Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Refined effective bounds for Bloch-Kato Selmer groups associated to hyperelliptic curves

Lee Berry

TL;DR

This work develops refined, computable bounds for Bloch-Kato Selmer groups attached to $ ext{CH}^2(J,1)$ of hyperelliptic curves by extending explicit $2$-descent to curves without rational Weierstrass points and by exploiting good ordinary reduction at $2$. It integrates the Chabauty-Kim framework with explicit local obstructions at $2$ and infinity through boundary maps and a de Rham regulator map $ heta_{ ext{dR}}$, yielding concrete upper bounds on $ ext{dim}_{Q_2} H^1_f(Q, igwedge^2 V_2J)$ and finiteness criteria for the depth $2$ Chabauty-Kim set $X(Q_2)_2$. The paper provides algorithmic procedures to compute $ ext{Ker}( heta_{ ext{dR}})$ and demonstrates substantial practical impact by verifying finiteness on many genus $2$ curves and several genus $3$ curves from the LMFDB, including cases with no rational Weierstrass point. Conditional results under Bloch-Kato conjectures further illuminate expected dimensions of global Selmer groups, connecting geometric properties of $J$ (e.g., Néron–Severi rank) to the effectiveness of the descent. Overall, the methods offer a constructive pathway to determine finiteness of $p$-adic Chabauty sets and rational points on broad families of hyperelliptic curves, with substantial computational feasibility for genus up to $3$.

Abstract

We develop refined methods to effectively bound the dimension of Bloch-Kato Selmer groups associated to the higher Chow group $\mathrm{CH}^2(J,1)$, where $J$ is the Jacobian of a hyperelliptic curve $X$. This extends the recent work of Dogra on explicit $2$-descent for these Selmer groups to include cases where $X$ does not have a rational Weierstrass point. Additionally, we develop methods for obtaining sharper dimension bounds under the assumption that $X$ has good ordinary reduction at $2$. As a consequence, we establish new criteria for deducing finiteness of the depth $2$ Chabauty-Kim set $X(\mathbb{Q}_2)_2$, and demonstrate the efficacy of these criteria on curves from the LMFDB.

Refined effective bounds for Bloch-Kato Selmer groups associated to hyperelliptic curves

TL;DR

This work develops refined, computable bounds for Bloch-Kato Selmer groups attached to of hyperelliptic curves by extending explicit -descent to curves without rational Weierstrass points and by exploiting good ordinary reduction at . It integrates the Chabauty-Kim framework with explicit local obstructions at and infinity through boundary maps and a de Rham regulator map , yielding concrete upper bounds on and finiteness criteria for the depth Chabauty-Kim set . The paper provides algorithmic procedures to compute and demonstrates substantial practical impact by verifying finiteness on many genus curves and several genus curves from the LMFDB, including cases with no rational Weierstrass point. Conditional results under Bloch-Kato conjectures further illuminate expected dimensions of global Selmer groups, connecting geometric properties of (e.g., Néron–Severi rank) to the effectiveness of the descent. Overall, the methods offer a constructive pathway to determine finiteness of -adic Chabauty sets and rational points on broad families of hyperelliptic curves, with substantial computational feasibility for genus up to .

Abstract

We develop refined methods to effectively bound the dimension of Bloch-Kato Selmer groups associated to the higher Chow group , where is the Jacobian of a hyperelliptic curve . This extends the recent work of Dogra on explicit -descent for these Selmer groups to include cases where does not have a rational Weierstrass point. Additionally, we develop methods for obtaining sharper dimension bounds under the assumption that has good ordinary reduction at . As a consequence, we establish new criteria for deducing finiteness of the depth Chabauty-Kim set , and demonstrate the efficacy of these criteria on curves from the LMFDB.

Paper Structure

This paper contains 21 sections, 21 theorems, 126 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. $2$-descent for Bloch-Kato Selmer groups
  3. Notation and conventions
  4. Acknowledgements
  5. Preliminaries in Galois cohomology
  6. The even degree case and the generalised Jacobian
  7. Hyperelliptic curves over $\mathbb{Q}_2$ with good ordinary reduction
  8. Descent in the Chabauty-Kim method
  9. Effective bounds and finiteness criteria
  10. Case 1: Hyperelliptic curves with a rational Weierstrass point
  11. Local obstructions at $2$ and $\infty$ via explicit boundary maps
  12. Statement of the main theorem in the odd degree case
  13. Case 2: Hyperelliptic curves without a rational Weierstrass point
  14. Low genus examples
  15. Conditional dimension formulae
  16. ...and 6 more sections

Key Result

Theorem 1

Suppose Then $X(\mathbb{Q}_p)_2$ is finite.

Theorems & Definitions (50)

  • Theorem : [Kim09, Theorem 1], [Dog23, Lemma 25]
  • Theorem A: Theorems \ref{['MainOddTheorem']} and \ref{['MainEvenThm']}
  • Theorem B: Theorem \ref{['ExampleComputation']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4: [DM23, Lemma 2.1]
  • Lemma 2.6
  • ...and 40 more