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 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 to the zero of the associated L-function at a special point.

Examples

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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  . Let   be defined by the Weierstrass equation

 

over the rational numbers. It has discriminant   (and this polynomial can be used to define a global model  ). It can be found[3]

 

through the following procedure. 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   is that the torsion of prime to  , for   having good reduction to  , denoted   injects into  , so

 

We check at two primes   and calculate the cardinality of the sets

 

note 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

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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)

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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-cocyle

 

for 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 groups

 

which using universal properties is the same as giving two maps  , hence we can write it as a map

 

where   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 geometry[4]pg 5. In particular, from universal properties of this construction,   is an abelian variety over   which is isomorphic to   after base-change to  .

Theorem

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For the setup given above,[5] there is an isomorphism of abelian groups

 

where   is the Jacobian of the curve  , and   is the 2-torsion subgroup of  .

See also

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References

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  1. ^ Tate, John T. (1974-09-01). "The arithmetic of elliptic curves". Inventiones Mathematicae. 23 (3): 179–206. Bibcode:1974InMat..23..179T. doi:10.1007/BF01389745. ISSN 1432-1297. S2CID 120008651.
  2. ^ Silverman, Joseph H., 1955– (2009). The arithmetic of elliptic curves (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-09494-6. OCLC 405546184.{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
  3. ^ a b Booher, Jeremy. "The Mordell–Weil theorem for elliptic curves" (PDF). Archived (PDF) from the original on 27 Jan 2021.
  4. ^ Weil, André, 1906-1998. (1982). "1.3". Adeles and algebraic groups. Boston: Birkhäuser. ISBN 978-1-4684-9156-2. OCLC 681203844.{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
  5. ^ Hazama, Fumio (1992). "The Mordell–Weil group of certain abelian varieties defined over the rational function field". Tohoku Mathematical Journal. 44 (3): 335–344. doi:10.2748/tmj/1178227300. ISSN 0040-8735.

Further examples and cases

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