Arithmetic aspects of strong F-Regularity

In this dissertation, we investigate the existence and abundance of finite torsors over the regular locus of strongly F-regular singularities. We do this by studying how the F-signature transforms under this type of finite cover. By restricting our attention to 'etale torsors, we prove that the local 'etale fundamental group of a strongly F-regular singularity is finite. In fact, we obtain effiective bounds on its order in terms of the F-signature. In the general case, we prove that any strongly F-regular singularity X admits a finite cover X? ! X, with X? strongly F-regular, such that the X? has the following property: for all finite group-schemes G whose connected component at the identity is either trigonalizable or nilpotent, we have that every G-torsor over the regular locus of X? is the restriction of a G-torsor over X?. As a consequence of that proof, we conclude that strongly F-regular singularities admit no nontrivial unipotent torsors. Along the way, we give a new Purity of the Branch Locus result for singularities with F-signature larger than 1=2. We also obtain eective bounds on the torsion of divisor class groups of strongly F-regular singularities, and globally F-regular varieties. Additionally, we prove that canonical covers of strongly F-regular (resp. F-pure) singularities are strongly F-regular (resp. F-pure).