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

Netan Dogra

Abstract

This paper introduces explicit Galois cohomological methods for determining the ranks of Bloch--Kato Selmer groups associated to the Tate twists of the 2-adic second étale cohomology of the Jacobian of a hyperelliptic curve with a rational Weierstrass point. In particular, this can give a method to determine the rational points on such curves via the Chabauty--Coleman--Kim method. This is applied to answer a question of Bugeaud, Mignotte, Siksek, Stoll and Tengely.

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

Abstract

This paper introduces explicit Galois cohomological methods for determining the ranks of Bloch--Kato Selmer groups associated to the Tate twists of the 2-adic second étale cohomology of the Jacobian of a hyperelliptic curve with a rational Weierstrass point. In particular, this can give a method to determine the rational points on such curves via the Chabauty--Coleman--Kim method. This is applied to answer a question of Bugeaud, Mignotte, Siksek, Stoll and Tengely.
Paper Structure (17 sections, 36 theorems, 155 equations)

This paper contains 17 sections, 36 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. $2$-descent for hyperelliptic curves
  3. Notation
  4. Galois cohomology of the wedge square
  5. The nonabelian $(x-T)$-map
  6. Explicit formula for the $H^1 (K,\wedge ^2 J[2])$ class
  7. $2$-descent for Selmer schemes: local aspects away from $2$
  8. Properties of $H^1 _f (K_v ,T)$
  9. $2$-descent for Selmer schemes: local aspects at $2$
  10. Integral mixed extensions
  11. The Chabauty--Coleman--Kim method
  12. Galois cohomology of number fields
  13. The Chabauty--Coleman--Kim sets $X(\mathbb Q _p )_n$
  14. Dimension bounds via descent
  15. Generalised heights
  16. ...and 2 more sections

Key Result

Theorem 1

Theorems & Definitions (65)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 55 more