Arithmetic of p-adic curves and sections of geometrically abelian\n fundamental groups

2020/05/11 by Mohamed Saı̈di, Saidi, Mohamed
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2005.04971

openalex publication_date 2020/05/11 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

Let X be a proper, smooth, and geometrically connected curve of genus\ng(X)\≥ 1 over a p-adic local field. We prove that there exists an\neffectively computable open affine subscheme U\⊂ X with the property\nthat period (X)=1, and index (X) equals 1 or 2 (resp. period(X)=index\n(X)=1, assuming period (X)=index (X)), if (resp. if and only if) the exact\nsequence of the geometrically abelian fundamental group of U splits. We\ncompute the torsor of splittings of the exact sequence of the geometrically\nabelian absolute Galois group associated to X, and give a new\ncharacterisation of sections of arithmetic fundamental groups of curves over\np-adic local fields which are orthogonal to Pic⁰ (resp. Pic ^wedge).\nAs a consequence we observe that the non-geometric (geometrically pro-p)\nsection constructed by Hoshi in [Hoshi] is orthogonal to Pic⁰.\n

Related

• Local sections of arithmetic fundamental groups of p-adic curves
• Abelian geometric fundamental groups for curves over a p-adic field
• On the period-index problem in light of the section conjecture
• Arithmetic in the fundamental group of a p-adic curve: On the p-adic section conjecture for curves
• Further examples of non-geometric sections of arithmetic fundamental\n groups