Finite descent obstructions and rational points on curves
Finite descent obstructions and rational points on curves
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DOI：
10.2140/ant.2007.1.349
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发表时间：
2006-06
影响因子：
1.3
通讯作者：
M. Stoll
中科院分区：
文献类型：
--
作者：
M. Stoll
设k是数域，X是光滑k-簇.本文研究了在有限k-群格式下通过挠曲线下降法得到的关于X上的k-有理点在顶点内的位置的信息。我们的主要结果是，如果曲线C/k非平凡地映射到阿贝尔簇A/k，使得A（k）是有限的，并且X（k，A）没有非平凡的可分元素，那么来自有限阿贝尔下降的信息精确地切掉了C的有理点。我们猜想这是所有亏格≥ 2的曲线的情况。我们与有限下降障碍的Brauer-Manin障碍，特别是，我们证明，在曲线上，Brauer集等于一套削减有限阿贝尔下降。因此，我们的猜想意味着Brauer-Manin对有理点的阻塞是曲线上唯一的一个。
Let k be a number field and X a smooth projective k-variety. In this paper, we study the information obtainable from descent via torsors under finite k-group schemes on the location of the k-rational points on X within the adelic points. Our main result is that if a curve C/k maps nontrivially into an abelian variety A/k such that A(k) is finite and X(k, A) has no nontrivial divisible elements, then the information coming from finite abelian descent cuts out precisely the rational points of C. We conjecture that this is the case for all curves of genus ≥ 2. We relate finite descent obstructions to the Brauer-Manin obstruction; in particular, we prove that on curves, the Brauer set equals the set cut out by finite abelian descent. Our conjecture therefore implies that the Brauer-Manin obstruction against rational points in the only one on curves.