$M_{0,5}$：迈向高维中的 Chabauty–Kim 方法
$M_{0,5}$: Towards the Chabauty-Kim method in higher dimensions
AI总结：
本文研究 Chabauty–Kim 方法向高维的推广，在半权重4的 ZZ[1/6] 情形下，利用权重优势构造并计算出曲面 $M_{0,5}$ 上首个非平凡 Kim 函数。
摘要：
如果 Z 是 Spec ZZ 的一个开子概形，X 是 QQ 上一条光滑曲线的足够良好的 Z-模型，并且 p 是 Z 的一个闭点，那么 Chabauty–Kim 方法会构造出 X(ZZ_p) 上在 X(Z) 处为零的局部解析函数；我们称这种函数为“Kim 函数”。至少从总体框架看，该方法很容易推广到更高维。事实上，在某种意义上，曲面 $M_{0,5}$ 应当比此前研究过的曲线 $M_{0,4}$ 更容易，因为它的点与 $M_{0,4}$ 的点密切相关，但还要满足一个额外的整性条件。这反映在我们遇到的某种“权重优势”上；由于这一优势，$M_{0,5}$ 拥有并非来自 $M_{0,4}$ 的新 Kim 函数。本文聚焦于“半权重 4 下的 ZZ[1/6]”情形，并在其中给出了曲面上 Kim 函数的第一个非平凡例子。我们处理 Chabauty–Kim 理论（由 S. Wewers、D. Corwin 和第一作者在相关工作中发展）的方法，其核心是能够把计算的几何部分与其算术背景分离开来。然而我们发现，在这一情形中，几何步骤的规模已经超出了当前计算机上标准算法的处理能力。因此，需要一些巧思才能解决这个看似简单直接的问题，而且我们得到的新 Kim 函数非常庞大。
英文摘要：
If Z is an open subscheme of Spec ZZ, X is a sufficiently nice Z-model of a smooth curve over QQ, and p is a closed point of Z, the Chabauty-Kim method leads to the construction of locally analytic functions on X(ZZ_p) which vanish on X(Z); we call such functions "Kim functions". At least in broad outline, the method generalizes readily to higher dimensions. In fact, in some sense, the surface M_{0,5} should be easier than the previously studied curve M_{0,4} since its points are closely related to those of M_{0,4}, yet they face a further condition to integrality. This is mirrored by a certain "weight advantage" we encounter, because of which, M_{0,5} possesses new Kim functions not coming from M_{0,4}. Here we focus on the case "ZZ[1/6] in half-weight 4", where we provide a first nontrivial example of a Kim function on a surface. Central to our approach to Chabauty-Kim theory (as developed in works by S. Wewers, D. Corwin, and the first author) is the possibility of separating the geometric part of the computation from its arithmetic context. However, we find that in this case the geometric step grows beyond the bounds of standard algorithms running on current computers. Therefore, some ingenuity is needed to solve this seemingly straightforward problem, and our new Kim function is huge.
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