Várilly-Alvarado, Anthony2022-09-262023-02-012022-082022-08-12August 202Wolff, Stephen Heinz. "The inverse Galois problem for del Pezzo surfaces of degree 1 and algebraic K3 surfaces." (2022) Diss., Rice University. <a href="https://hdl.handle.net/1911/113396">https://hdl.handle.net/1911/113396</a>.https://hdl.handle.net/1911/113396In this thesis we study the inverse Galois problem for algebraic K3 surfaces and for del Pezzo surfaces of degree one. We begin with an overview of how the question of the existence of k-points on a nice k-variety leads, via Brauer groups, to the inverse Galois problem. We then discuss an algorithm to compute all finite subgroups of the general linear group GL(n,Z), up to conjugacy. The first cohomology of these subgroups are a superset of the target groups of the inverse Galois problem for any family of nice k-varieties whose geometric Picard group is free and of finite rank. We apply these results to algebraic K3 surfaces defined over the rational numbers, providing explicit equations for a surface solving the only nontrivial instance of the inverse Galois problem in geometric Picard rank two. We then study the inverse Galois problem for representatives from three families of del Pezzo surfaces of degree one, searching for 5-torsion in the Brauer group. For two of the three surfaces, we show that the Brauer group is trivial when the surface is defined over the rational numbers, but becomes isomorphic to Z/5Z or (Z/5Z)^2 when the base field is raised to a suitable number field. For the third surface, we show that its splitting field has degree 2400 as an extension of the rational numbers, a degree consistent with 5-torsion in the Brauer group.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.Inverse Galois problemBrauer groupThesis2022-09-26