TY - JOUR

T1 - M0, 5

T2 - Toward the Chabauty–Kim method in higher dimensions

AU - Dan-Cohen, Ishai

AU - Jarossay, David

N1 - Funding Information:
We would like to thank Jennifer Balakrishnan, Amnon Besser, and Hidekazu Furusho for their interest and encouragement as well as for helpful conversations. In particular, we thank Balakrishnan for her participation in our attempts to approach the “geometric step” via computer computation. Finally, we are grateful to the referee for many helpful comments and suggestions. Both authors were supported by ISF grants 726/17 and 621/21.
Publisher Copyright:
© 2023 The Authors. Mathematika is copyright © University College London and published by the London Mathematical Society on behalf of University College London.

PY - 2023/10/1

Y1 - 2023/10/1

N2 - If Z is an open subscheme of (Figure presented.), X is a sufficiently nice Z-model of a smooth curve over (Figure presented.), and p is a closed point of Z, the Chabauty–Kim method leads to the construction of locally analytic functions on (Figure presented.) which vanish on (Figure presented.); 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 M0, 5 should be easier than the previously studied curve (Figure presented.) since its points are closely related to those of M0, 4, yet they face a further condition to integrality. This is mirrored by a certain weight advantage we encounter, because of which, M0, 5 possesses new Kim functions not coming from M0, 4. Here we focus on the case “ (Figure presented.) 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.

AB - If Z is an open subscheme of (Figure presented.), X is a sufficiently nice Z-model of a smooth curve over (Figure presented.), and p is a closed point of Z, the Chabauty–Kim method leads to the construction of locally analytic functions on (Figure presented.) which vanish on (Figure presented.); 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 M0, 5 should be easier than the previously studied curve (Figure presented.) since its points are closely related to those of M0, 4, yet they face a further condition to integrality. This is mirrored by a certain weight advantage we encounter, because of which, M0, 5 possesses new Kim functions not coming from M0, 4. Here we focus on the case “ (Figure presented.) 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.

UR - http://www.scopus.com/inward/record.url?scp=85166536386&partnerID=8YFLogxK

U2 - 10.1112/mtk.12215

DO - 10.1112/mtk.12215

M3 - Article

AN - SCOPUS:85166536386

SN - 0025-5793

VL - 69

SP - 1011

EP - 1059

JO - Mathematika

JF - Mathematika

IS - 4

ER -