Brauer groups of a family of nonnegative Kodaira dimension elliptic surfaces

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dc.contributor.advisorVárilly-Alvarado, Anthony
dc.creatorZheng, Ken
dc.date.accessioned2023-08-09T17:00:08Z
dc.date.available2023-08-09T17:00:08Z
dc.date.created2023-05
dc.date.issued2023-04-12
dc.date.submittedMay 2023
dc.date.updated2023-08-09T17:00:08Z
dc.description.abstractWe explore the Brauer groups of the elliptic surfaces given by y² = x³ + t⁶ᵐ + 1 over Q for m = 2, 3. When m = 2, the resulting surface is K3, and when m = 3, the surface is honestly elliptic with Kodaira dimension 1. We compute the algebraic Brauer groups of these surfaces by studying the action of Gal(̅Q/Q) on their Neron-Severi groups. Following the work of Gvirtz, Loughran, and Nakahara, we find bounds for the exponents of transcendental Brauer groups of these surfaces. The transcendental Brauer group is closely related to the transcendental lattice. The argument begins with an explicit description of the basis of the respective transcendental lattices and reinterpreting elements of these lattices as elements in rings of integers. From this, we bound the transcendental Brauer group. These bounds apply more generally to the surfaces given by y² = x³ + A₁t⁶ᵐ + A₂ for integers Aᵢ and m = 2, 3.
dc.format.mimetypeapplication/pdf
dc.identifier.citationZheng, Ken. "Brauer groups of a family of nonnegative Kodaira dimension elliptic surfaces." (2023) Diss., Rice University. <a href="https://hdl.handle.net/1911/115125">https://hdl.handle.net/1911/115125</a>.
dc.identifier.urihttps://hdl.handle.net/1911/115125
dc.language.isoeng
dc.rightsCopyright 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.subjectBrauer group
dc.subjectarithmetic geometry
dc.subjectalgebraic geometry
dc.subjectnumber theory
dc.subjectBrauer groups
dc.subjectelliptic surface
dc.subjectKodaira dimension
dc.subjectrational point
dc.subjectrational points
dc.subjectsurfaces
dc.subjectalgebraic surfaces
dc.subjecttranscendental
dc.subjecttranscendental lattice
dc.titleBrauer groups of a family of nonnegative Kodaira dimension elliptic surfaces
dc.typeThesis
dc.type.materialText
thesis.degree.departmentMathematics
thesis.degree.disciplineNatural Sciences
thesis.degree.grantorRice University
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy
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