数论问题：Diopthantine几何几个定理和Diophantine集合

[changbaihou]

Real number是local field，我觉得挺费解的。Real number和p-adic number有什么共同点呢？

而complex number不是local field。那么就是global field了？

No. R 不是个local field。

[TheMatrix]

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。

嗯。不错。这就可以了。

[TheMatrix]

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.

[forecasting]

我只说说我印象中知道的，如果不准确的话见谅。好多年前当研究生时做过椭圆曲线，后来就没太关心过了。

关于找Q上曲线的有理点，genus 0的曲线不用说，太简单了。印象中genus大于等于2的曲线也有有效算法。椭圆曲线独成特殊一档，因为可能有无穷多有理点，但又不像二次曲线那么简单。Torsion part通过Nagell-Lutz定理可以很轻松地确定，自由部分没有有效可行的办法，我印象中就是蛮干。当然“蛮干”中肯定也有不少trick(比如递降法，local方法，。。。)，我不做计算也不熟。我一个师弟专做计算，以前用MAGMA和SAGE算椭圆曲线算得很来劲。

给定一条椭圆曲线，其L-函数在s=1的阶是可以确定的。所以，假设BSD猜想的话，我们起码知道蛮干到啥地步停下来。从理论上来说，如果知道椭圆曲线的Mordell-Weil rank大于零的话，那曲线上(canonical) height\leq X的有理点个数是有渐近公式的(depending on the rank though)，而且(heuristically) 在一定X界下的有理点应该包含了所有free part的一组生成元。但是这些渐近结果并不是effective的，所以对一条给定elliptic curve，目前并没有个确定的界X，说是找出height小于等于X的有理点就搞定了。所以只能边找有理点边检测己有点的linear dependence，直到找到一组生成元。

我们常常看到说有人确定某条椭圆曲线rank大于等于8 (for example)，而不是说等于8。有可能是计算太复杂，找到一组8个点的线性无关组就停了。但也有可能其实解析rank就是8，但是因为是基于未证实的BSD猜想，严谨点就只能说是大于等于8。如果愿意的话，很多情况下其实是可以确定rank的。比如，如果一条椭圆曲线有trivial的Tate-Shafarevich group的话，它的Selmer group的rank和Mordell-Weil rank是一样的。而Selmer group好算多了。

以上仅根据我的记忆，可能有不准确的地方。

这些得查阅实现一下。但是在stackoverflow上顺便问这个问题时，有回答者说这是个开问题，因为没学习过，所以不知道他的回答是否正确。