Mordell–Weil group

In arithmetic geometry, the Mordell–Weil group is an abelian group associated to any abelian variety $A$ defined over a number field $K$ . It is an arithmetic invariant of the Abelian variety. It is simply the group of $K$ -points of $A$ , so $A(K)$ is the Mordell–Weil group^[1]^[2]^{pg 207}. The main structure theorem about this group is the Mordell–Weil theorem which shows this group is in fact a finitely-generated abelian group. Moreover, there are many conjectures related to this group, such as the Birch and Swinnerton-Dyer conjecture which relates the rank of $A(K)$ to the zero of the associated L-function at a special point.

Examples

Constructing^[3] explicit examples of the Mordell–Weil group of an abelian variety is a non-trivial process which is not always guaranteed to be successful, so we instead specialize to the case of a specific elliptic curve $E/\mathbb {Q}$ . Let $E$ be defined by the Weierstrass equation

$y^{2}=x(x-6)(x+6)$

over the rational numbers. It has discriminant $\Delta _{E}=2^{12}\cdot 3^{6}$ (and this polynomial can be used to define a global model ${\mathcal {E}}/\mathbb {Z}$ ). It can be found^[3]

$E(\mathbb {Q} )\cong \mathbb {Z} /2\times \mathbb {Z} /2\times \mathbb {Z}$

through the following procedure. First, we find some obvious torsion points by plugging in some numbers, which are

$\infty ,(0,0),(6,0),(-6,0)$

In addition, after trying some smaller pairs of integers, we find $(-3,9)$ is a point which is not obviously torsion. One useful result for finding the torsion part of $E(\mathbb {Q} )$ is that the torsion of prime to $p$ , for $E$ having good reduction to $p$ , denoted $E(\mathbb {Q} )_{\mathrm {tors} ,p}$ injects into $E(\mathbb {F} _{p})$ , so

$E(\mathbb {Q} )_{\mathrm {tors} ,p}\hookrightarrow E(\mathbb {F} _{p})$

We check at two primes $p=5,7$ and calculate the cardinality of the sets

${\begin{aligned}\#E(\mathbb {F} _{5})&=8=2^{3}\\\#E(\mathbb {F} _{7})&=12=2^{2}\cdot 3\end{aligned}}$

note that because both primes only contain a factor of $2^{2}$ , we have found all the torsion points. In addition, we know the point $(-3,9)$ has infinite order because otherwise there would be a prime factor shared by both cardinalities, so the rank is at least $1$ . Now, computing the rank is a more arduous process consisting of calculating the group $E(\mathbb {Q} )/2E(\mathbb {Q} )\cong (\mathbb {Z} /2)^{r+2}$ where $r=\operatorname {rank} (E(\mathbb {Q} ))$ using some long exact sequences from homological algebra and the Kummer map.

Theorems concerning special cases

There are many theorems in the literature about the structure of the Mordell–Weil groups of abelian varieties of specific dimension, over specific fields, or having some other special property.

Abelian varieties over the rational function field k(t)

For a hyperelliptic curve $C$ and an abelian variety $A$ defined over a fixed field $k$ , we denote the $A_{b}$ the twist of $A|_{k(t)}$ (the pullback of $A$ to the function field $k(t)=k(\mathbb {P} ^{1})$ ) by a 1-cocyle

$b\in Z^{1}(\operatorname {Gal} (k(C)/k(t)),{\text{Aut}}(A))$

for Galois cohomology of the field extension associated to the covering map $f:C\to \mathbb {P} ^{1}$ . Note $G=\operatorname {Gal} (k(C)/k(t)\cong \mathbb {Z} /2$ which follows from the map being hyperelliptic. More explicitly, this 1-cocyle is given as a map of groups

$G\times G\to \operatorname {Aut} (A)$

which using universal properties is the same as giving two maps $G\to {\text{Aut}}(A)$ , hence we can write it as a map

$b=(b_{id},b_{\iota })$

where $b_{id}$ is the inclusion map and $b_{\iota }$ is sent to negative $\operatorname {Id} _{A}$ . This can be used to define the twisted abelian variety $A_{b}$ defined over $k(t)$ using general theory of algebraic geometry^[4]^{pg 5}. In particular, from universal properties of this construction, $A_{b}$ is an abelian variety over $k(t)$ which is isomorphic to $A|_{k(C)}$ after base-change to $k(C)$ .