Mordell–Weil group

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, :E(\mathbb{Q}) \cong \mathbb{Z}/2\times \mathbb{Z}/2 \times \mathbb{Z} through the following procedure: first, we find 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). We check at two primes p = 5,7 and calculate the cardinality of the sets :\begin{align} \# E(\mathbb{F}_5) &= 8 = 2^3 \\ \# E(\mathbb{F}_{7}) &= 12 = 2^2\cdot 3 \end{align} Note that because both primes only contain a common 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-cocyleb \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 groupsG\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 mapb = (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 geometrypg 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). Theorem For the setup given above, there is an isomorphism of abelian groupsA_b(k(t)) \cong \operatorname{Hom}_k(J(C), A)\oplus A_2(k)where J(C) is the Jacobian of the curve C, and A_2 is the 2-torsion subgroup of A. == See also ==