Chabauty--Kim, finite descent, and the Section Conjecture for locally geometric sections

2023/05/16 by L. Alexander Betts, Betts, L. Alexander, Theresa Kumpitsch +3
Arts and Humanities · Mathematics · Social Sciences · #14H25 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Historical Studies and Socio-cultural Analysis #Number Theory (math.NT) #Primary: 14H30. Secondary: 11G30 #Vietnamese History and Culture Studies

paper · pdf · doi:10.48550/arxiv.2305.09462

openalex publication_date 2023/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let X be a smooth projective curve of genus ≥2 over a number field. A natural variant of Grothendieck's Section Conjecture postulates that every section of the fundamental exact sequence for X which everywhere locally comes from a point of X in fact globally comes from a point of X. We show that X/ℚ satisfies this version of the Section Conjecture if it satisfies Kim's Conjecture for almost all choices of auxiliary prime p, and give the appropriate generalisation to S-integral points on hyperbolic curves. This gives a new "computational" strategy for proving instances of this variant of the Section Conjecture, which we carry out for the thrice-punctured line over ℤ[1/2].

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