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arXiv · 2601.16975

Explicit Brauer-Manin obstructions on plane quartics

Nils Bruin·Brendan Creutz
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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.

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Nils Bruin, Brendan Creutz. 2026-01-23. Explicit Brauer-Manin obstructions on plane quartics. https://arxiv.org/abs/2601.16975

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