Mordell–Weil group
In arithmetic geometry, the Mordell–Weil group is an abelian group associated to any abelian variety formula_1 defined over a number field formula_2, it is an arithmetic invariant of the Abelian variety. It is simply the group of formula_2-points of formula_1, so formula_5 is the Mordell–Weil group. 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 formula_5 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 formula_7. Let formula_8 be defined by the Weierstrass equationformula_9over the rational numbers. It has discriminant formula_10 (and this polynomial can be used to define a global model formula_11). It can be foundformula_12through the following procedure. First, we find some obvious torsion points by plugging in some numbers, which areformula_13In addition, after  trying some smaller pairs of integers, we find formula_14 is a point which is not obviously torsion. One useful result for finding the torsion part of formula_15 is that the torsion of prime to formula_16, for formula_8 having good reduction to formula_16, denoted formula_19 injects into formula_20, soformula_21We check at two primes formula_22 and calculate the cardinality of the setsformula_23note that because both primes "only" contain a factor of formula_24, we have found all the torsion points. In addition, we know the point formula_14 has infinite order because otherwise there would be a prime factor shared by both cardinalities, so  the rank is at least formula_26. Now, computing the rank is a more arduous process consisting of calculating the group formula_27 where formula_28 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 formula_29 and an abelian variety formula_1 defined over a fixed field formula_31, we denote the formula_32 the twist of  formula_33 (the pullback of formula_1 to the function field formula_35) by a 1-cocyleformula_36for Galois cohomology of the field extension associated to the covering map formula_37. Note formula_38 which follows from the map being hyperelliptic. More explicitly, this 1-cocyle is given as a map of groupsformula_39where using universal, properties, this is the same as giving two maps formula_40, hence we can write it as a mapformula_41where formula_42 is the inclusion map and formula_43 is sent to negative formula_44. This can be used to define the twisted abelian variety formula_32 defined over formula_46 using general theory of algebraic geometry. In  particular, from universal properties of this construction, formula_32 is an abelian variety over formula_46 which is isomorphic to formula_49 after base-change to formula_50. Theorem. For the setup given above, there is an isomorphism of abelian groupsformula_51where formula_52 is the Jacobian of the curve formula_29, and formula_54 is the 2-torsion subgroup of formula_1.