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

2012/09/04 by Jennifer S. Balakrishnan, Ishai Dan‐Cohen, Balakrishnan, Jennifer +5 · 2 citations
Mathematics · #14G05 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1209.0640

openalex publication_date 2012/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We state a conjectural criterion for identifying global integral points on a hyperbolic curve over ℤ in terms of Selmer schemes inside non-abelian cohomology functors with coefficients in ℚ_p-unipotent fundamental groups. For ℙ¹∖ \0,1,∞\ 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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