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On Birch and Swinnerton-Dyer's cubic surfaces

Mckenzie West

Abstract

In a 1975 paper of Birch and Swinnerton-Dyer, a number of explicit norm form cubic surfaces are shown to fail the Hasse Principle. They make a correspondence between this failure and the Brauer--Manin obstruction, recently discovered by Manin. We generalize their work, making use of modern computer algebra software to show that a larger set of cubic surfaces have a Brauer--Manin obstruction to the Hasse principle, thus verifying the Colliot-Thélène--Sansuc conjecture for infinitely many cubic surfaces.

On Birch and Swinnerton-Dyer's cubic surfaces

Abstract

In a 1975 paper of Birch and Swinnerton-Dyer, a number of explicit norm form cubic surfaces are shown to fail the Hasse Principle. They make a correspondence between this failure and the Brauer--Manin obstruction, recently discovered by Manin. We generalize their work, making use of modern computer algebra software to show that a larger set of cubic surfaces have a Brauer--Manin obstruction to the Hasse principle, thus verifying the Colliot-Thélène--Sansuc conjecture for infinitely many cubic surfaces.

Paper Structure

This paper contains 7 sections, 9 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. The Hasse Principle for Cubic Surfaces
  3. Outline
  4. Setup and background
  5. Proof of Main Theorem
  6. Invariant map computations
  7. Examples

Key Result

Theorem 1.1

Suppose $k$ is a number field and $X$ is as in BSDeqn.

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Definition
  • Lemma 3.1
  • proof : Proof of \ref{['thm:main-1']}.
  • proof : Proof of \ref{['thm:main-2']}
  • Lemma 3.2: sd99, corn
  • Corollary 3.3
  • ...and 12 more