M0, 5: Toward the Chabauty–Kim method in higher dimensions

Abstract

If Z is an open subscheme of $SpecZ$, X is a sufficiently nice Z-model of a smooth curve over $Q$, and p is a closed point of Z, the Chabauty–Kim method leads to the construction of locally analytic functions on $X\left(Z_p\right)$ which vanish on $X\left(Z\right)$; 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 $M_{0,4}=P^1\setminus \{0,1,\infty \}$ 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 “$Z[1/6]$ 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 Wewers, 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.