4.2 Mordell-Weil theorem

The Mordell-Weil theorem is a cornerstone of elliptic curve theory. It states that the group of rational points on an elliptic curve over a number field is finitely generated. This powerful result connects algebra, geometry, and number theory.

Understanding the Mordell-Weil theorem is crucial for studying elliptic curves' structure and applications. It impacts areas like Diophantine equations, cryptography, and number theory, providing a framework for exploring rational points on these fascinating mathematical objects.

Statement of Mordell-Weil theorem

• Fundamental theorem in the study of elliptic curves over global fields such as the rational numbers $\mathbb{Q}$ or number fields
• Asserts that the group of rational points on an elliptic curve, denoted $E(\mathbb{Q})$ or more generally $E(K)$ for a number field $K$, is finitely generated
• Has important implications for understanding the structure and properties of elliptic curves and their rational points

For elliptic curves over Q

• When $E$ is defined over the rational numbers $\mathbb{Q}$, the Mordell-Weil theorem states that $E(\mathbb{Q})$ is a finitely generated abelian group
• This means $E(\mathbb{Q})$ is isomorphic to $\mathbb{Z}^r \oplus E(\mathbb{Q})_{\text{tors}}$, where $r$ is a non-negative integer called the rank and $E(\mathbb{Q})_{\text{tors}}$ is the torsion subgroup
• The torsion subgroup $E(\mathbb{Q})_{\text{tors}}$ is finite and consists of all points of finite order, while the rank $r$ measures the "size" of the infinite part of $E(\mathbb{Q})$

For elliptic curves over number fields

• The Mordell-Weil theorem extends to elliptic curves defined over arbitrary number fields $K$
• In this case, $E(K)$ is also a finitely generated abelian group isomorphic to $\mathbb{Z}^r \oplus E(K)_{\text{tors}}$
• The rank $r$ and the structure of the torsion subgroup $E(K)_{\text{tors}}$ depend on the number field $K$ and can be more complex than in the case of $\mathbb{Q}$

Finitely generated abelian groups

• Algebraic structures that play a central role in the statement and proof of the Mordell-Weil theorem
• Can be decomposed into a direct sum of a free abelian group of finite rank and a finite torsion subgroup
• Understanding their properties is crucial for studying the structure of elliptic curve groups

Definition and properties

• An abelian group $G$ is finitely generated if there exists a finite set $\{g_1, \ldots, g_n\} \subseteq G$ such that every element of $G$ can be expressed as a linear combination of the $g_i$ with integer coefficients
• The rank of a finitely generated abelian group is the size of a maximal linearly independent subset, or equivalently, the dimension of the tensor product $G \otimes_{\mathbb{Z}} \mathbb{Q}$ as a $\mathbb{Q}$-vector space
• Every finitely generated abelian group $G$ is isomorphic to a direct sum of the form $\mathbb{Z}^r \oplus G_{\text{tors}}$, where $r$ is the rank and $G_{\text{tors}}$ is the torsion subgroup

Torsion subgroup

• The torsion subgroup of a finitely generated abelian group $G$ is the subgroup consisting of all elements of finite order
• It is a finite abelian group and can be decomposed into a direct sum of cyclic groups of prime power order
• For elliptic curves over $\mathbb{Q}$, the possible structures of the torsion subgroup are classified by Mazur's theorem, which lists 15 possible isomorphism classes

Rank of group

• The rank of a finitely generated abelian group $G$ measures the size of the free part of $G$
• It is a non-negative integer and can be defined as the dimension of the $\mathbb{Q}$-vector space $G \otimes_{\mathbb{Z}} \mathbb{Q}$
• For elliptic curves, the rank is a key invariant that is often difficult to compute and is related to important conjectures like the Birch and Swinnerton-Dyer conjecture

Proof of Mordell-Weil theorem

• The proof of the Mordell-Weil theorem is a deep and technical result that combines ideas from algebraic geometry, number theory, and group theory
• It proceeds in two main steps: first establishing the weak Mordell-Weil theorem, and then using height functions to prove the full statement
• The proof is a cornerstone of the theory of elliptic curves and has led to the development of important techniques and concepts

Weak Mordell-Weil theorem

• States that for an elliptic curve $E$ over a number field $K$, the quotient group $E(K)/mE(K)$ is finite for every positive integer $m$
• Proved using techniques from Galois cohomology and the Kummer exact sequence
• Provides a first step towards the full Mordell-Weil theorem by showing that $E(K)$ is "almost" finitely generated

Height functions on elliptic curves

• A key tool in the proof of the Mordell-Weil theorem, used to measure the "size" or "complexity" of rational points on elliptic curves
• Can be defined using the canonical height, which is a quadratic form on $E(K)$ that behaves well under multiplication by integers
• Satisfy important properties like the Northcott property, which states that there are only finitely many points of bounded height

Descent via height functions

• A technique used to prove the full Mordell-Weil theorem by combining the weak Mordell-Weil theorem with height functions
• Involves constructing a suitable height function and using it to show that the quotient group $E(K)/mE(K)$ is not only finite but also of bounded height
• Leads to the conclusion that $E(K)$ is finitely generated by applying the Northcott property and the ascending chain condition for finitely generated abelian groups

Computing Mordell-Weil groups

• The Mordell-Weil theorem guarantees that the group of rational points on an elliptic curve is finitely generated, but computing the rank and generators can be a challenging problem
• Various methods and algorithms have been developed to compute Mordell-Weil groups, both in theory and in practice
• Computing Mordell-Weil groups is an active area of research with connections to computational number theory and cryptography

Computation of torsion subgroup

• The torsion subgroup of an elliptic curve over $\mathbb{Q}$ can be effectively computed using division polynomials and Mazur's theorem
• Over number fields, the computation of the torsion subgroup is more complex but can be achieved using techniques from algebraic number theory (Lutz-Nagell theorem)
• Efficient algorithms for computing torsion subgroups have been implemented in computer algebra systems like Sage and Magma

Computation of rank

• Computing the rank of an elliptic curve is a more difficult problem than computing the torsion subgroup
• Techniques for bounding the rank include descent methods (2-descent, 3-descent), the use of L-functions and the Birch and Swinnerton-Dyer conjecture, and the method of Selmer groups
• In practice, a combination of these methods is often used to obtain upper and lower bounds on the rank, but determining the exact rank can be challenging

Algorithms for finding generators

• Once the rank of an elliptic curve is known, finding a set of generators for the Mordell-Weil group is the next step
• Methods for finding generators include the use of height functions and the Mordell-Weil sieve, which systematically searches for points of bounded height
• The Heegner point construction can be used to find generators in some cases, particularly when the rank is equal to the analytic rank predicted by the Birch and Swinnerton-Dyer conjecture

Applications of Mordell-Weil theorem

• The Mordell-Weil theorem has numerous applications in various areas of mathematics, including Diophantine equations, cryptography, and number theory
• It provides a powerful tool for studying the structure and properties of elliptic curves and their rational points
• Many important problems and conjectures in number theory are related to the Mordell-Weil theorem and the behavior of elliptic curve groups

In Diophantine equations

• The Mordell-Weil theorem can be used to study the solvability of certain Diophantine equations, particularly those that can be related to elliptic curves
• For example, the theorem can be applied to prove the non-existence of rational solutions to certain cubic equations or to find all rational points on specific elliptic curves
• The Mordell-Weil theorem is also connected to the study of integral points on elliptic curves and the Siegel-Mahler theorem

In cryptography

• Elliptic curve cryptography (ECC) is a widely used form of public-key cryptography that relies on the difficulty of the elliptic curve discrete logarithm problem (ECDLP)
• The Mordell-Weil theorem guarantees that the group of rational points on an elliptic curve used in ECC is finitely generated, which is important for the security and efficiency of the cryptosystem
• Understanding the structure and properties of Mordell-Weil groups can help in the design and analysis of ECC protocols and in the study of potential attacks

In number theory

• The Mordell-Weil theorem is closely related to many important problems and conjectures in number theory, such as the Birch and Swinnerton-Dyer conjecture and the Tate-Shafarevich group
• It provides a framework for studying the arithmetic of elliptic curves and their L-functions, which encode deep information about the distribution of rational points
• The theorem also has connections to other areas of number theory, such as modular forms, Galois representations, and the Langlands program

Examples and exercises

• To gain a deeper understanding of the Mordell-Weil theorem and its applications, it is helpful to work through concrete examples and exercises
• These can range from simple computations of torsion subgroups and ranks to more advanced problems involving Diophantine equations and cryptographic protocols
• Examples and exercises also serve to illustrate the key concepts and techniques used in the study of elliptic curves and Mordell-Weil groups

Elliptic curves with various ranks

• Consider the elliptic curve $E: y^2 = x^3 - x$ over $\mathbb{Q}$. Compute its torsion subgroup and rank, and find a set of generators for the Mordell-Weil group $E(\mathbb{Q})$
• Investigate the elliptic curve $E: y^2 = x^3 - 4x$ over $\mathbb{Q}$, which has rank 0. Prove that $E(\mathbb{Q})$ consists only of the point at infinity and the two torsion points $(0, 0)$ and $(2, 0)$
• Study the elliptic curve $E: y^2 + y = x^3 - 7x + 6$ over $\mathbb{Q}$, which has rank 3. Find a set of generators for $E(\mathbb{Q})$ using the method of descent

Computing Mordell-Weil groups step-by-step

• For the elliptic curve $E: y^2 = x^3 + 2x - 1$ over $\mathbb{Q}$, compute the torsion subgroup using division polynomials and Mazur's theorem
• Apply the method of 2-descent to find an upper bound on the rank of the elliptic curve $E: y^2 = x^3 - 5x + 4$ over $\mathbb{Q}$
• Use the Heegner point construction to find a non-torsion point on the elliptic curve $E: y^2 = x^3 - 4x$ over $\mathbb{Q}(\sqrt{-7})$, and use this point to generate a subgroup of the Mordell-Weil group

Challenge problems and open questions

• Investigate the Birch and Swinnerton-Dyer conjecture for specific elliptic curves, and compute the analytic rank using L-functions and modular symbols
• Study the Tate-Shafarevich group of an elliptic curve and its relationship to the Mordell-Weil group and the Cassels-Tate pairing
• Explore unsolved problems related to the Mordell-Weil theorem, such as the existence of elliptic curves with arbitrarily large rank or the distribution of ranks in families of elliptic curves