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Abelian variety. In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in ...
Rank of an abelian group. In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. [1] The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the rational numbers of ...
Abelian group. In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian ...
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 group [1][2]pg 207. The main structure theorem about this group is the ...
Free abelian group. In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as an integer ...
The k-th homology group of an n-torus is a free abelian group of rank n choose k. It follows that the Euler characteristic of the n -torus is 0 for all n . The cohomology ring H • ( T n {\displaystyle \mathbb {T} ^{n}} , Z ) can be identified with the exterior algebra over the Z - module Z n {\displaystyle \mathbb {Z} ^{n}} whose generators ...
In mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group A. Often, this construction is made in the following situation: G is a commutative group scheme over a field K, Ks is the separable closure of K, and A = G (Ks) (the Ks -valued points of G). In this case, the Tate module of A is ...
In arithmetic geometry, the Tate–Shafarevich group ะจ(A/K) of an abelian variety A (or more generally a group scheme) defined over a number field K consists of the elements of the Weil–Châtelet group (/) = (,), where = (/) is the absolute Galois group of K, that become trivial in all of the completions of K (i.e., the real and complex completions as well as the p-adic fields obtained from ...