Papers by Michael Rapoport
arXiv (Cornell University), Feb 13, 2013
Proceedings of symposia in pure mathematics, 1997
arXiv (Cornell University), Apr 3, 2008
The supersingular locus in the fiber at p of a Shimura variety attached to a unitary similitude g... more The supersingular locus in the fiber at p of a Shimura variety attached to a unitary similitude group GU(1, n -1) over Q is uniformized by a formal scheme N . In the case when p is an inert prime, we define special cycles Z(x) in N , associated to collections x of m 'special homomorphisms' with fundamental matrix T ∈ Herm m (O k ). When m = n and T is nonsingular, we show that the cycle Z(x) is either empty or is a union of components of the Ekedahl-Oort stratification, and we give a necessary and sufficient condition, in terms of T , for Z(x) to be irreducible. When Z(x) is zero dimensional -in which case it reduces to a single point -we determine the length of the corresponding local ring by using a variant of the theory of quasi-canonical liftings. We show that this length coincides with the derivative of a representation density for hermitian forms.
arXiv (Cornell University), Jun 15, 2021
We establish basic results on p-adic shtukas and apply them to the theory of local and global Shi... more We establish basic results on p-adic shtukas and apply them to the theory of local and global Shimura varieties, and on their interrelation. We construct canonical integral models for (local, and global) Shimura varieties of Hodge type with parahoric level structure.
arXiv (Cornell University), Jul 10, 2020
We prove p-adic uniformization for Shimura curves attached to the group of unitary similitudes of... more 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 Vv is anisotropic, let ν be an extension of v to the reflex field E. We define an integral model of the corresponding Shimura curve over Spec O E,(ν) 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 uniformization of the formal completion of this model along its special fiber is given in terms of the formal Drinfeld upper half plane Ω Fv for Fv. The proof relies on the construction of the contracting functor which relates a relative Rapoport-Zink space for strict formal O Fv -modules with a Rapoport-Zink space of p-divisible groups which arise from the moduli problem, where the O Fv -action is usually not strict when Fv = Qp. Our main tool is the theory of displays, in particular the Ahsendorf functor.
Cambridge University Press eBooks, Jul 20, 2011
Computer Science and Information Systems (FedCSIS), 2019 Federated Conference on, Sep 26, 2019
The physical design placement problem is one of the hardest and most important problems in micro ... more The physical design placement problem is one of the hardest and most important problems in micro chips production. The placement defines how to place the electrical components on the chip. We consider the problem as a combinatorial optimization problem, whose instance is defined by a set of 2dimensional rectangles, with various sizes and wire connectivity requirements. We focus on minimizing the placement area and the total wire-length. We propose a local-search method for coping with the problem, based on natural dynamics common in game theory. Specifically, we suggest to perform variants of Best-Response Dynamics (BRD). In our method, we assume that every component is controlled by a selfish agent, who aim at minimizing his individual cost, which depends on his own location and the wire-length of his connections. We suggest several BRD methods, based on selfish migrations of a single or a cooperative of components. We performed a comprehensive experimental study on various test-benches, and compared our results with commonly known algorithms, in particular, with simulated annealing. The results show that selfish local-search, especially when applied with cooperatives of components, may be beneficial for the placement problem.
Advanced studies in pure mathematics, Mar 7, 2019
Pacific Journal of Mathematics, Nov 30, 2012
2002) in moduli theoretic terms, as a construction of certain families of polarized abelian varie... more 2002) in moduli theoretic terms, as a construction of certain families of polarized abelian varieties of Picard type. We show that these period maps are morphisms defined over their natural field of definition.
The new edition of this celebrated and long-unavailable book preserves the original book's co... more The new edition of this celebrated and long-unavailable book preserves the original book's content and structure and its unrivalled presentation of a universal method for the resolution of a class of singularities in algebraic geometry. At the same time, the book has been completely re-typeset, errors have been eliminated, proofs have been streamlined, the notation has been made consistent and uniform, an index has been added, and a guide to recent literature has been added. The book brings together ideas from algebraic geometry, differential geometry, representation theory and number theory, and will continue to prove of value for researchers and graduate students in these areas.
Advances in Mathematics, Sep 1, 2008
Cambridge University Press eBooks, Jul 20, 2011
BRILL eBooks, Feb 7, 2023
BRILL eBooks, Feb 7, 2023
arXiv (Cornell University), Sep 27, 2022
We consider parahoric Bruhat-Tits group schemes over a smooth projective curve and torsors under ... more We consider parahoric Bruhat-Tits group schemes over a smooth projective curve and torsors under them. If the characteristic of the ground field is either zero or positive but not too small and the generic fiber is absolutely simple and simply-connected, we show that such group schemes can be written as invariants of reductive group schemes over a tame cover of the curve. We relate the torsors under the Bruhat-Tits group scheme and torsors under the reductive group scheme over the cover which are equivariant for the action of the covering group. For this, we develop a theory of local types for such equivariant torsors. We also relate the moduli stacks of torsors under the Bruhat-Tits group scheme and equivariant torsors under the reductive group scheme over the cover.
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Papers by Michael Rapoport