Explicit Brauer-Manin obstructions on plane quartics

TL;DR

The paper presents a new method to determine the absence of rational points on plane quartics over number fields, improving previous techniques by avoiding complex computations and applying to various curve types.

Contribution

It introduces an explicit approach to Brauer-Manin obstructions on plane quartics, extending to other curves and surpassing prior 2-cover descent methods.

Findings

Successfully applied to examples showing no rational points

Determined indices of plane quartics in specific cases

Enhanced applicability over previous methods

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 \'etale 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.