Weil-Châtelet group
Encyclopedia
In arithmetic geometry, the Weil–Châtelet group of an abelian variety
A defined over a field
K is the abelian group
of principal homogeneous space
s for A, defined over K. It is named for André Weil
, who introduced the general group operation in it, and François Châtelet. It plays a basic role in the arithmetic of abelian varieties
, in particular for elliptic curve
s, because of its connection with infinite descent
.
It can be defined directly from Galois cohomology
, as H1(GK,A), where GK is the absolute Galois group
of K. It is of particular interest for local field
s and global field
s, such as algebraic number field
s. For K a finite field
, it was proved that the group is trivial.
The Tate–Shafarevich group, named for John Tate
and Igor Shafarevich
, of an abelian variety A defined over a number field K consists of the elements of the Weil–Châtelet group that become trivial in all of the completions of K (i.e. the p-adic fields obtained from K, as well as its real and complex completions). Thus, in terms of Galois cohomology, in can be written as
It is often denoted Ш(A/K), where Ш is the Cyrillic letter "Sha", for Shafarevich.
Geometrically, the non-trivial elements of the Tate-Shafarevich group can be thought of as the homogeneous spaces of A that have Kv-rational point
s for every place v of K, but no K-rational point. Thus, the group measures the extent to which the Hasse principle
fails to hold for rational equations with coefficients in the field K. The Tate-Shafarevich group is conjectured to be finite; the first results on this were obtained by Karl Rubin
.
The Selmer group, named after Ernst S. Selmer, of A with respect to an isogeny
f:A→B of abelian varieties is a related group which can be defined in terms of Galois cohomology as
where Av[f] denotes the f-torsion of Av and
is the local Kummer map
. Note that 
is isomorphic to
. Geometrically, the principal homogeneous spaces coming from elements of the Selmer group have Kv-rational points for all places v of K. The Selmer group is finite. This has implications to the conjecture that the Tate-Shafarevich group is finite due to the following exact sequence
Ralph Greenberg has generalized the notion of Selmer group to more general p-adic Galois representations and to p-adic variations of motive
s in the context of Iwasawa theory
.
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and 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...
A defined over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K is the abelian group
Abelian group
In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...
of principal homogeneous space
Principal homogeneous space
In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G such that the stabilizer subgroup of any point is trivial...
s for A, defined over K. It is named for André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
, who introduced the general group operation in it, and François Châtelet. It plays a basic role in the arithmetic of abelian varieties
Arithmetic of abelian varieties
In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or family of those. It goes back to the studies of Fermat on what are now recognised as elliptic curves; and has become a very substantial area both in terms of results and conjectures...
, in particular for elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s, because of its connection with infinite descent
Infinite descent
In mathematics, a proof by infinite descent is a particular kind of proof by contradiction which relies on the fact that the natural numbers are well ordered. One typical application is to show that a given equation has no solutions. Assuming a solution exists, one shows that another exists, that...
.
It can be defined directly from Galois cohomology
Galois cohomology
In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups...
, as H1(GK,A), where GK is the absolute Galois group
Absolute Galois group
In mathematics, the absolute Galois group GK of a field K is the Galois group of Ksep over K, where Ksep is a separable closure of K. Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K. The absolute Galois group is unique up to isomorphism...
of K. It is of particular interest for local field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
s and global field
Global field
In mathematics, the term global field refers to either of the following:*an algebraic number field, i.e., a finite extension of Q, or*a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq, the field of rational functions...
s, such as algebraic number field
Algebraic number field
In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
s. For K a finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
, it was proved that the group is trivial.
The Tate–Shafarevich group, named for John Tate
John Tate
John Torrence Tate Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry.-Biography:...
and Igor Shafarevich
Igor Shafarevich
Igor Rostislavovich Shafarevich is a Soviet and Russian mathematician, founder of a school of algebraic number theory and algebraic geometry in the USSR, and a political writer. He was also an important dissident figure under the Soviet regime, a public supporter of Andrei Sakharov's Human Rights...
, of an abelian variety A defined over a number field K consists of the elements of the Weil–Châtelet group that become trivial in all of the completions of K (i.e. the p-adic fields obtained from K, as well as its real and complex completions). Thus, in terms of Galois cohomology, in can be written as
It is often denoted Ш(A/K), where Ш is the Cyrillic letter "Sha", for Shafarevich.
Geometrically, the non-trivial elements of the Tate-Shafarevich group can be thought of as the homogeneous spaces of A that have Kv-rational point
Rational point
In number theory, a K-rational point is a point on an algebraic variety where each coordinate of the point belongs to the field K. This means that, if the variety is given by a set of equationsthen the K-rational points are solutions ∈Kn of the equations...
s for every place v of K, but no K-rational point. Thus, the group measures the extent to which the Hasse principle
Hasse principle
In mathematics, Helmut Hasse's local-global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number...
fails to hold for rational equations with coefficients in the field K. The Tate-Shafarevich group is conjectured to be finite; the first results on this were obtained by Karl Rubin
Karl Rubin
Karl Rubin is an American mathematician at University of California, Irvine as Thorp Professor of Mathematics. His research interest is in elliptic curves. He was the first mathematician to show that some elliptic curves over the rationals have finite Tate-Shafarevich groups...
.
The Selmer group, named after Ernst S. Selmer, of A with respect to an isogeny
Isogeny
In mathematics, an isogeny is a morphism of varieties between two abelian varieties that is surjective and has a finite kernel....
f:A→B of abelian varieties is a related group which can be defined in terms of Galois cohomology as
where Av[f] denotes the f-torsion of Av and
is the local Kummer map. Note that is isomorphic to. Geometrically, the principal homogeneous spaces coming from elements of the Selmer group have Kv-rational points for all places v of K. The Selmer group is finite. This has implications to the conjecture that the Tate-Shafarevich group is finite due to the following exact sequenceExact sequence
An exact sequence is a concept in mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory...
- 0→B(K)/f(A(K))→Sel(f)(A/K)→Ш(A/K)[f]→0.
Ralph Greenberg has generalized the notion of Selmer group to more general p-adic Galois representations and to p-adic variations of motive
Motive (algebraic geometry)
In algebraic geometry, a motive denotes 'some essential part of an algebraic variety'. To date, pure motives have been defined, while conjectural mixed motives have not. Pure motives are triples , where X is a smooth projective variety, p : X ⊢ X is an idempotent correspondence, and m an integer...
s in the context of Iwasawa theory
Iwasawa theory
In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa, in the 1950s, as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur...
.