$M_{0,5}$: Towards the Chabauty-Kim method in higher dimensions

TL;DR

This paper extends the Chabauty-Kim method to higher-dimensional varieties, specifically constructing a Kim function on the surface M_{0,5} over ZZ[1/6], demonstrating the method's potential beyond curves.

Contribution

It introduces the first nontrivial Kim function on a higher-dimensional surface, showcasing the method's applicability and highlighting computational challenges involved.

Findings

Constructed a Kim function on M_{0,5} over ZZ[1/6]

Demonstrated the method's extension from curves to surfaces

Identified computational limitations in geometric steps

Abstract

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.