Person: Kisin, Mark
Loading...
Email Address
AA Acceptance Date
Birth Date
Research Projects
Organizational Units
Job Title
Last Name
Kisin
First Name
Mark
Name
Kisin, Mark
12 results
Search Results
Now showing 1 - 10 of 12
Publication Mod $p$ points on Shimura varieties of abelian type(American Mathematical Society (AMS), 2017-01-11) Kisin, MarkWe show that the mod p points on a Shimura variety of abelian type with hyperspecial level have the form predicted by the conjectures of Kottwitz and Langlands-Rapoport. Along the way we show that the isogeny class of a mod p point contains the reduction of a special point.Publication D-modules and finite monodromy(Springer Nature, 2016) Esnault, Hélène; Kisin, MarkWe investigate an analogue of the Grothendieck p-curvature conjecture, where the vanishing of the p-curvature is replaced by the stronger condition, that the module with connection mod p underlies a XDX -module structure. We show that this weaker conjecture holds in various situations, for example if the underlying vector bundle is finite in the sense of Nori, or if the connection underlies a ℤZ -variation of Hodge structure. We also show isotriviality assuming a coprimality condition on certain mod p Tannakian fundamental groups, which in particular resolves in the projective case a conjecture of Matzat–van der Put.Publication Moduli of finite flat group schemes, and modularity(Annals of Mathematics, Princeton U, 2009) Kisin, MarkWe prove that, under some mild conditions, a two dimensional p-adic Galois representation which is residually modular and potentially Barsotti-Tate at p is modular. This provides a more conceptual way of establishing the Shimura-Taniyama-Weil conjecture, especially for elliptic curves which acquire good reduction over a wildly ramified extension of ℚ3. The main ingredient is a new technique for analyzing flat deformation rings. It involves resolving them by spaces which parametrize finite flat group scheme models of Galois representations.Publication Galois representations and Lubin-Tate groups(Universität Bielefeld, Fakultät für Mathematik, 2009) Kisin, Mark; Ren, WeiUsing Lubin-Tate groups, we develop a variant of Fontaine's theory of $(\varphi, \Gamma)$-modules, and we use it to give a description of the Galois stable lattices inside certain crystalline representations.Publication Modularity of 2-Adic Barsotti-Tate Representations(Springer Verlag, 2009) Kisin, MarkWe prove a modularity lifting theorem for two dimensional, 2-adic, potentially Barsotti-Tate representations. This proves hypothesis (H) of Khare-Wintenberger, and completes the proof of Serre’s conjecture. The main new ingredient is a classification of connected finite flat group schemes over rings of integers of finite extensions of ℚ2.Publication Integral canonical models of Shimura varieties(Cellule MathDoc/CEDRAM, 2009) Kisin, MarkThe aim of these notes is to provide an introduction to the subject of integral canonical models of Shimura varieties, and then to sketch a proof of the existence of such models for Shimura varieties of Hodge and, more generally, abelian type. For full details the reader is refered to [Ki 3].Publication The Fontaine-Mazur conjecture for {GL}_2(American Mathematical Society (AMS), 2009) Kisin, MarkWe prove new cases of the Fontaine-Mazur conjecture, that a 2 -dimensional p -adic representation rho of G_{{Q}, S} which is potentially semi-stable at p with distinct Hodge-Tate weights arises from a twist of a modular eigenform of weight k>= 2 . Our approach is via the Breuil-Mezard conjecture, which we prove (many cases of) by combining a global argument with recent results of Colmez and Berger-Breuil on the p -adic local Langlands correspondence.Publication Rigidity, Locally Symmetric Varieties, and the Grothendieck-Katz Conjecture(Oxford University Press (OUP), 2009) Farb, B.; Kisin, MarkUsing Margulis's results on lattices in semisimple Lie groups, we prove the Grothendieck–Katz p-curvature conjecture for many locally symmetric varieties, including Hilbert–Blumenthal modular varieties and the moduli space of abelian varieties Graphic when g > 1.Publication Connected components of affine Deligne–Lusztig varieties in mixed characteristic(Oxford University Press (OUP), 2015) Chen, Miaofen; Kisin, Mark; Viehmann, EvaWe determine the set of connected components of minuscule affine Deligne–Lusztig varieties for hyperspecial maximal compact subgroups of unramified connected reductive groups. Partial results are also obtained for non-minuscule closed affine Deligne–Lusztig varieties. We consider both the function field case and its analog in mixed characteristic. In particular, we determine the set of connected components of unramified Rapoport–Zink spaces.Publication The Structure of Potentially Semi-Stable Deformation Rings(Zürich : European Mathematical Society, 2011-04-04) Kisin, MarkInside the universal deformation space of a local Galois representation one has the set of deformations which are potentially semi-stable of given p-adic Hodge and Galois type. It turns out these points cut out a closed subspace of the deformation space. A deep conjecture due to Breuil-M´ezard predicts that part of the structure of this space can be described in terms of the local Langlands correspondence. For 2-dimensional representations the conjecture can be made precise. We explain some of the progress in this case, which reveals that the conjecture is intimately connected to the p-adic local Langlands correspondence, as well as to the Fontaine-Mazur conjecture.