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

Netan Dogra

Abstract

We give refined methods for proving finiteness of the Chabauty--Coleman--Kim set $X(\mathbb{Q}_2 )_2 $, when $X$ is a hyperelliptic curve with a rational Weierstrass point. The main developments are methods for computing Selmer conditions at $2$ and $\infty$ for the mod 2 Bloch--Kato Selmer group associated to the higher Chow group $\mathrm{CH}^2 (\mathrm{Jac}(X),1)$. 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(\mathbb{Q}_2 )_2 <\infty $. We also obtain a field-theoretic description of second descent on the Jacobian of a hyperelliptic curve (under some conditions).

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

Abstract

We give refined methods for proving finiteness of the Chabauty--Coleman--Kim set , when is a hyperelliptic curve with a rational Weierstrass point. The main developments are methods for computing Selmer conditions at and for the mod 2 Bloch--Kato Selmer group associated to the higher Chow group . 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 . We also obtain a field-theoretic description of second descent on the Jacobian of a hyperelliptic curve (under some conditions).
Paper Structure (22 sections, 33 theorems, 152 equations)

This paper contains 22 sections, 33 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Relation to elliptic curve Chabauty
  3. Arithmetic statistics of Bloch--Kato Selmer groups
  4. Notation
  5. A review of $2$-descent for Bloch--Kato Selmer groups
  6. Boundary maps for $J[4]$
  7. Strategy of proof of Proposition \ref{['prop:everything']}
  8. Field-theoretic description of $H^1 (K,\mathrm{End} (J[2]))$ and the cup product map
  9. Describing the class of $J[4]$
  10. The boundary map for $\wedge ^2 J[4]$
  11. A field-theoretic description of $H^1 (K,\mathrm{End} (\wedge ^2 J[2]))$
  12. Field-theoretic description of the map $H^1 (K,\mathrm{End} (J[2]))\to H^1 (K,\mathrm{End} (\wedge ^2 J[2]))$
  13. The boundary map for $\wedge ^2 J[4]$ in terms of cup products
  14. The boundary map for local fields
  15. Lifting obstructions at $\infty$
  16. ...and 7 more sections

Key Result

Theorem 1

Of the 6,603 curves in the LMFDB with Mordell--Weil rank 2 and exactly one rational Weierstrass point, at least $3,323$ satisfy $\# X(\mathbb Q _2 )_2 <\infty$. Moreover, for each such $X$, $X'(\mathbb Q _2 )_2$ is finite whenever $X'$ is a quadratic twist of $X$ with Jacobian of Mordell--Weil rank

Theorems & Definitions (58)

  • Theorem 1
  • Lemma 1: BKdescent1, Lemma 25
  • Conjecture 1
  • Lemma 2
  • Proposition 1: BKdescent1, Proposition 1
  • Proposition 2: BKdescent1
  • proof
  • Proposition 3
  • Remark 1
  • Lemma 3
  • ...and 48 more