Purity results for pp-divisible groups and abelian schemes over regular bases of mixed characteristic

Let $p$ be a prime. Let (R,\ideal{m}) be a regular local ring of mixed characteristic $(0,p)$ and absolute index of ramification $e$. We provide general criteria of when each abelian scheme over \Spec R\setminus\{\ideal{m}\} extends to an abelian scheme over \Spec R. We show that such extensions always exist if $e\le p-1$, exist in most cases if $p\le e\le 2p-3$, and do not exist in general if $e\ge 2p-2$. The case $e\le p-1$ implies the uniqueness of integral canonical models of Shimura varieties over a discrete valuation ring $O$ of mixed characteristic $(0,p)$ and index of ramification at most $p-1$. This leads to large classes of examples of N\'eron models over $O$. If $p>2$ and index $p-1$, the examples are new.Comment: 28 pages. Final version identical (modulo style) to the galley proofs. To appear in Doc. Mat

Families of p-divisible groups with constant Newton polygon

A p-divisible group over a base scheme in characteristic p in general does not admit a slope filtration. Let X be a p-divisible group with constant Newton polygon over a normal noetherian scheme S; we prove that there exists an isogeny from X to Y such that Y admits a slope filtration. In case S is regular this was proved by N. Katz for dim(S) = 1 and by T. Zink for dim(S) > 0. We give an example of a p-divisible group over a non-normal base which does not admit an isogeny to a p-divisible group with a slope filtration.Comment: To be published in Documenta Mathematic

Music and Medicine

Music therapy is not a new concept, although its acceptance by the medical community as a clinical modality is just beginning to grow. This newfound acceptance is the result of recently emerging empirical evidence supporting the efficacy of music in a range of applications. Using music to aid learning, either in recovery from brain damage or to overcome neurological disorders is widely accepted. For instance, music has been used to help patients learn to speak after traumatic brain injury (Schlaug, 2009). Much of these music learning programs are based off the Tomatis method that uses specifically adapted music tracks to stimulate cerebral blood flow and facilitate the formation of new neural pathways (Thompson, 2000). Music therapy is also used regularly to treat anxiety. These applications are easily accepted as nearly everyone has grown up with intimate musical experiences; music’s power to affect mood and set a tone are well recognized because it is a part of daily life

A comparison of logarithmic overconvergent de Rham-Witt and log-crystalline cohomology for projective smooth varieties with normal crossing divisor

This is the author accepted manuscript. The final version is available from Università di Padova via the DOI in this record.In this note we derive for a smooth projective variety X with normal crossing divisor Z an integral comparison between the log-crystalline cohomology of the associated log-scheme and the logarithmic overconvergent de Rham–Witt cohomology de fined by Matsuue. This extends our previous result that in the absence of a divisor Z the crystalline cohomology and overconvergent de Rham–Witt cohomology are canonically isomorphic

Overconvergent Wittvectors

Copyright © 2012 by Walter de Gruyter. The final publication is available at www.degruyter.comLet A be a finitely generated algebra over a field K of characteristic p > 0. We introduce a subring W†(A) ⊂ W(A), which we call the ring of overconvergent Witt vectors, and prove its basic properties. In a subsequent paper we use the results to define an overconvergent de Rham–Witt complex for smooth varieties over K whose hypercohomology is the rigid cohomology

Overconvergent de Rham-Witt cohomology

Copyright © 2011 Société mathématique de FranceThe goal of this work is to construct, for a smooth variety X over a perfect field k of finite characteristic p > 0, an overconvergent de Rham-Witt complex WyX=k as a suitable sub-complex of the de Rham-Witt complex of Deligne-Illusie. This complex, which is functorial in X, is a complex of etale sheaves and a differential graded algebra over the ring Wy( OX) of Overconvergent Witt-vectors. If X is affine one proves that there is a isomorphism between Monsky-Washnitzer cohomology and (rational) overconvergent de Rham-Witt cohomology. Finally we define for a quasiprojective X an isomorphism between the rational overconvergent de Rham-Witt cohomology and the rigid cohomology

On the p-adic uniformization of unitary Shimura curves

We prove $p$-adic uniformization for Shimura curves attached to the group of unitary similitudes of certain binary skew hermitian spaces $V$ with respect to an arbitrary CM field $K$ with maximal totally real subfield $F$. For a place $v|p$ of $F$ that is not split in $K$ and for which $V_v$ is anisotropic, let $\nu$ be an extension of $v$ to the reflex field $E$. We define an integral model of the corresponding Shimura curve over ${\rm Spec}\, O_{E, (\nu)}$ by means of a moduli problem for abelian schemes with suitable polarization and level structure prime to $p$. The formulation of the moduli problem involves a Kottwitz condition, an Eisenstein condition, and an adjusted invariant. The first two conditions are conditions on the Lie algebra of the abelian varieties; the last condition is a condition on the Riemann form of the polarization. The uniformization of the formal completion of this model along its special fiber is given in terms of the formal Drinfeld upper half plane for $F_v$. The proof relies on the construction of the contracting functor which relates a relative Rapoport-Zink space for strict formal $O_{F_v}$-modules with a Rapoport-Zink space of $p$-divisible groups which arise from the moduli problem, where the $O_{F_v}$-action is usually not strict when $F_v\ne \mathbb {Q}_p$. Our main tool is the theory of displays, in particular the Ahsendorf functor.Comment: 145 page