In
arithmetic geometry
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.
...
, the Tate–Shafarevich group of an
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 functi ...
(or more generally a
group scheme
In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups ha ...
) defined over a number field consists of the elements of the
Weil–Châtelet group
In arithmetic geometry, the Weil–Châtelet group or WC-group of an algebraic group such as an abelian variety ''A'' defined over a field ''K'' is the abelian group of principal homogeneous spaces for ''A'', defined over ''K''. named it for who ...
that become trivial in all of the completions of (i.e. the
-adic fields obtained from , as well as its real and complex completions). Thus, in terms of
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. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a nat ...
, it can be written as
:
This group was introduced by
Serge Lang and
John Tate and
Igor Shafarevich
Igor Rostislavovich Shafarevich (russian: И́горь Ростисла́вович Шафаре́вич; 3 June 1923 – 19 February 2017) was a Soviet and Russian mathematician who contributed to algebraic number theory and algebraic geometr ...
.
Cassels introduced the notation , where is the
Cyrillic letter "
Sha", for Shafarevich, replacing the older notation or .
Elements of the Tate–Shafarevich group
Geometrically, the non-trivial elements of the Tate–Shafarevich group can be thought of as the homogeneous spaces of that have -
rational point
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the fie ...
s for every
place
Place may refer to:
Geography
* Place (United States Census Bureau), defined as any concentration of population
** Census-designated place, a populated area lacking its own municipal government
* "Place", a type of street or road name
** Ofte ...
of , but no -rational point. Thus, the group measures the extent to which the
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 di ...
fails to hold for rational equations with coefficients in the field . Carl-Erik Lind gave an example of such a homogeneous space, by showing that the genus 1 curve has solutions over the reals and over all -adic fields, but has no rational points.
Ernst S. Selmer gave many more examples, such as .
The special case of the Tate–Shafarevich group for the finite group scheme consisting of points of some given finite order of an abelian variety is closely related to the
Selmer group.
Tate-Shafarevich conjecture
The Tate–Shafarevich conjecture states that the Tate–Shafarevich group is finite.
Karl Rubin proved this for some elliptic curves of rank at most 1 with
complex multiplication
In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visibl ...
. Victor A. Kolyvagin extended this to modular elliptic curves over the rationals of analytic rank at most 1 (The
modularity theorem
The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Weil conjecture or modularity conjecture for elliptic curves) states that elliptic curves over the field of rational numbers are related to modular forms. ...
later showed that the modularity assumption always holds).
Cassels–Tate pairing
The Cassels–Tate pairing is a
bilinear pairing , where is an abelian variety and is its dual. Cassels introduced this for
elliptic curves, when can be identified with and the pairing is an alternating form. The kernel of this form is the subgroup of divisible elements, which is trivial if the Tate–Shafarevich conjecture is true. Tate extended the pairing to general abelian varieties, as a variation of
Tate duality. A choice of polarization on ''A'' gives a map from to , which induces a bilinear pairing on with values in , but unlike the case of elliptic curves this need not be alternating or even skew symmetric.
For an elliptic curve, Cassels showed that the pairing is alternating, and a consequence is that if the order of is finite then it is a square. For more general abelian varieties it was sometimes incorrectly believed for many years that the order of is a square whenever it is finite; this mistake originated in a paper by Swinnerton-Dyer, who misquoted one of the results of Tate. Poonen and Stoll gave some examples where the order is twice a square, such as the Jacobian of a certain genus 2 curve over the rationals whose Tate–Shafarevich group has order 2, and Stein gave some examples where the power of an odd prime dividing the order is odd. If the abelian variety has a principal polarization then the form on is skew symmetric which implies that the order of is a square or twice a square (if it is finite), and if in addition the principal polarization comes from a rational divisor (as is the case for elliptic curves) then the form is alternating and the order of is a square (if it is finite).
See also
*
Birch and Swinnerton-Dyer conjecture
Citations
References
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* English translation in his collected mathematical papers
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{{DEFAULTSORT:Tate-Shafarevich group
Algebraic geometry
Number theory