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If Z is an open subscheme of Spec ZZ, X is a sufficiently nice Z-model of asmooth curve over QQ, and p is a closed point of Z, the Chabauty-Kim methodleads to the construction of locally analytic functions on X(ZZ_p) which vanishon X(Z); we call such functions "Kim functions". At least in broad outline, themethod generalizes readily to higher dimensions. In fact, in some sense, thesurface 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 afurther condition to integrality. This is mirrored by a certain "weightadvantage" we encounter, because of which, M_{0,5} possesses new Kim functionsnot 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 thegeometric part of the computation from its arithmetic context. However, we findthat in this case the geometric step grows beyond the bounds of standardalgorithms running on current computers. Therefore, some ingenuity is needed tosolve this seemingly straightforward problem, and our new Kim function is huge.

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