Perinormality in Polynomial and Module-Finite Ring Extensions

In this dissertation we investigate some open questions posed by Epstein and Shapiro in [9] regarding perinormal domains. More specifically, we focus on the ascent/descent property of perinormality between "canonical" integral domain extensions, in particular, R [superscript] R[X] and R [suberscript] Rb. We give special conditions under which perinormality ascends from R to the polynomial ring R[X] in the case that R is a universally catenary domain. Whereas we have a characterizing result for when perinormality descends from R[X] to R, the sufficient condition for the descent is cumbersome to check. For this reason, we turn to special cases for which perinormality descends from R[X] to R. In the case of an analytically irreducible local domain (R, m) and its m-adic completion (R, b mRb), we refer to a technique for generating examples in which perinormality fails to ascend. When Rb is perinormal, we explore hypotheses under which R must be normal, perinormal, or weakly normal.

Rapoport–Zink spaces for spinor groups

After the work of Kisin, there is a good theory of canonical integral models of Shimura varieties of Hodge type at primes of good reduction. The first part of this paper develops a theory of Hodge type Rapoport–Zink formal schemes, which uniformize certain formal completions of such integral models. In the second part, the general theory is applied to the special case of Shimura varieties associated with groups of spinor similitudes, and the reduced scheme underlying the Rapoport–Zink space is determined explicitly.

Galois Sections in Absolute Anabelian Geometry

We show that isomorphisms between arithmetic fundamental groups of hyperbolic curves over p-adic local fields preserve the decomposition groups of all closed points (respectively, closed points arising from torsion points of the underlying elliptic curve), whenever the hyperbolic curves in question are isogenous to hyperbolic curves of genus zero defined over a number field (respectively, are once-punctured elliptic curves [which are not necessarily defined over a number field]). We also show that, under certain conditions, such isomorphisms preserve certain canonical “integral structures” at the cusps [i.e., points at infinity] of the hyperbolic curve.