changbaihou 写了： 2024年 1月 17日 18:11
本身就不是简单的东西。泛泛地讲就是关于离local-global principle成立有多远的问题。粗略地说，Mordell-Weil group可以认为是Selmer group的一个子群，其商群就是Sha。
CHAT:
The Tate-Shafarevich group is a mathematical object that arises in the study of abelian varieties, which are generalizations of elliptic curves. It is named after John Tate and Igor Shafarevich, who introduced and studied it in the 1950s and 1960s.
The Tate-Shafarevich group measures the failure of the Hasse principle, which roughly states that a geometric object defined over a number field has a rational point if and only if it has a point over every completion of the field. The Hasse principle is known to hold for some classes of objects, such as quadratic forms, but not for others, such as cubic curves.
The Tate-Shafarevich group is defined as the kernel of a map from the Weil-Chatelet group, which consists of all principal homogeneous spaces of a given abelian variety, to the product of the local cohomology groups, which capture the local information of the abelian variety. The elements of the Tate-Shafarevich group are those homogeneous spaces that have points everywhere locally, but no global point. Thus, the Tate-Shafarevich group encodes the global obstruction to the Hasse principle.
One of the major open problems in number theory is the Tate-Shafarevich conjecture, which asserts that the Tate-Shafarevich group is always finite. This conjecture is closely related to the Birch and Swinnerton-Dyer conjecture, which predicts the rank of the Mordell-Weil group of an abelian variety from the analytic properties of its L-function. The Tate-Shafarevich conjecture implies that the Mordell-Weil group is finitely generated, which is a necessary condition for the Birch and Swinnerton-Dyer conjecture.
The Tate-Shafarevich group is a subtle and mysterious invariant that is very hard to compute in general. However, there are some cases where it is known to be trivial or finite, such as for some elliptic curves with complex multiplication or modular forms. There are also some examples where it is known to be infinite, such as for some Jacobians of hyperelliptic curves.