Generalised height pairings and the Albanese kernel

2025/07/14 by Dogra, Netan · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.10111

Abstract

The Chabauty--Coleman--Kim method in depth two describes the rational points on a curve in terms of a generalisation of Nekovář's p-adic height pairing which replaces \mathbbG_m with a higher Chow group. It is unclear both what the domain of definition of this pairing is, and how to compute it. This paper explores the relevance of the Beilinson--Bloch conjectures to this problem. In particular, it is shown that if X is a smooth projective curve and the Albanese kernel of X× X is torsion, then there is an algorithm to compute the generalised height pairing on a pair of rational points on the Jacobian. This leads to the consideration of certain `motivic refinements' of the nonabelian cohomology varieties which arise in nonabelian Chabauty.

Citations

• O-minimal geometry of higher Albanese manifolds
• Refined effective bounds for Bloch-Kato Selmer groups associated to hyperelliptic curves
• Periods of Real Biextensions
• 2-descent for Bloch--Kato Selmer groups and rational points on hyperelliptic curves I
• Hyperelliptic curves mapping to abelian varieties and applications to Beilinson's conjecture for zero-cycles
• Quadratic Chabauty and rational points II: Generalised height functions on Selmer varieties
• The unit theorem for finite-dimensional algebras
• Remarks on Non-Abelian Cohomology of Proalgebraic Groups
• Remark on fundamental groups and effective Diophantine methods for hyperbolic curves
• Motivic interpretation of Milnor K-groups attached to Jacobian varieties
• The unipotent Albanese map and Selmer varieties for curves
• Galois actions on fundamental groups of curves and the cycle C-C^-
• Groupes fondamentaux motiviques de Tate mixte

Cited by

• Affine Chabauty I

Related

• A generalization of Beilinson's geometric height pairing
• Quadratic Chabauty and rational points II: Generalised height functions on Selmer varieties
• Quadratic Chabauty: p-adic height pairings and integral points on hyperelliptic curves
• Asymptotics of local height pairing
• Bounds on the Chabauty--Kim Locus of Hyperbolic Curves