Local Langlands Correspondences and Vanishing Cycles on Shimura Varieties

2001 ◽  
pp. 407-427 ◽  
Author(s):  
Michael Harris
Keyword(s):  
Shimura Varieties ◽  
Vanishing Cycles

scholarly journals Conjecture de monodromie-poids pour quelques variétés de Shimura unitaires

2010 ◽  
Vol 146 (2) ◽  
pp. 367-403 ◽  
Author(s):  
Pascal Boyer
Keyword(s):  
Explicit Description ◽  
Shimura Varieties ◽  
Perverse Sheaves ◽  
Automorphic Representations ◽  
Local Systems ◽  
Langlands Correspondence ◽  
Princeton University ◽  
Vanishing Cycles ◽  
Local Langlands Correspondence ◽  
Invent Math

AbstractIn Boyer [Monodromy of perverse sheaves on vanishing cycles on some Shimura varieties, Invent. Math. 177 (2009), 239–280 (in French)], a sheaf version of the monodromy-weight conjecture for some unitary Shimura varieties was proved by giving explicitly the monodromy filtration of the complex of vanishing cycles in terms of local systems introduced in Harris and Taylor [The geometry and cohomology of some simple Shimura varieties (Princeton University Press, Princeton, NJ, 2001)]. The main result of this paper is the cohomological version of the monodromy-weight conjecture for these Shimura varieties, which we prove by means of an explicit description of the groups of cohomology in terms of automorphic representations and the local Langlands correspondence.

scholarly journals Vertical Distribution Relations for Special Cycles on Unitary Shimura Varieties

2018 ◽  
Vol 2020 (13) ◽  
pp. 3902-3926
Author(s):  
Réda Boumasmoud ◽  
Ernest Hunter Brooks ◽  
Dimitar P Jetchev
Keyword(s):  
Vertical Distribution ◽  
Three Dimensional ◽  
Complex Multiplication ◽  
Horizontal Distribution ◽  
Unitary Groups ◽  
Shimura Varieties ◽  
Heegner Points ◽  
Universal Norm

Abstract We consider cycles on three-dimensional Shimura varieties attached to unitary groups, defined over extensions of a complex multiplication (CM) field $E$, which appear in the context of the conjectures of Gan et al. [6]. We establish a vertical distribution relation for these cycles over an anticyclotomic extension of $E$, complementing the horizontal distribution relation of [8], and use this to define a family of norm-compatible cycles over these fields, thus obtaining a universal norm construction similar to the Heegner $\Lambda $-module constructed from Heegner points.

scholarly journals Conjectures on Stably Newton Degenerate Singularities

2021 ◽  
Author(s):  
Jan Stevens
Keyword(s):  
Characteristic Zero ◽  
Arbitrary Number ◽  
Milnor Number ◽  
Plane Curves ◽  
Newton Diagram ◽  
Finite Characteristic ◽  
Vanishing Cycles

AbstractWe discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions.We review the various non-degeneracy concepts in the literature. For finite characteristic, we conjecture that there are no wild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, $$x^p+x^q$$ x p + x q in characteristic p, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate.

scholarly journals On a conjecture of Pappas and Rapoport about the standard local model for GL_d

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Dinakar Muthiah ◽  
Alex Weekes ◽  
Oded Yacobi
Keyword(s):  
Type A ◽  
Positive Answer ◽  
Local Model ◽  
Schubert Varieties ◽  
Shimura Varieties ◽  
Local Models ◽  
Full Generality ◽  
Frobenius Splitting ◽  
Affine Grassmannians ◽  
Main Ideas

AbstractIn their study of local models of Shimura varieties for totally ramified extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of {n\times n} matrices. We give a positive answer to their conjecture in full generality. Our main ideas follow naturally from two of our previous works. The first is our proof of a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman on the equations defining type A affine Grassmannians. The second is the work of the first two authors and Kamnitzer on affine Grassmannian slices and their reduced scheme structure. We also present a version of our argument that is almost completely elementary: the only non-elementary ingredient is the Frobenius splitting of Schubert varieties.

scholarly journals Serre–Tate theory for Shimura varieties of Hodge type

2020 ◽  
Author(s):  
Ananth N. Shankar ◽  
Rong Zhou
Keyword(s):  
Shimura Varieties

scholarly journals RAPOPORT–ZINK UNIFORMIZATION OF HODGE-TYPE SHIMURA VARIETIES

2018 ◽  
Vol 6 ◽  
Author(s):  
WANSU KIM
Keyword(s):  
Shimura Varieties ◽  
Good Reduction ◽  
Canonical Models

We show that the integral canonical models of Hodge-type Shimura varieties at odd good reduction primes admits ‘$p$-adic uniformization’ by Rapoport–Zink spaces of Hodge type constructed in Kim [Forum Math. Sigma6(2018) e8, 110 MR 3812116].

Arithmetic divisors on orthogonal and unitary Shimura varieties

Astérisque ◽  
2020 ◽  
Vol 421 ◽  
Author(s):  
Jan H. Bruinier ◽  
Benjamin Howard ◽  
Stephen S. Kudla ◽  
Michael Rapoport ◽  
Tonghai Yang ◽  
...  
Keyword(s):  
Shimura Varieties
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