{"id":{"repo_id":"rice","oai_identifier":"oai:repository.rice.edu:1911/114191"},"canonical_url":"https://search.dev.ndltd.org/etd/rice/oai:repository.rice.edu:1911/114191","repository":{"repo_id":"rice","name":"Rice University","base_url":"https://repository.rice.edu/server/oai/request"},"display":{"title":"A Bayesian approach to computing Brauer groups of cubic surfaces","abstract":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["James, Austen A"],"institution":"Rice University","degree_name":"Doctor of Philosophy","degree_level":"Doctoral","degree_discipline":"Natural Sciences","degree_department":null,"school":null,"contributors":[],"advisors":["Varilly-Alvarado, Anthony"],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-12-02","date_published":"2022-12-02","updated_at":"2026-07-24T04:10:15Z","subjects":["arithmetic geometry","brauer group","cubic surfaces","del pezzo surfaces","algebraic geometry","rational points"],"languages":["eng"],"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."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1911/114191","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Varilly-Alvarado, Anthony"]},{"key":"dc:creator","label":"Author","values":["James, Austen A"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2023-01-03T22:33:25Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2023-01-03T22:33:25Z"]},{"key":"dc:date.issued","label":"Date","values":["2022-12-02"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Natural Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Rice University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["arithmetic geometry","brauer group","cubic surfaces","del pezzo surfaces","algebraic geometry","rational points"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["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."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/1911/114191"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["A Bayesian approach to computing Brauer groups of cubic surfaces"]}]}],"canonical_facts":{"dc:contributor.advisor":["Varilly-Alvarado, Anthony"],"dc:creator":["James, Austen A"],"dc:date.accessioned":["2023-01-03T22:33:25Z"],"dc:date.available":["2023-01-03T22:33:25Z"],"dc:date.issued":["2022-12-02"],"dc:description.abstract":["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."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["https://hdl.handle.net/1911/114191"],"dc:language.iso":["eng"],"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."],"dc:subject":["arithmetic geometry","brauer group","cubic surfaces","del pezzo surfaces","algebraic geometry","rational points"],"dc:title":["A Bayesian approach to computing Brauer groups of cubic surfaces"],"dc:type":["Thesis"],"thesis:degree_discipline":["Natural Sciences"],"thesis:degree_level":["Doctoral"],"thesis:degree_name":["Doctor of Philosophy"],"thesis:institution_name":["Rice University"]},"updated_at":"2026-07-24T04:10:15Z"}