2-descent for Bloch--Kato Selmer groups and rational points on hyperelliptic curves II

2024/03/12 by Netan Dogra, Dogra, Netan
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2403.07476

openalex publication_date 2024/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give refined methods for proving finiteness of the Chabauty--Coleman--Kim set X(ℚ₂ )₂ , when X is a hyperelliptic curve with a rational Weierstrass point. The main developments are methods for computing Selmer conditions at 2 and ∞ for the mod 2 Bloch--Kato Selmer group associated to the higher Chow group CH² (Jac(X),1). As a result we show that most genus 2 curves in the LMFDB of Mordell--Weil rank 2 with exactly one rational Weierstrass point satsify # X(ℚ₂ )₂ <∞ . We also obtain a field-theoretic description of second descent on the Jacobian of a hyperelliptic curve (under some conditions).

Related

• 2-descent for Bloch--Kato Selmer groups and rational points on hyperelliptic curves I
• Refined effective bounds for Bloch-Kato Selmer groups associated to hyperelliptic curves
• Bounds on the Chabauty--Kim Locus of Hyperbolic Curves
• Two-descent on some genus two curves
• 2-Selmer groups of hyperelliptic curves with two marked points