3,643 research outputs found
Harris-Viehmann conjecture for Hodge-Newton reducible Rapoport-Zink spaces
Rapoport-Zink spaces, or more generally local Shimura varieties, are expected
to provide geometric realization of the local Langlands correspondence via
their -adic cohomology. Along this line is a conjecture by Harris and
Viehmann, which roughly says that when the underlying local Shimura datum is
not basic, the -adic cohomology of the local Shimura variety is
parabolically induced.
We verify this conjecture for Rapoport-Zink spaces which are Hodge type and
Hodge-Newton reducible. The main strategy is to embed such a Rapoport-Zink
space into an appropriate space of EL type, for which the conjecture is already
known to hold by the work of Mantovan.Comment: 22 pages, to appear in the Journal of the London Mathematical Societ
Classification of quotient bundles on the Fargues-Fontaine curve
We completely classify all quotient bundles of a given vector bundle on the
Fargues-Fontaine curve. As consequences, we have two additional classification
results: a complete classification of all vector bundles that are generated by
a fixed number of global sections and a nearly complete classification of
subbundles of a given vector bundle. For the proof, we combine the dimension
counting argument for moduli of bundle maps developed in [BFH+17] with a series
of reduction arguments based on some reinterpretation of the classifying
conditions.Comment: 40 pages, 15 figure
On the Hodge-Newton filtration for p-divisible groups of Hodge type
A p-divisible group, or more generally an F-crystal, is said to be
Hodge-Newton reducible if its Hodge polygon passes through a break point of its
Newton polygon. Katz proved that Hodge-Newton reducible F-crystals admit a
canonical filtration called the Hodge-Newton filtration. The notion of
Hodge-Newton reducibility plays an important role in the deformation theory of
p-divisible groups; the key property is that the Hodge-Newton filtration of a
p-divisible group over a field of characteristic p can be uniquely lifted to a
filtration of its deformation.
We generalize Katz's result to F-crystals that arise from an unramified local
Shimura datum of Hodge type. As an application, we give a generalization of
Serre-Tate deformation theory for local Shimura data of Hodge type. We also
apply our deformation theory to study some congruence relations on Shimura
varieties of Hodge type.Comment: 31 page
On the Hodge-Newton filtration for p-divisible groups of Hodge type
A p-divisible group, or more generally an F-crystal, is said to be Hodge–Newton reducible if its Newton polygon and Hodge polygon have a nontrivial contact point. Katz proved that Hodge–Newton reducible F-crystals admit a canonical filtration called the Hodge–Newton filtration. The notion of Hodge–Newton reducibility plays an important role in the deformation theory of p-divisible groups; the key property is that the Hodge–Newton filtration of a p-divisible group over a field of characteristic p can be uniquely lifted to a filtration of its deformation. We generalize Katz’s result to F-crystals that arise from an unramified local Shimura datum of Hodge type. As an application, we give a generalization of Serre–Tate deformation theory for local Shimura data of Hodge type
On nonemptiness of Newton strata in the -Grassmannian for
We study the Newton stratification in the -Grassmannian for
associated to an arbitrary (possibly nonbasic) element of
. Our main result classifies all nonempty Newton strata in an
arbitrary minuscule Schubert cell. For a large class of elements in
, our classification is given by some explicit conditions in
terms of Newton polygons. For the proof, we proceed by induction on n using a
previous result of the author that classifies all extensions of two given
vector bundles on the Fargues-Fontaine curve.Comment: 20 pages, 8 figure
Classification of subbundles on the Fargues-Fontaine curve
We completely classify all subbundles of a given vector bundle on the
Fargues-Fontaine curve. Our classification is given in terms of a simple and
explicit condition on Harder-Narasimhan polygons. Our proof is inspired by the
proof of the main theorem in [Hon19], but also involves a number of nontrivial
adjustments.Comment: 26 page
On Hodge-Newton Reducible Local Shimura Data of Hodge Type
Rapoport-Zink spaces are formal moduli spaces of p-divisible groups which give rise to local analogues of certain Shimura varieties. In particular, one can construct them from purely group theoretic data called local Shimura data.
The primary purpose of this dissertation is to study Rapoport-Zink spaces whose underlying local Shimura datum is of Hodge type and Hodge-Newton reducible. Our study consists of two main parts: the study of the l-adic cohomology of Rapoport-Zink spaces in relation to the local Langlands correspondence and the study of deformation spaces of p-divisible groups via the local geometry of Rapoport-Zink spaces.
The main result of the first part is a proof of the Harris-Viehmann conjecture in our setting; in particular, we prove that the l-adic cohomology of Rapoport-Zink spaces contains no supercuspidal representations under our assumptions. In the second part, we obtain a generalization of Serre-Tate deformation theory for Shimura varieties of Hodge type.</p
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