TY - UNPB

T1 - M0,5

T2 - Towards the Chabauty-Kim method in higher dimensions

AU - Dan-Cohen, Ishai

AU - Jarossay, David

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - Mathematics - Algebraic Geometry

KW - Mathematics - Number Theory

M3 - ???researchoutput.researchoutputtypes.workingpaper.preprint???

T3 - Arxiv preprint

BT - M0,5

ER -