论文标题
相对完成的chabauty-kim方法
The Chabauty-Kim Method for Relative Completions
论文作者
论文摘要
在本文中，我们为动机基本群体的相对完成开发了一个chabauty-kim理论，包括塞尔默堆栈和可允许的托架的模量空间，以相对完成de rham基本群体的相对完成。一方面，这项工作通过添加基本组的还原商来概括了金（以及chabauty）的结果。从这个角度来看，添加还原部分使人们可以将chabauty-type方法应用于具有微不足道单功能完成的基本组，例如$ sl_2（\ mathbb {z}）$。 另一方面，单身部分自然地扩展了劳伦斯和Venkatesh的最新作品。我们表明，随着一个人向上移动一层塔，远离旗帜品种的还原世界和高斯 - 曼宁的连接，他们对Frobenius的中央器的关注消失了。一个人希望相对完成将提供莫德尔猜想的统一证明，以利用这两种方法。为此，我们将作品应用于Legendre家族的投影线减去三分，这是劳伦斯和Venkatesh方法失败的一个特殊示例。
In this thesis we develop a Chabauty-Kim theory for the relative completion of motivic fundamental groups, including Selmer stacks and moduli spaces of admissible torsors for the relative completion of the de Rham fundamental group. On one hand, this work generalizes results of Kim (and therefore Chabauty) in the unipotent case by adding a reductive quotient of the fundamental group. From this perspective, the addition of a reductive part allows one to apply Chabauty-type methods to fundamental groups with trivial unipotent completion, such as $SL_2(\mathbb{Z})$. On the other hand, the unipotent part provides a natural extension of the recent work of Lawrence and Venkatesh. We show that their concern with the centralizer of Frobenius goes away as one moves up the unipotent tower and away from the reductive world of flag varieties and the Gauss-Manin connection. One is tempted to hope that the relative completion will provide a unified proof of Mordell's conjecture that takes advantage of the two methods. Toward this end, we apply our work to the Legendre family on the projective line minus three points, a particular example where the method of Lawrence and Venkatesh fails.
