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

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.

arXiv commentImprovements to the exposition and numerous minor corrections throughout

References in corpus (4)

• p-adic multiple zeta values I -- p-adic multiple polylogarithms and the p-adic KZ equation
• Massey products for elliptic curves of rank 1
• Selmer varieties for curves with CM Jacobians
• Explicit Coleman integration for hyperelliptic curves

Cited by in corpus (2)

• Quadratic Chabauty and rational points I: p-adic heights
• Computing integral points on hyperelliptic curves using quadratic Chabauty