Mordell–Weil theorem

Mordell–Weil theorem. In mathematics, specifically in the branch of number theory known as arithmetic geometry, the Mordell–Weil theorem is a foundational result concerning the structure of the set of rational points on an abelian variety defined over a number field. Proven by Louis Mordell for elliptic curves over the rational numbers and later generalized by André Weil to abelian varieties over arbitrary number fields, it states that this set of points forms a finitely generated abelian group. This theorem connects the discrete arithmetic of Diophantine equations with the continuous theory of Lie groups and algebraic geometry, providing a crucial framework for modern research.

Statement of the theorem

The theorem formally states that if $A$ is an abelian variety defined over a number field $K$, then the group $A(K)$ of $K$-rational points on $A$ is a finitely generated abelian group. By the Structure theorem for finitely generated abelian groups, this means $A(K)$ is isomorphic to $\backslash mathbb\{Z\}^r\; \backslash oplus\; T$, where $r$ is a non-negative integer called the Mordell–Weil rank and $T$ is the finite torsion subgroup. The proof relies on combining the theory of height functions, specifically the Néron–Tate height, with Faltings's theorem on the Mordell conjecture, and techniques from Galois cohomology such as the Selmer group and the Tate–Shafarevich group.

Examples

The most classical example is an elliptic curve over the rational numbers, such as the curve defined by $y^2\; =\; x^3\; -\; x$. Here, the group of rational points is finitely generated; for this specific curve, the Mordell–Weil rank is zero and the torsion subgroup is isomorphic to $\backslash mathbb\{Z\}/2\backslash mathbb\{Z\}\; \backslash times\; \backslash mathbb\{Z\}/2\backslash mathbb\{Z\}$, as shown by work following Leonhard Euler. For the curve $y^2\; +\; y\; =\; x^3\; -\; x^2$, known as the elliptic curve 11a1 in Cremona database, the work of John Coates and Andrew Wiles shows it has rank zero and torsion $\backslash mathbb\{Z\}/5\backslash mathbb\{Z\}$. In contrast, the curve defined by $y^2\; +\; xy\; =\; x^3\; +\; x^2\; -\; 2x$ has rank 1, with a generator found using algorithms based on the Birch and Swinnerton-Dyer conjecture.

Proof sketch

The proof, following the strategy refined by André Weil and later mathematicians, proceeds in two main steps. First, one proves the Weak Mordell–Weil theorem, which states that for any positive integer $m$, the quotient group $A(K)/mA(K)$ is finite. This step uses the Kummer theory sequence in Galois cohomology, relating points to elements of the Selmer group, and relies on the finiteness of ideal class groups and the Dirichlet's unit theorem in the ring of integers of $K$. The second step introduces a height function, specifically the canonical Néron–Tate height, which is a positive-definite quadratic form on the real vector space $A(K)\; \backslash otimes\; \backslash mathbb\{R\}$. The combination of the height's properties and the finiteness result from the first step forces $A(K)$ to be finitely generated.

Generalizations and related results

The theorem has been vastly generalized. André Weil extended it to abelian varieties over global fields, including function fields over finite fields. A profound related result is Faltings's theorem, which proves the Mordell conjecture and states that a curve of genus greater than one over a number field has only finitely many rational points. The structure of the Selmer group and the conjectured finiteness of the Tate–Shafarevich group are central to understanding the Mordell–Weil rank, as seen in the Birch and Swinnerton-Dyer conjecture. Furthermore, the theorem applies in the context of Jacobians of curves over number fields, and analogues exist for abelian varieties over certain p-adic fields studied by Jean-Pierre Serre.

Applications

The theorem is fundamental to the arithmetic study of Diophantine equations. It underpins the method of infinite descent on elliptic curves, allowing the resolution of equations like the Congruent number problem via the Tunnell's theorem. It provides the structural framework for the Birch and Swinnerton-Dyer conjecture, one of the Clay Mathematics Institute's Millennium Prize Problems. Computationally, algorithms for determining the Mordell–Weil group, such as those using Heegner points developed by Benedict Gross and Don Zagier, or 2-descent methods implemented in software like PARI/GP and Magma, rely on the theorem. Its principles also influence Iwasawa theory for elliptic curves, as in the work of Barry Mazur and Ralph Greenberg.

Category:Diophantine geometry Category:Theorems in number theory Category:Algebraic geometry