states that every such group is isomorphic to a direct product of cyclic groups
• The Mordell-Weil theorem implies that the group of rational points on an elliptic curve has this structure (e.g., $E(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}$)

Torsion subgroup

• The torsion subgroup of $E(K)$ consists of all points of finite order, i.e., points $P$ such that $nP = \mathcal{O}$ for some positive integer $n$
• The torsion subgroup is always finite, and its structure is well-understood for many classes of elliptic curves
• For example, over $\mathbb{Q}$, the torsion subgroup is isomorphic to one of 15 possible groups (e.g., $\mathbb{Z}/2\mathbb{Z}$, $\mathbb{Z}/4\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$)

Rank of elliptic curve

• The rank of an elliptic curve $E$ over a field $K$ is the rank of the free abelian part of $E(K)$
• In other words, it is the number of independent points of infinite order in $E(K)$
• Determining the rank of an elliptic curve is a difficult problem, and there is no known algorithm for computing it in general (e.g., the Birch and Swinnerton-Dyer conjecture relates the rank to the behavior of the L-function of the curve)

Proof of Mordell-Weil theorem

• The proof of the Mordell-Weil theorem is a major result in arithmetic geometry and involves several deep ideas and techniques
• The key steps in the proof include the use of height functions, the weak Mordell-Weil theorem, and the descent procedure
• Understanding the proof provides insight into the structure of rational points on elliptic curves and the methods used to study them

Height functions on elliptic curves

• Height functions are a way to measure the "size" or "complexity" of rational points on an elliptic curve
• The canonical height is a quadratic form on $E(K)$ that behaves like a logarithmic version of the naive height on projective space
• The height functions play a crucial role in the proof of the Mordell-Weil theorem, as they allow for the construction of a finite-index subgroup of $E(K)$

Descent procedure

• The descent procedure is a method for studying the rational points on an elliptic curve by relating them to points on simpler curves
• It involves constructing an auxiliary curve (often a twist of the original curve) and using information about its rational points to deduce information about the original curve
• The descent procedure is used in the proof of the weak Mordell-Weil theorem and can also be used to compute bounds on the rank of an elliptic curve

Weak Mordell-Weil theorem

• The weak Mordell-Weil theorem states that for any integer $n \geq 2$, the quotient group $E(K)/nE(K)$ is finite
• This means that the group of rational points modulo $n$-torsion is finite, which is a key step in proving the full Mordell-Weil theorem
• The proof of the weak Mordell-Weil theorem uses the descent procedure and the Kummer exact sequence to reduce the problem to studying the Galois cohomology of the curve

Applications of Mordell-Weil theorem

• The Mordell-Weil theorem has numerous applications in number theory, algebraic geometry, and cryptography
• It provides a powerful tool for studying the arithmetic properties of elliptic curves and related Diophantine equations
• Some of the most notable applications include the study of integral points, elliptic curve cryptography, and the resolution of certain Diophantine equations

Integral points on elliptic curves

• An integral point on an elliptic curve is a rational point whose coordinates are integers
• The Mordell-Weil theorem can be used to show that the set of integral points on an elliptic curve is finite
• Studying integral points is important for understanding the arithmetic of elliptic curves and has connections to Diophantine equations and the ABC conjecture (e.g., the equation $y^2 = x^3 + 1$ has only four integral solutions: $(0, \pm 1)$ and $(-1, 0)$)

Elliptic curve cryptography

• Elliptic curve cryptography (ECC) is a public-key cryptography system that uses the group of rational points on an elliptic curve as the underlying mathematical structure
• ECC relies on the difficulty of the elliptic curve discrete logarithm problem, which is believed to be harder than the discrete logarithm problem in finite fields
• The Mordell-Weil theorem ensures that the group of rational points is finite, which is necessary for the security of ECC (e.g., the curve $y^2 = x^3 - x + 1$ over a finite field is used in some ECC implementations)

Diophantine equations

• The Mordell-Weil theorem can be used to study certain types of Diophantine equations, which are polynomial equations in integers
• Many Diophantine equations can be transformed into questions about rational points on elliptic curves, allowing the theorem to be applied
• For example, the Fermat equation $x^n + y^n = z^n$ for $n \geq 3$ can be studied using elliptic curves, and the Mordell-Weil theorem played a role in Andrew Wiles' proof of Fermat's Last Theorem

Examples and computations

• Concrete examples and computations are essential for understanding the Mordell-Weil theorem and its applications
• By studying specific elliptic curves and their rational points, one can gain insight into the general theory and its practical implications
• In this section, we will look at some examples of curves with torsion points, curves of rank 0 and 1, and infinite families of curves

Curves with torsion points

• Torsion points on elliptic curves are points of finite order, i.e., points that generate a finite subgroup of the group of rational points
• The torsion subgroup of an elliptic curve over $\mathbb{Q}$ is well-understood and can be classified into 15 possible isomorphism classes
• Examples of curves with torsion points include:
  • $y^2 = x^3 - x$, which has torsion subgroup isomorphic to $\mathbb{Z}/4\mathbb{Z}$, generated by the point $(0,0)$
  • $y^2 = x^3 + 1$, which has torsion subgroup isomorphic to $\mathbb{Z}/6\mathbb{Z}$, generated by the point $(0,1)$

Curves of rank 0 and 1

• The rank of an elliptic curve is the number of independent points of infinite order in the group of rational points
• Curves of rank 0 have a finite number of rational points, while curves of rank 1 have infinitely many rational points that can be generated by a single point
• Examples of curves of rank 0 and 1 include:
  • $y^2 = x^3 - x$, which has rank 0 and only four rational points: $(0,0)$, $(1,0)$, $(-1,0)$, and the point at infinity
  • $y^2 = x^3 - 2x + 1$, which has rank 1 and infinitely many rational points generated by the point $(1,1)$

Infinite families of curves

• Some elliptic curves belong to infinite families with specific properties, such as having a certain torsion subgroup or rank
• Studying these families can provide insight into the behavior of elliptic curves and their rational points
• Examples of infinite families of curves include:
  • The Mordell curves, defined by the equation $y^2 = x^3 + k$ for integer values of $k$, which have been studied extensively due to their connection to the Mordell-Weil theorem
  • The Hessian family, defined by the equation $x^3 + y^3 + z^3 = dxyz$ for integer values of $d$, which includes curves with various torsion subgroups and ranks