Explicit Brauer-Manin obstructions on plane quartics
Number Theory
2026-05-15 v2
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 \'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.
Cite
@article{arxiv.2601.16975,
title = {Explicit Brauer-Manin obstructions on plane quartics},
author = {Nils Bruin and Brendan Creutz},
journal= {arXiv preprint arXiv:2601.16975},
year = {2026}
}
Comments
v2: minor corrections and added references. Magma code for the examples is now available as ancillary files