In this thesis, we set up the basic theory of p-adic modular forms over Shimura curves over Q, parallel to the classical case over modular curves. We define and study the structure of the spaces of p-adic modular forms with respect to certain quaternion algebras over Q. We study the relation of these modular forms with classical quaternionic modular forms. We prove a canonical subgroup theorem ...

We will formulate and prove a certain reciprocity law relating certain residues of the differential symbol dlog2 from the K2 of a Mumford curve to the rigid analytic regulator constructed by the author in a previous paper. We will use this result to deduce some consequences on the kernel and image of the rigid analytic regulator analogous to some old conjectures of Beilinson and Bloch on the co...

We give an example of an affinoid curve without analytic continuation. We use this to produce an example of an affinoid morphism that cannot be flattened by a finite sequence of local blow-ups. Thus the global rigid analogue, [7], Theorem 2.3, of Hironaka’s complex analytic flattening theorem is not true. Since this is a key step in the proof of the affinoid elimination theorem, [7], Theorem 3....

The geometry of the Lubin-Tate space of deformations of a formal group is studied via an étale, rigid analytic map from the deformation space to projective space. This leads to a simple description of the equivariant canonical bundle of the deformation space which, in turn, yields a formula for the dualizing complex in stable homotopy theory. Introduction Ever since Quillen [22, 1 ] discovered ...

Transformable polyhedral surfaces with rigid facets, i.e., rigid origami, are useful for designing kinetic and deployable structures. In order to apply rigid origami to various architectural and other engineering design purposes, it is essential to consider the geometry of origami in kinetic motion and provide sufficiently generalized methods to produce controlled variations of shapes that suit...