The inverse Galois problem for del Pezzo surfaces of degree 1 and algebraic K3 surfaces

Wolff, Stephen Heinz

The inverse Galois problem for del Pezzo surfaces of degree 1 and algebraic K3 surfaces

dc.contributor.advisor	Várilly-Alvarado, Anthony
dc.creator	Wolff, Stephen Heinz
dc.date.accessioned	2022-09-26T19:25:09Z
dc.date.available	2023-02-01T06:01:17Z
dc.date.created	2022-08
dc.date.issued	2022-08-12
dc.date.submitted	August 2022
dc.date.updated	2022-09-26T19:25:09Z
dc.description.abstract	In 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.
dc.embargo.terms	2023-02-01
dc.format.mimetype	application/pdf
dc.identifier.citation	Wolff, 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>.
dc.identifier.uri	https://hdl.handle.net/1911/113396
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	Inverse Galois problem
dc.subject	Brauer group
dc.title	The inverse Galois problem for del Pezzo surfaces of degree 1 and algebraic K3 surfaces
dc.type	Thesis
dc.type.material	Text
thesis.degree.department	Mathematics
thesis.degree.discipline	Natural Sciences
thesis.degree.grantor	Rice University
thesis.degree.level	Doctoral
thesis.degree.name	Doctor of Philosophy