Dualizing complexes on the moduli of parabolic bundles
-
Linus Hamann
Abstract
For a non-archimedean local field đč and a connected reductive group đș over đč equipped with a parabolic subgroup đ, we show that the dualizing complex on
1 Introduction
Let
It is a natural question to wonder what the
Proposition 1.1 (Proposition 3.18)
For
This is claimed in a footnote in [8, Section V.5] indicating an argument, which is close to the one in Remark 3.19.
The idea for proving this is to note that, on the semistable locus
and the dualizing complex on such classifying stacks
Proposition 1.3 (Proposition 3.16)
For
where
A proof of this result and the previous result through a different but less explicit method has been carried out for
The restriction to the subcategory
The proof we give of Proposition 1.3 is in fact related to the computation of the dualizing object on certain â-cohomologically smooth charts
where
One can wonder whether this relationship between the dualizing complex on
We write
Theorem 1.6 (Theorem 3.20)
For
of objects in
In other words, the dualizing complex is the geometrization of the modulus character in the representation theory of đ-adic reductive groups.
Using Proposition 1.3, Theorem 1.6 is reduced to determining the dualizing object on
under
When đș is quasi-split and đ is a Borel, this recovers [12, Theorem 1.2], where this result was used as a fundamental computational input in the theory of geometric Eisenstein series over the FarguesâFontaine curve in the principal case.
The result we prove here should serve a similar role in building an analogous theory for general parabolics.
We partially carry this out in Section 4.3 by defining the Eisenstein functor with the appropriate modulus character twists in light of Theorem 1.6 (Definition 4.17), proving some of its basic properties (Lemmas 4.19 and 4.20), and computing its values after restricting to the semistable locus of
The importance of this result is also emphasized by its relation to various other computations occurring in the FarguesâScholze geometric Langlands program and the theory of local Shimura varieties.
Namely, we already mentioned above that, in the quasi-split case, the â-cohomologically smooth charts for
respectively, where
Corollary 1.7 (Proposition 3.15)
For
where
For the rest of our results, we work in the larger level of generality where Î is a
Corollary 1.8 (Proposition 4.4)
The compactly supported cohomology (resp. dualizing complex) of
The character appearing in this compactly supported cohomology appears in several places in the literature; namely in [19, 13] in the guise of Manotovanâs product formula, and similarly in [10, 12] in the guise of comparing Hecke correspondences on
Corollary 1.9 (Corollary 4.2 and Corollary 4.6)
The following is true.
The character đ appearing in [12, Lemma 11.1] is equal to
As an application, we show the HarrisâViehmann conjecture [24, Conjecture 8.4] in the HodgeâNewton reducible case.
Assume that Î admits a fixed choice of square root of đ.
Let
Proposition 1.10 (Proposition 4.11)
Let đ be a character of
in the derived category of
Using Corollary 1.9 and Proposition 1.10 as well as results in [10], we show the following form of the HarrisâViehmann conjecture (cf. [24, Conjecture 8.4], where we note that there are missing Tate twists here as first observed by Bertoloni Meli; see [2, Conjecture 3.2.1] for the fixed conjecture and the discussion around [2, Example 3.2.5] for an explanation).[1]
Theorem 1.11 (Theorem 4.13)
Assume that đč is đ-adic, đș is quasi-split,
in the derived category of
In Section 2, we recall the notion of the Picard đŁ-groupoid attached to a vector bundle â° on the FarguesâFontaine đč and prove several results on its dualizing complex. In Section 3, we describe the geometric structure of the moduli stack of parabolic structures and prove Proposition 1.1, Proposition 1.3, Theorem 1.6, and Corollary 1.7. In Section 4, we deduce the applications to the HarrisâViehmann conjecture and geometric Eisenstein series.
Notation
For a field đč, let
For a non-archimedean local field đč, let
Let đč be a non-archimedean local field with residue characteristic đ. Let
Let
We let
Let Î be a torsion ring of order prime to đ unless otherwise stated.
For a fine map (see [9, Definition 1.3])
constructed in [9], which, together with
We let
For
Let đș be a connected reductive algebraic group over đč. Let đ be a parabolic subgroup of đș, and đż the Levi factor of đ.
For a connected reductive group
We let
We let
for the normalized parabolic induction functor.
For
Let
For
For
We write
We write
for the natural pairing induced by the pairing between cocharacters and characters.
We recall that the Kottwitz set
the slope homomorphism
where
the Kottwitz invariant
where
If đș is quasi-split with a fixed Borel pair
Throughout, we will use the terminology that đ is an iterated fibration of đŁ-stacks in the đŁ-stacks
such that
2 Dualizing complexes on Picard đŁ-groupoids
For
Let
â° is a vector bundle on đ that is of positive (resp. negative) slopes.
is fully faithful, and the adjunction morphism
Proof
The first claim follows from [8, Proposition II.3.5].
The second claim is [8, Proposition V.2.1].
The third claim is reduced to the case where the torsor is split and
Let
Proof
This is proved in the same way as Lemma 2.1. â
The calculation of the dualizing complex on
Let
which we refer to as the Picard đŁ-groupoid of
We will only consider this in the case that
We consider the dualizing complex
where the last map is given by taking the product of the elements and
We have an isomorphism
We first need some preliminaries.
For
and let
Let
such that the action of
Proof
We follow the proof of [8, Proposition II.2.5â(iv)].
In the đ-adic case, the question is reduced to the case where
Assume that đč is of equal characteristic.
By the proof of [27, Proposition 3.1.3â(iii)],
Under identification (2.2), the natural action of
We recall that the map
Let
be the structure maps, where
If đ is positive, then
as sheaves with
If
as sheaves with
If đ is negative, then
as sheaves with
Proof
The first claims in 1 and 3, and the first two claims in 2 are proven in [8, Proposition II.2.5].
The later two claims in 2 follow from [16, Example 4.2.4].
For the remaining claims, it suffices to construct isomorphisms as sheaves that are compatible with the actions of
We prove the second claim in 1
in a way similar to [10, Lemma 4.17].
The claims on
using Lemma 2.5.
By [26, Theorem 24.1], we know that
that is compatible with the action of
We can see that the action of
By [26, Lemma 15.6], it suffices to show the same claim for
Since any automorphism of
Assume that
induced by
as in [28, Example 15.2.9.4], where
Let
by [18, Lemma 1.3].
The action of
We show that
By Lemma 2.1, we may check this after the pullback under
where we use (2.7) at the second isomorphism and (2.5) at the third isomorphism.
The claim on
We prove the second claim in 3.
Assume that
as in [28, p.â138]. Hence we have
by (2.5) and (2.6). Therefore, the claim follows from claims 1 and 2.
Assume that
as in [28, p.â139].
Hence the claim is reduced to the case of
With this in hand, we can prove Proposition 2.4.
Proof of Proposition 2.4
By [8, Proposition II.2.5], we have identifications
Therefore, we can factorize the structure map
Since all these maps are â-cohomologically smooth by the slope decomposition of â° and [8, Proposition II.2.5], we can use the formula
for a pair of â-cohomologically smooth maps đ and đ, as follows from the definition [8, Definition IV.1.15]. This reduces us to showing the claim in three cases.
The slopes of â° are strictly positive.
The slopes of â° are all 0.
The slopes of â° are strictly negative.
3 Structure of the moduli space of đ-structures
3.1 Modulus character
We make use of the following notation.
Let đ be a parabolic with Levi factor đż. We set
which factors through
Recall that
Given
where
For a locally free sheaf â° over a scheme đ, we define the group scheme
Let đ be a parabolic subgroup of a reductive group scheme đș over a scheme đ with a Levi subgroup đż.
Then there is a filtration
the action of đ on
there is an isomorphism
there is an isomorphism
of schemes with an action of đż, and
if đ is connected semi-local, there is unique
where
Proof
We do the argument as in [7, ExposĂ© XXVI, 2.1.1] for a pinning of đș adapted to đ such that the maximal torus is contained in đż. Then there is an isomorphism
compatible with the actions of đż for a locally free sheaf
In the argument of [7, ExposĂ© XXVI, 2.1.2], different pinnings are conjugate by đż since we use only the pinnings such that the maximal tori are contained in đż.
Hence
Assume that đ is connected semi-local.
We note that đż is the centralizer of
The modulus character
Proof
Since the both characters are trivial on
Since the modulus character of đż is trivial, we have
We take an
3.2 Connected components of
Bun
P
We have a natural diagram
given by the corresponding maps of reductive groups.
By [8, Corollary IV.1.23], we have decompositions
into connected components via the Kottwitz invariant.
We let
Consider a parabolic
is Cartesian.
Proof
This follows from the fact that the fibers of
For
Let
We have
We have an isomorphism
of đŁ-stacks. This gives the commutative diagram
Proof
Since
We now have our key structural result on the components
The map
Proof
The first part follows from [11, Proposition 3.16] and its proof. The second part follows from combining [11, Lemma 3.8, Proposition 4.7] and [8, Theorem IV.1.19] when đș is quasi-split. The general case is reduced to the quasi-split case using Lemma 3.3 and Lemma 3.4. â
In particular, we deduce the following from this.
We have a decomposition into connected components
Proof
This follows by combining with the previous proposition together with the fact that â-cohomologically smooth maps are universally open [26, Proposition 23.1].[3] â
We consider the semistable locus
where
3.3 Description of the preimage HN-strata
Let
We write
which we view as group schemes over
We now come to the key description of the space
Let
Proof
The fiber
Let
Proof
This follows from the proof of Lemma 3.7, since
We now prove the following proposition using Lemma 3.7.
We have an isomorphism
Proof
The adjoint action of đ on
which factors through
where
We consider the natural map
given by the left conjugation action, and the element
composed with the right action of
It is convenient to slightly reformulate this. We define the following.
We let
whose value on each connected component is given by
We let
We have an isomorphism
of objects in
We have the following lemma, where the maps
For
are Cartesian.
Proof
Let
By the proof of [8, Proposition V.3.5], the fiber of
Let
Proof
The second bijection can be checked easily (cf. [18, Lemma 2.2]).
We define đ» as in the proof of Lemma 3.12 using đș and đ.
We define
We put
We take a quasi-split inner form
where
We have isomorphisms
3.4 Local systems on
Bun
G
and reduction to the semistable locus
For an Artin đŁ-stack đ, we write
and that
We start by deducing some interesting consequences of Corollary 3.11.
There is a well-defined equivalence of categories
where
Proof
The claim that this is well-defined will follow from the proof.
In particular, we will show that
By induction and excision on
To do this, we recall the â-cohomologically smooth chart
Applying smooth base change with respect to
where we note that
For đ non-basic, we have that
unless
Proof
By Proposition 3.15, it follows that we have an isomorphism
using that the dualizing complex on
Since
We can apply excision with respect to the pair
This completes the proof of Proposition 3.16. â
From this, we can deduce the following.
We have an isomorphism
Proof
We know that
Note that it is easy to describe the dualizing complex on
For completeness, we also provide a sketch for the alternative proof of Proposition 3.16 and Proposition 3.18, explained by Wild for
One then invokes the very soft claim that
We now have the following.
We have an isomorphism
of objects in
Proof
By Proposition 3.18 and its proof, we have isomorphisms
We apply
It suffices to restrict to a connected component indexed by
where we have used proper base change for the last identification, and
Using Corollary 3.11 and the fact that the dualizing complex on
but using Proposition 3.16 [4], this is easily identified with
as desired, where
is the natural map induced by
4 Applications
4.1 Connection with other results
We now discuss the relation of this paper with various other results in the literature.
For
considered in [8, Proposition V.2.2] defines via pullback an equivalence
where
For all
Proof
This follows from Proposition 3.15
together with the unimodularity of
The action of
The sheaf
Now we will briefly change the assumption on Î, and let Î be a
Let
â° is a vector bundle on đ that is of positive (resp. negative) slopes.
Proof
This is proven in the same way as [8, Proposition V.2.1], using [8, Proposition VII.3.1] and [18, Lemma 1.3]. â
For an Artin v-stack đ, let
For an â-cohomologically smooth morphism
which is an invertible object by the smoothness, and
for
Consider
Proof
Let
be the natural morphisms. By Proposition 4.1, we have an isomorphism
Hence we have
Let
for
Consider
Proof
This is proved in the same way as Proposition 4.4 using the natural morphisms
Proposition 4.4 and Proposition 4.5 are compatible with [12, Proposition 11.3], as well as compatible with [10, Lemma 4.18].
The following is true.
Proposition 4.1 also appeared in [15, Proposition 1.14], where an argument was mentioned that used properties of the FarguesâScholze local Langlands correspondence. We now fill in the details and show that those properties are compatible with what we have proved.
Another proof of Proposition 4.1
Let đž be a finite extension of
As in the proof of [8, Proposition V.2.2], we recall that the map
We will now compute
Let
For Î a
where
where
Proof
The Schur irreducibility follows easily from the isomorphisms
coming from adjunction, and the irreducibility of đ (recalling that
when passing to the endomorphism algebras.
Now, since the action of the excursion algebra factors through a map to endomorphisms of the identity functor on
We note that we have
Now, it follows by the cohomological smoothness of
where
of conjugacy classes of parameters. In particular, substituting in the relationship
we obtain that
and the left-hand side identifies with
From here, we conclude that
4.2 HarrisâViehmann conjecture
We show the HarrisâViehmann conjecture in the HodgeâNewton reducible case using our results and results in [10].
In this subsection, we assume that Î is a
For a cocharacter đ of đș, let
Let
For a finite separable extension đž of đč in
Let
We put
Let đ be a torus over đč, đ a cocharacter of đ, đž a finite separable extension of đč, and đ a character of
Proof
Under the natural morphism
We consider the diagram
where the right two squares are Cartesian.
By
Hence we have
This implies
Therefore, we obtain
Let
Proof
Since we have a natural morphism
Let
Proof
We note that
where we use Lemma 4.9 at the second isomorphism. â
Let
Let đ be a smooth representation of
in the derived category of
Proof
We put
The lower right square in (4.2) is Cartesian because the natural map
is an isomorphism.
Hence the upper right square is also Cartesian since
Let
using [8, Proposition VII.3.1]. We have the diagram
where
be the natural projections. Then we have
since
Therefore, we have
using [8, Proposition VII.3.1]. By Lemma 4.8, we have
Let
We take a compact open subgroup đŸ of
and consider the diagram
given by the natural morphisms. Both squares in (4.8) are Cartesian. In the Cartesian squares
the morphism
where the second morphism is the natural open and closed immersion.
Hence
Let
where we use [18, Lemma 1.6] at the second isomorphism, Lemma 4.9 at the third isomorphism, the â-cohomologically smoothness of
be the base change of đ and
the natural morphism. Then we have
where we use Proposition 4.4 at the first and last isomorphism, and (4.7), (4.3), (4.5), Lemma 4.10, and (4.6) at the second, third, fourth, fifth, and sixth isomorphisms respectively. â
Assume that đč is đ-adic, đș is quasi-split,
We have an isomorphism
which is compatible with the actions of
Proof
This follows from [10, Corollary 4.14, Proposition 4.24] and Corollary 4.6, as in the proof of [10, Theorem 4.26]. In particular, we note that we have isomorphisms
since the restriction of the sheaf
Similarly, we have an identification
Then, using the identity
Assume that đ is minuscule.
Let đ be a smooth representation of
as objects in the derived category of
in the Grothendieck group of finite length admissible representations.
Proof
We have
where we use Proposition 4.11 at the last isomorphism.
We have
is equal to
In the above claim, we used the notation
(see [8, Remark V.7.3] and the proof of [8, Proposition V.7.2] for details).
Here
given by internal Hom in the category of smooth
given by tensoring the global section functor
which shows that the above operation also respects the continuous
4.3 Geometric Eisenstein series
We again consider the diagram
and attach to it the unnormalized Eisenstein functor
where
We assume now that Î admits a square root of đ, which we now fix. This allows us to make the following definition.
We define the sheaf
We note the following corollary of Theorem 3.20.
The sheaf
In particular, this motivates the morally correct definition of the Eisenstein functor.
We define the normalized Eisenstein functor to be
We note that we have an isomorphism
We write
for the direct sum decompositions induced by Corollary 3.6. These functors are compatible with inclusions of parabolics in the obvious sense.
Let
of objects in
More precisely, given
unless
The analogous claim also holds for the unnormalized functors.
Proof
We treat the case of the normalized functors, with the case of the unnormalized functors being strictly easier.
It is easy to see that
unless
The key point can be found in the proof of [6, Lemma 6.4].
We recall this now.
For
of schemes over
in the Ă©tale topology on
If we are given a
It follows that, for a fixed đș-bundle
a
a
where top left square is Cartesian by the previous equivalence. Restricting to connected components, this gives
It follows that, for all
where the last isomorphism follows from proper base change and
is the natural map. We now note that we have an equality
Indeed, this follows from breaking up
To deal with the modulus character twists, consider the following commutative diagram:
This tells us that
as desired. Here we have used a projection formula in the fifth isomorphism. â
We assume from now on that the group đș is quasi-split with a choice of Borel
Let đ„ denote the set of vertices in the (relative) Dynkin diagram. For each
For a Levi subgroup
Let
Let
of objects in
Proof
We note that
Therefore, we reduce to showing that
We write
Using the isomorphisms
as desired. â
In general the values of these geometric Eisenstein functors are very complicated and difficult to compute (see for example [15, Example 3.3.2]), but there are certain simple contributions coming from the split reductions.
This is the locus of
We define
We will now study the set
Let
be the subset of elements which pair non-negatively with the image of
We say an element
For
where
Assume that
The inclusion
The parabolic subgroup đ of đș transfer to a parabolic subgroup of
Proof
Claim 1 follows from [21, Sections 1.4 and 3.6]. Claim 2 follows from 1 and [6, p.â259]. â
Let đș be a connected reductive group over a field đč.
We fix a maximal split torus đ and a minimal parabolic subgroup
Then we have a natural bijection
Proof
The map is given by taking a representative of an element of
is a bijection. Therefore, the first surjective map is also bijective. â
Let đș be a connected reductive group over a field đč.
Let đ be a parabolic subgroup of đș with a Levi subgroup đż.
Then the inclusions
Proof
Injectivity of the second map follows from [5, 4.13 ThĂ©orĂšme] (cf. [29, p.â136, Exercice 1]).
The map
We now work again with our fixed quasi-split đș.
We recall again that đ admits a reduction to a basic element
For every element
that sends đ to the unique
Proof
We first start by constructing the map and showing it has the desired properties. By [1, Lemma 2.11], we have a bijection
where
Now let
We now show the injectivity of the map.
Suppose we have elements đ and
Since
Hence we can reduce the problem to showing that the map
and this is injective by Lemma 4.24.
We saw in the previous paragraph that the map
We will also need the following result.
An element
Proof
This follows from combining Lemma 4.25 with Lemma 4.22 applied to the basic reduction
We now turn to computing our Eisenstein functors. Let us start with the simplest family of cases.
Suppose that
where
Proof
The second isomorphism follows from the first using the relationship
as in Remark 4.18.
To prove the first isomorphism, by an application of proper base change, we are interested in computing
Note that, since đ maps to đ, we have an equality
Therefore, the previous diagram just becomes
where đ and đ are induced by the natural maps of groups.
If đ is a smooth representation of
as desired. â
With this warmup out of the way, we come to the following more general claim.
For all
for the image under the equivalence of categories induced by
in
of objects in
Proof
The second claim follows immediately from the first by noting that, in the case that đ is HN-dominant with respect to the Borel đ” for which đ is standard, the space
For the first claim, we set
We let
By definition of
By Lemma 4.27, this is isomorphic to
Therefore, we have reduced to showing the following.
For all
of objects in
Proof
Let
We write
with maps as in the diagram
We look at
for all roots đŒ occurring in
where
The closed substack
It follows that
is a classifying stack attached to a group diamond
We claim we can now reduce to the following.
The compactly supported cohomology of
as a
We note that we have a chain of equalities
by definition of
For the second isomorphism, we have used that đź can be identified with the composition
This completes the proof of Theorem 4.28. â
Proof of Lemma 4.30
Let
as in the equality
The claim now follows from the previous description of
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: 390685813
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: 22H00093
Funding statement: This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germanyâs Excellence Strategy â EXC-2047/1 â 390685813, and it was also supported by JSPS KAKENHI Grant Number 22H00093.
Acknowledgements
We would like to thank Laurent Fargues, David Hansen, Teruhisa Koshikawa, Peter Scholze, and Ruth Wild for helpful discussions. Many of the key ideas surrounding this project were found during the Hausdorff Trimester program âArithmetic of the Langlands Programâ at the Hausdorff Research Institute for Mathematics, and we would like to thank the organizers for the fantastic learning environment. We thank Konrad Zou for helpful comments. We also thank the referee for helpful suggestions to improve the manuscript. This work was carried out while L. Hamann was a NSF postdoctoral scholar, and he thanks them for their support.
References
[1]
I.âN. BernĆĄteÄn and A.âV. ZelevinskiÄ,
Representations of the group
[2] A. Bertoloni Meli, The cohomology of unramified RapoportâZink spaces of EL-type and Harrisâs conjecture, J. Inst. Math. Jussieu 21 (2022), no. 4, 1163â1218. 10.1017/S1474748020000535Search in Google Scholar
[3] A. Bertoloni Meli, L. Hamann and K.âH. Nguyen, Compatibility of the FarguesâScholze correspondence for unitary groups, Math. Ann. 390 (2024), no. 3, 4729â4787. 10.1007/s00208-024-02877-xSearch in Google Scholar
[4]
A. Bertoloni Meli and M. Oi,
The
[5] A. Borel and J. Tits, Groupes rĂ©ductifs, Publ. Math. Inst. Hautes Ătudes Sci. 27 (1965), 55â150. 10.1007/BF02684375Search in Google Scholar
[6] M. Chen, L. Fargues and X. Shen, On the structure of some đ-adic period domains, Camb. J. Math. 9 (2021), no. 1, 213â267. 10.4310/CJM.2021.v9.n1.a4Search in Google Scholar
[7] M. Demazure and A. Grotjendieck, Schémas en groupes. III: Structure des schémas en groupes réductifs, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3), Lecture Notes in Math. 153, Springer, Berlin 1970. 10.1007/BFb0059027Search in Google Scholar
[8] L. Fargues and P. Scholze, Geometrization of the local Langlands correspondence, preprint (2021), https://arxiv.org/abs/2102.13459. Search in Google Scholar
[9] D. Gulotta, D. Hansen and J. Weinstein, An enhanced six-functor formalism for diamonds and đŁ-stacks, preprint (2022), https://arxiv.org/abs/2201.12467. Search in Google Scholar
[10] I. Gaisin and N. Imai, Non-semi-stable loci in Hecke stacks and Farguesâ conjecture, preprint (2016), https://arxiv.org/abs/1608.07446. Search in Google Scholar
[11] L. Hamann, A Jacobian criterion for Artin đŁ-stacks, preprint (2021), https://arxiv.org/abs/2209.07495. Search in Google Scholar
[12] L. Hamann, Geometric Eisenstein series, intertwining operators, and Shinâs averaging formula, preprint (2022), https://arxiv.org/abs/2209.08175. Search in Google Scholar
[13] L. Hamann and S. Y. Lee, Torsion vanishing for some Shimura varieties, preprint (2023), https://arxiv.org/abs/2309.08705. Search in Google Scholar
[14] D. Hansen, Moduli of local shtukas and Harrisâs conjecture, Tunis. J. Math. 3 (2021), no. 4, 749â799. 10.2140/tunis.2021.3.749Search in Google Scholar
[15] D. Hansen, Beijing notes on the categorical local Langlands conjecture, preprint (2023), https://arxiv.org/abs/2310.04533. Search in Google Scholar
[16] D. Hansen, T. Kaletha and J. Weinstein, On the Kottwitz conjecture for local shtuka spaces, Forum Math. Pi 10 (2022), Paper No. e13. 10.1017/fmp.2022.7Search in Google Scholar
[17] R. Huber, Ătale cohomology of rigid analytic varieties and adic spaces, Aspects of Math. E30, Friedrich Vieweg & Sohn, Braunschweig 1996. 10.1007/978-3-663-09991-8Search in Google Scholar
[18] N. Imai, Convolution morphisms and Kottwitz conjecture, preprint (2019), https://arxiv.org/abs/1909.02328. Search in Google Scholar
[19] T. Koshikawa, On the generic part of the cohomology of local and global Shimura varieties, preprint (2021), https://arxiv.org/abs/2106.10602. Search in Google Scholar
[20] R.âE. Kottwitz, Shimura varieties and twisted orbital integrals, Math. Ann. 269 (1984), no. 3, 287â300. 10.1007/BF01450697Search in Google Scholar
[21] R.âE. Kottwitz, Isocrystals with additional structure. II, Compos. Math. 109 (1997), no. 3, 255â339. 10.1023/A:1000102604688Search in Google Scholar
[22]
K.âH. Nguyen and E. Viehmann,
A HarderâNarasimhan stratification of the
[23] M. Rapoport and M. Richartz, On the classification and specialization of đč-isocrystals with additional structure, Compos. Math. 103 (1996), no. 2, 153â181. Search in Google Scholar
[24] M. Rapoport and E. Viehmann, Towards a theory of local Shimura varieties, MĂŒnster J. Math. 7 (2014), no. 1, 273â326. Search in Google Scholar
[25] S. Schieder, The HarderâNarasimhan stratification of the moduli stack of đș-bundles via Drinfeldâs compactifications, Selecta Math. (N.âS.) 21 (2015), no. 3, 763â831. 10.1007/s00029-014-0161-ySearch in Google Scholar
[26] P. Scholze, Etale cohomology of diamonds, preprint (2017), https://arxiv.org/abs/1709.07343. 10.2139/ssrn.3542189Search in Google Scholar
[27] P. Scholze and J. Weinstein, Moduli of đ-divisible groups, Camb. J. Math. 1 (2013), no. 2, 145â237. 10.4310/CJM.2013.v1.n2.a1Search in Google Scholar
[28] P. Scholze and J. Weinstein, Berkeley lectures on đ-adic geometry, Ann. of Math. Stud. 207, Princeton University, Princeton 2020. 10.23943/princeton/9780691202082.001.0001Search in Google Scholar
[29] J.-P. Serre, Cohomologie galoisienne, 2th ed., Lecture Notes in Math. 5, Springer, Berlin 1994. 10.1007/BFb0108758Search in Google Scholar
[30]
M.-F. Vignéras,
ReprĂ©sentations đ-modulaires dâun groupe rĂ©ductif đ-adique avec
[31] R. Wild, Ătale cohomology of the moduli stack of rank two vectorbundles on the FarguesâFontaine curve, Masterâs Thesis, University of Bonn, 2024. Search in Google Scholar
© 2025 the author(s), published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
Articles in the same Issue
- Frontmatter
- Irreducible symplectic varieties with a large second Betti number
- Splitting of almost ordinary abelian surfaces in families and the đ-integrality conjectures
- Applications of perverse sheaves in commutative algebra
- Dualizing complexes on the moduli of parabolic bundles
- Dynamics of convex mean curvature flow
- ColdingâMinicozzi entropies in CartanâHadamard manifolds
- Asymptotic directions in the moduli space of curves
- Existence of arithmetic degrees for generic orbits and dynamical LangâSiegel problem
Articles in the same Issue
- Frontmatter
- Irreducible symplectic varieties with a large second Betti number
- Splitting of almost ordinary abelian surfaces in families and the đ-integrality conjectures
- Applications of perverse sheaves in commutative algebra
- Dualizing complexes on the moduli of parabolic bundles
- Dynamics of convex mean curvature flow
- ColdingâMinicozzi entropies in CartanâHadamard manifolds
- Asymptotic directions in the moduli space of curves
- Existence of arithmetic degrees for generic orbits and dynamical LangâSiegel problem
