Mordell–Weil group
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. It is simply the group of -points of , so  is the Mordell–Weil grouppg 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  to the zero of the associated L-function at a special point. Examples 
Constructing 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 . Let  be defined by the Weierstrass equationover the rational numbers. It has discriminant  (and this polynomial can be used to define a global model ). It can be foundthrough the following procedure. First, we find some obvious torsion points by plugging in some numbers, which areIn 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  is that the torsion of prime to , for  having good reduction to , denoted  injects into , soWe check at two primes  and calculate the cardinality of the setsnote that because both primes only contain a factor of , we have found all the torsion points. In addition, we know the point  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  where  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  and an abelian variety  defined over a fixed field , we denote the  the twist of  (the pullback of  to the function field ) by a 1-cocylefor Galois cohomology of the field extension associated to the covering map . Note  which follows from the map being hyperelliptic. More explicitly, this 1-cocyle is given as a map of groupswhich using universal properties is the same as giving two maps , hence we can write it as a mapwhere  is the inclusion map and  is sent to negative . This can be used to define the twisted abelian variety  defined over  using general theory of algebraic geometrypg 5. In particular, from universal properties of this construction,  is an abelian variety over  which is isomorphic to  after base-change to . Theorem 
For the setup given above, there is an isomorphism of abelian groupswhere  is the Jacobian of the curve , and  is the 2-torsion subgroup of . See also 

 Mordell–Weil theorem

References

Further examples and cases 

 The Mordell–Weil Group of Curves of Genus 2
 Determining the Mordell–Weil group of a universal elliptic curve
 Galois descent and twists of an abelian variety
 On Mordell–Weil groups of Jacobians over function fields

Diophantine geometry
Elliptic curves
Abelian varieties