Diophantine approximations on projective spaces
Diophantine approximations on projective spaces
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射影空间上的丢番图近似
DOI：
10.1007/bf01231559
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发表时间：
1994
影响因子：
3.1
通讯作者：
G. Wüstholz
中科院分区：
文献类型：
--
作者：
G. Faltings;G. Wüstholz
本文研究任意维射影空间上的丢番图逼近。特别是对WM施密特的结果给出了一个新的证明.历史上，丢番图逼近一直是证明丢番图几何中有限性结果的方法。在投影线的情况下，基本结果是罗斯定理。这已经被施密特推广到更高的维度。他的基本结果是子空间定理（见[$2]）。然而，证明是非常复杂和复杂的。它涉及到深刻的结果在几何的号码。最近一个新的发展理论开始Vojta导致突破丢番图近似的交换品种。他发现了一种新的方法来使用高度和定理的莫德尔和韦尔。他的主要见解是，在产品的阿贝尔品种之一，有更多的线束不仅仅是产品束。他的发现启发了Faltings在丢番图近似理论中找到了一个新的一般结果，即乘积定理。它非常简化了理论和它的应用阿贝尔品种导致一些推广的莫德尔猜想（见[-法]）。作为第二个应用法尔明斯还提出了一个下界的距离合理的一点，以超曲面的交换品种。在本文中，我们研究的后果的产品定理在更经典的情况下，射影空间。这里没有那么多的线束可作为阿贝尔品种。因此，结果一般会弱得多。然而，在施密特所考虑的情况下，结果最好可能达到e。在目前的情况下的优点是，我们不必科普复杂的线丛，在阿贝尔的情况下，这大大简化了证明。另一方面，人们需要的估计必须更加精确。为了在逼近定理中得到尽可能好的指数，我们必须非常小心。现在让我们解释一下我们的结果。我们固定一个数域K，选取定义在K上的一个子簇Ecn~”，研究K-有理点的距离d1，（E; x）
In this paper we study diophantine approximations on projective spaces of arbitrary dimension. Especially we give a new proof for the results of WM Schmidt. Historically diophantine approximation has been the method for proving finiteness results in diophantine geometry. In the case of the projective line the fundamental result is Roth's theorem. This has been generalized to higher dimension by Schmidt. His basic result is the subspace theorem (see [$2]). However the proof is very complicated and involved. It relates on deep results in the geometry of numbers.Recently a new development in the theory was started by Vojta which lead to a breakthrough for diophantine approximations on abelian varieties. He found a new way to use heights and the theorem of Mordell and Weil. His main insight was that on products of abelian varieties one has many more line bundles than just the product bundles. His discovery inspired Faltings to find a new general result in the theory of diophantine approximations, the product theorem. It very much simplifies the theory and its application to abelian varieties lead to some generalization of Mordell's conjecture (see [-Fa]). As a second application Faltings also gave a lower bound for the distance of a rational point to a hypersurface on an abelian variety. In the present paper we study the consequences of the product theorem in the more classical case of projective spaces. Here not so many line bundles are available as for abelian varieties. Therefore the results will be much weaker in general. However in the case Schmidt is considering the result is best possible up to an e. The advantage in the present situation is that we do not have to cope with complicated line bundles as in the abelian case; this simplifies the proofs considerably. On the other hand the estimates one needs have to be much more precise. In order to obtain the best possible exponents in the approximation theorems we have to be very careful. Now let us explain our results. We fix a number field K and chose a subvariety Ecn~" defined over K and study the distance d,,(E; x) of a K-rational point