In arithmetic geometry, the Mordell–Weil group is an abelian group associated to any abelian variety
defined over a number field
It is an arithmetic invariant of the Abelian variety.
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
to the zero of the associated L-function at a special point.
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
be defined by the Weierstrass equation
(and this polynomial can be used to define a global model
First, we find some obvious torsion points by plugging in some numbers, which are
In addition, after trying some smaller pairs of integers, we find
is a point which is not obviously torsion.
One useful result for finding the torsion part of
and calculate the cardinality of the sets
note that because both primes only contain a factor of
, we have found all the torsion points.
has infinite order because otherwise there would be a prime factor shared by both cardinalities, so the rank is at least
Now, computing the rank is a more arduous process consisting of calculating the group
using some long exact sequences from homological algebra and the Kummer map.
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.
and an abelian variety
defined over a fixed field
for Galois cohomology of the field extension associated to the covering map
More explicitly, this 1-cocyle is given as a map of groups
which using universal properties is the same as giving two maps
This can be used to define the twisted abelian variety
using general theory of algebraic geometry[4]pg 5.
In particular, from universal properties of this construction,
is an abelian variety over
For the setup given above,[5] there is an isomorphism of abelian groups