Várilly-Alvarado, Anthony2023-08-092023-08-092023-052023-04-12May 2023Zheng, 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>.https://hdl.handle.net/1911/115125We 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.application/pdfengCopyright 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.Brauer grouparithmetic geometryalgebraic geometrynumber theoryBrauer groupselliptic surfaceKodaira dimensionrational pointrational pointssurfacesalgebraic surfacestranscendentaltranscendental latticeThesis2023-08-09