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arXiv · 1209.0640

A non-abelian conjecture of Tate-Shafarevich type for hyperbolic curves

Jennifer Balakrishnan·Ishai Dan-Cohen·Minhyong Kim·Stefan Wewers
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Abstract

We state a conjectural criterion for identifying global integral points on a hyperbolic curve over $\mathbb{Z}$ in terms of Selmer schemes inside non-abelian cohomology functors with coefficients in $\mathbb{Q}_p$-unipotent fundamental groups. For $\mathbb{P}^1\setminus \{0,1,\infty\}$ and the complement of the origin in semi-stable elliptic curves of rank 0, we compute the local image of global Selmer schemes, which then allows us to numerically confirm our conjecture in a wide range of cases.

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Jennifer Balakrishnan, Ishai Dan-Cohen, Minhyong Kim, Stefan Wewers. 2012-09-04. A non-abelian conjecture of Tate-Shafarevich type for hyperbolic curves. https://arxiv.org/abs/1209.0640

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