A Bayesian approach to computing Brauer groups of cubic surfaces

We present an algorithm for computing the Brauer groups of cubic surfaces. The algorithm takes as input an equation for a cubic surface X and a confidence threshold 0.5 < r < 1 and outputs a candidate for the Brauer group of X and a confidence level > r for the result. The algorithm runs by sampling lifts of Frobenius at many primes of good reduction and relies on Chebotarev’s density theorem and Bayesian inference to produce, with confidence level > r, a subgroup of the Weyl group of E_6. This subgroup represents the action of Galois on the geometric Picard group of X, from which we compute the Brauer group of X. We give a description of this algorithm and a proof that it terminates, as well as an implementation in Magma. We also examine the speed of such an approach relative to existing methods and explore how the Bayesian technique of this algorithm can be applied to answer questions concerning the Galois and Brauer groups of other classes of surfaces.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Natural Sciences

Grantor

Rice University

Year dc:date.issued

2022

Author dc:creator

• James, Austen A

Advisor dc:contributor.advisor

• Varilly-Alvarado, Anthony

• arithmetic geometry
• brauer group
• cubic surfaces
• del pezzo surfaces
• algebraic geometry
• rational points

Statement dc:rights

• Copyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.

Language dc:language.iso

eng

Handle dc:identifier.uri

https://hdl.handle.net/1911/114191

OAI identifier oai:identifier

oai:repository.rice.edu:1911/114191