Search results
Results From The WOW.Com Content Network
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 A ( K ) {\displaystyle A(K)} to the zero of the associated L ...
This is the fundamental theorem of finitely generated abelian groups. The existence of algorithms for Smith normal form shows that the fundamental theorem of finitely generated abelian groups is not only a theorem of abstract existence, but provides a way for computing expression of finitely generated abelian groups as direct sums. [14]: 26–27
The case with an elliptic curve and the field of rational numbers is Mordell's theorem, answering a question apparently posed by Henri Poincaré around 1901; it was proved by Louis Mordell in 1922. It is a foundational theorem of Diophantine geometry and the arithmetic of abelian varieties .
The fundamental theorem of finitely generated abelian groups can be stated two ways, generalizing the two forms of the fundamental theorem of finite abelian groups.The theorem, in both forms, in turn generalizes to the structure theorem for finitely generated modules over a principal ideal domain, which in turn admits further generalizations.
Every elementary abelian p-group is a vector space over the prime field with p elements, and conversely every such vector space is an elementary abelian group. By the classification of finitely generated abelian groups, or by the fact that every vector space has a basis, every finite elementary abelian group must be of the form (Z/pZ) n for n a ...
There is a complement to this theorem, first stated by Leo Zippin (1935) and proved in Kurosh (1960), which addresses the existence of an abelian p-group with given Ulm factors. Let τ be an ordinal and { A σ } be a family of countable abelian p - groups indexed by the ordinals σ < τ such that the p - heights of elements of each A σ are ...
One important structure theorem of abelian varieties is Matsusaka's theorem. It states that over an algebraically closed field every abelian variety A {\displaystyle A} is the quotient of the Jacobian of some curve; that is, there is some surjection of abelian varieties J → A {\displaystyle J\to A} where J {\displaystyle J} is a Jacobian.
is an abelian group NS(V), called the Néron–Severi group of V. This is a finitely-generated abelian group by the Néron–Severi theorem, which was proved by Severi over the complex numbers and by Néron over more general fields. In other words, the Picard group fits into an exact sequence