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Explicit Brauer-Manin obstructions on plane quartics

Nils Bruin, Brendan Creutz

TL;DR

This work develops a practical, unconditional obstruction for plane quartics over number fields to possess rational points by employing a 2-descent framework tied to the 28 bitangents. Central to the approach is an explicit Hilbert-symbol pairing on an étale algebra $L$ associated to the bitangents, and a descent map $C_f$ that sends Picard groups to $(L\otimes K)^{\times}/(K^{\times}(L\otimes K)^{\times 2})$, enabling the construction of obstruction sets $V_{B,S}(Y)$. A key advance is the ability to obtain nontrivial obstructions without fully determining the $S$-unit group, using partial $S$-unit information and local-global pairings; the framework extends to arbitrary smooth projective curves via hyperelliptic and theta-characteristic data. The paper provides an algorithm and concrete computational demonstrations, including curves with index $2$ or $4$ and cases where the maximum local index is strictly smaller, and connects the obstruction to Brauer-Manin and Selmer-set viewpoints, enriching both theory and practical computations.

Abstract

We describe a method to show a plane quartic over a number field has no rational points. The method can be adapted to show that a curve does not have divisors of degree 1 or 2 and can be generalized to arbitrary smooth projective curves. Our approach significantly improves on the applicability over previous 2-cover descent methods by not requiring the computation of the full $S$-unit group of the étale algebras involved. We illustrate the practicality with several examples, including examples where we determine plane quartics to be of index 2 or 4 when the maximum local index is strictly smaller.

Explicit Brauer-Manin obstructions on plane quartics

TL;DR

This work develops a practical, unconditional obstruction for plane quartics over number fields to possess rational points by employing a 2-descent framework tied to the 28 bitangents. Central to the approach is an explicit Hilbert-symbol pairing on an étale algebra associated to the bitangents, and a descent map that sends Picard groups to , enabling the construction of obstruction sets . A key advance is the ability to obtain nontrivial obstructions without fully determining the -unit group, using partial -unit information and local-global pairings; the framework extends to arbitrary smooth projective curves via hyperelliptic and theta-characteristic data. The paper provides an algorithm and concrete computational demonstrations, including curves with index or and cases where the maximum local index is strictly smaller, and connects the obstruction to Brauer-Manin and Selmer-set viewpoints, enriching both theory and practical computations.

Abstract

We describe a method to show a plane quartic over a number field has no rational points. The method can be adapted to show that a curve does not have divisors of degree 1 or 2 and can be generalized to arbitrary smooth projective curves. Our approach significantly improves on the applicability over previous 2-cover descent methods by not requiring the computation of the full -unit group of the étale algebras involved. We illustrate the practicality with several examples, including examples where we determine plane quartics to be of index 2 or 4 when the maximum local index is strictly smaller.
Paper Structure (19 sections, 13 theorems, 21 equations)

This paper contains 19 sections, 13 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Étale algebras
  3. Finding elements in $\left(L^\times/L^{\times 2}\right)_{S}$
  4. An obstruction to rational points on plane quartics
  5. Generalization to arbitrary curves
  6. Examples of obstructions to rational points on quartic curves
  7. An example from BPS revisited
  8. A database of plane quartics of small discriminant
  9. Using good primes
  10. Testing
  11. Computing the index of a plane quartic
  12. Examples
  13. Local index $4$
  14. Local index $2$
  15. Local index $1$
  16. ...and 4 more sections

Key Result

Theorem 2.2

Let $L/k$ be an étale algebra over a number field $k$ and let $S$ be a finite set of primes of $k$.

Theorems & Definitions (35)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Theorem 3.3
  • proof
  • Remark 3.5
  • Remark 3.6
  • ...and 25 more