摘要翻译:
给定数域$k$上亏格的光滑射影曲线$x$至少为2,Grothendieck的截面猜想预言了从$x$的基本群到绝对Galois群的正则投影有截面当且仅当该曲线有有理点。我们证明了存在这样的曲线:上面的映射在每完成$k$上有一个截面,但不超过$k$。在附录中,Victor Flynn在属2中给出了明确的例子。我们的结果是一个更一般的研究的结果,它是关于几何部分为abelianized的Etale基本群在Galois群上的投影的截面的存在性。给出了任意维数和任意完美域上存在截面的一个判据,并进一步研究了局部域和整体域上曲线的情形。我们还指出了Colliot-th\'el\'ene、Sansuc与初等梗阻的关系。
---
英文标题:
《Galois sections for abelianized fundamental groups》
---
作者:
David Harari, Tamas Szamuely
---
最新提交年份:
2009
---
分类信息:
一级分类:Mathematics 数学
二级分类:Algebraic Geometry 代数几何
分类描述:Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇,叠,束,格式,模空间,复几何,量子上同调
--
一级分类:Mathematics 数学
二级分类:Number Theory 数论
分类描述:Prime numbers, diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory
素数,丢番图方程,解析数论,代数数论,算术几何,伽罗瓦理论
--
---
英文摘要:
Given a smooth projective curve $X$ of genus at least 2 over a number field $k$, Grothendieck's Section Conjecture predicts that the canonical projection from the \'etale fundamental group of $X$ onto the absolute Galois group of $k$ has a section if and only if the curve has a rational point. We show that there exist curves where the above map has a section over each completion of $k$ but not over $k$. In the appendix Victor Flynn gives explicit examples in genus 2. Our result is a consequence of a more general investigation of the existence of sections for the projection of the \'etale fundamental group `with abelianized geometric part' onto the Galois group. We give a criterion for the existence of sections in arbitrary dimension and over arbitrary perfect fields, and then study the case of curves over local and global fields more closely. We also point out the relation to the elementary obstruction of Colliot-Th\'el\`ene and Sansuc.
---
PDF链接:
https://arxiv.org/pdf/0808.2556