Introduction
The Integral Coefficient Geometric Satake Equivalence in Mixed
A Geometric Jacquet-Langlands Transfer
Notations
Summary of the Contents
The Mixed Characteristic Affine Grassmannians
Preliminaries
The Mixed Characteristic Affine Grassmannian
Consider the following closure relations as in [Zhu17, Proposition 2.5] (the equally characteristic analog of this statement is proved in [MV07, Proposition 3.1). Finally, based on the exactness of the global cohomology functor H∗(•), we conclude that H∗(F ) is a direct summand of H∗(𝑃𝑍(Λ)⊕𝑚) and is thus a projectiveΛ-module. Note that as for H∗𝐿+𝐺(•), the monoidal structure(6.16) is defined through the composition of the following isomorphisms.
As explained in [Zhu16, §5.5], the Galois groupGal(F¯𝑝/F𝑝) acts on the Satake category Sat𝐺 ,Λ by tensor autoequivalences. For three sequences of dominant weight 𝜇1•, 𝜇2•, and𝜇3•, the following lemma is an immediate consequence of Theorem 8.0.13. 10.4) The Satake correspondences for bounded local Hecke stacks and the Satake correspondences for bounded local Shtukas are related by the restriction morphisms and summarized in the following diagram.
If we pull the cohomological correspondence on the right-hand side of (11.1) back to the upper edge and precompose it with (11.1), we get a map. Let 𝐾 ⊂ 𝐺(A𝑓) be a (sufficiently small) open compact subgroup and denote by Sh𝐾(𝐺 , 𝑋) the corresponding Shimur variety defined over 𝐸.
The Satake Category
The Satake Category as an Abelian Category
The Monoidal Structure of the Satake Category
The following two results are analogous to the equal characteristic counterparts and can be proved in exactly the same way. The restriction of weight function CT𝜈 to 𝑃𝐿+𝐺(𝑍 ,Λ) is represented by the projective object𝑃𝑍(𝜈,Λ) in𝑃𝐿+𝐺(𝑍 ,Λ). Note that by Proposition 4.2.3 we have the decomposition of the total weight function into direct sum of the weight functions H∗(𝐺 𝑟𝐺,F ★G) ' ⊕𝜆H∗𝑐(𝑆𝜆,F ★G).
According to [Zhu17, Theorem 2.7], the spectral sequence on the 𝐸1 page degenerates and the filtration Fil≥𝜇1, 𝜇2 thus moves up to a new filtration of H∗𝑇. Note that the monoidal structure of the total weight functor CT is compatible with that of the hypercohomological functor H∗ according to Theorem 6.3. By the construction of the commutativity constraint in 𝑃𝐿+𝐺(𝐺 𝑟𝐺,Q¯ℓ) in [Zhu17] we conclude that 𝑑 is the null map and that the multiplication map in 𝐵(Λ) is therefore commutative.
Then for a regular𝜆∈X+•(𝑇), the orbit𝑆𝐺·𝜆 ⊂𝑊·𝜆 is the subset of𝑊𝐺·𝜆 consisting of elements 𝜇 such that the line segment connecting𝜆and 𝐺and 𝐺 of 𝜇x. We need the following result of Xiao-Zhu [XZ17, Proposition 7.2.4] for our proof of the main theorem.
Semi-Infinite Orbits and Weight Functors
The Geometry of Semi-infinite Orbits
The Weight Functors
The proof is similar to the equal characteristic case (cf. [MV07, Theorem 3.5]) since the dimension estimation of the intersections of the semi-infinite orbits and Schubert manifolds are fixed in [Zhu17, Corollary 2.8]. Let ModΛ denote the category of finitely generatedΛ-modules and Mod(X•)denote the category of X•-graded finitely generatedΛ-modules. By the definition of the semi-infinite orbits and the Iwasawa decomposition, we obtain two stratifications of 𝐺 𝑟𝐺 by {𝑆𝜇 | 𝜇 ∈ X•} and {𝑆−.
According to Proposition 4.2.1, the two filtrations are complementary and together they define the decomposition H∗(𝐺 𝑟𝐺,•) 'É. To do this, it suffices to show that the weight functor CT𝜇 is exact for every 𝜇∈X•. Since CT is exact, it suffices to prove that CT maps non-zero objects to non-zero objects.
We prove by induction on the number of 𝐿+𝐺 orbits in 𝑋 as in the proof of [MV07, Proposition A.1]. Considering the action of 𝐿+𝐺 on 𝑋eand𝑈e upon left multiplication on the second factor, we can define categories𝑃𝐿+𝐺(𝑋 ,e Λ)and𝑃𝐿+𝐺(𝑈 ,e Λ).
Notations
Then argue as before and use the monoidal structure and the fidelity of the functor. We will make use of the following theorem for quasi-reductive group schemes proved inloc.cit. Then the dimension estimate (8.1) concludes the proof of the lemma in the semisimple or adjoint type case.
The group-theoretic description of the moduli of restricted local Shtukas (cf. [XZ implies that Shtloc0 (𝑚,𝑛) is perfectly smooth. Considering the Satake transfer of the image of Zℓ basis of Zℓ[𝐺ˆ](𝐺ˆ) inZ ℆][𝐺ˆ) inZ℆] (𝐺ˆ) ⊗Zℓ𝑄, we conclude the proof of The assumption guarantees the existence of the ind scheme Sh𝜇1|𝜇2 which fits into the following commutative diagram.
By our construction of the map Spc, this function is given by the retraction of a function 𝑓0 on Shtloc𝜇|𝜇 = 𝐺(Z𝑝)\𝐺(Q𝑝)/𝐺(Z𝑝).
Representability of Weight Functors and the Structure of Repre-
The Monoidal Structure of H ∗
Weight Functors on the Convolution Grassmannian
Through our discussion on the G𝑚 action on 𝐺 𝑟𝐺ט𝐺 𝑟𝐺 above, the first isomorphism can be obtained by applying Braden's hyperbolic localization theorem as in [DG14]. Therefore, we are left to prove the second isomorphism and the vanishing property of the cohomology. The idea of constructing this isomorphism is completely similar to the one presented in [Zhu17, Coro.2.17], and we outline it here.
Since Note that there is always a map from H∗𝑐(𝑆𝜇 .. 2−𝜇1,G) to be two arbitrary elements in cohomology groups. Since F G is centered on non-positive perverse degrees, we can compose the above morphism with the natural truncation morphism to obtain the following element.
Since the functor 𝑝H0(•G) is exactly correct, we get the following exact sequence. 6.3) Consider the diagonal action of G𝑚 on 𝐺 𝑟𝐺 ×𝐺 𝑟𝐺. The previous lemma motivates us to study the analog of the total weight function CT0:= Ê. The two filters are complementary to each other according to Lemma 6.1.1 and so is the theorem.
Monoidal Structure of the Hypercohomology Functor
Taking the quotient of the Satake correspondence (2.3) of affine Grassmannians by 𝐿+𝐺, we get the Satake correspondence for local Hecke stacks.
Tannakian Construction
Identification of Group Schemes
Local Hecke Stacks
Torsors over the Local Hecke Stacks
It is compatible with the restriction maps in (9.4) in the sense that the following diagram is commutative.
Perverse Sheaves on the Moduli of Local Hecke Stacks
In the rest of this thesis we will make use of the theory of cohomological correspondences between perfect schemes and perfect pfp algebraic spaces. In this chapter we state and prove the key theorem for our construction of the Jacquet-Langlands transfer. Let (𝑚, 𝑛) be some 𝑉•and 𝑊•-large integers, we can define the modules of bounded local Shtukas Shtloc𝑉 (𝑚,𝑛).
For each representation𝑉1, 𝑉2, 𝑉3, let We write Sh𝜇𝑖for mod 𝑝the fiber of the canonical integral model of the change of basis in In the case that(𝐺1, 𝑋1)= (𝐺2, 𝑋2),Sh𝜇1|𝜇2 is the refinement of the fiber mod p of a model natural integral of some Hecke correspondence.
Recalling our construction ℭ𝑊, the cohomological correspondence ˜𝑓𝑛 is given by the finite direct sum of the function loc★𝑝(SO(𝑓)), because the Shimura variety we are considering is a set of discrete points.
Moduli of Local Shtukas
Moduli of Restricted Local Shtukas
Let 𝜇• = 0 or, in general, the central cocharacter, Shtloc𝜇• ' B𝐺(O), which is not a completely finite representation as a prestack. To use the ℓ-adic formalism, it is therefore desirable to study the following approximation Shtloc𝜇•. We define the stack of modules Shtloc(𝜇• 𝑚,𝑛) (𝑚, 𝑛)-bounded local iterated shtukas as the stack that sorts for every complete𝑘-algebra 𝑅,.
The natural forgetful morphism 𝜓loc(𝑚,𝑛) : Shtloc(𝜇• 𝑚,𝑛) → Hkloc(𝜇• 𝑚) is a perfectly smooth morphism of relative dimension 𝑛dim𝐺.
Perverse Sheaves on the Moduli of Local Shtukas
For a pair of non-negative integers (𝑚, 𝑛), we can generalize the notion 𝜇•-large and define the notion 𝑉•-large. Recall from Remark 8.0.14 that the dual Langlands group ˆ𝐺 is naturally endowed with an operation of the Frobenius arithmetic 𝜎. Recall the definition of the Borel-Moore homology HBM𝑖 (𝑋) for a perfect algebraic space pfp which is defined on a closed algebraic field (cf. [XZ17, A.1.3]).
In this section, we adapt the machinery developed in previous sections and apply it to the study of the cohomological correspondences between different Hodge-type Shimura manifolds following the idea of [XZ17]. In the Siegel case, this can be understood as the perfection of the morphism that sends an abelian variety to its underlying 𝑝-divisible group. It can be checked that L𝑊 ,Qℓ :=L𝑊 ,Zℓ⊗Q is an étaleQℓ-local system on Sh𝜇,𝐾 which is independent of the choice ofΛℓ.
Let 𝜈 | 𝑝be a place of the composite of reflex fields of (𝐺𝑖, 𝑋𝑖), determined by our choice of isomorphism𝜄. Denote the global section of the structure sheaf on the quotient stack [𝐺 𝜎ˆ /𝐺ˆ] with J, and the prime-to-𝑝 Hecke algebra with H𝑝.
Key Theorem for Constructing the Jacquet-Langlands Transfer . 64
Key Theorem
We reason that the composition of maps in (12.11) factors through CorrShtloc(𝑆(𝑉e1), 𝑆(𝑉e2)). 2) be a quadruple of non-negative integers satisfying appropriate acceptance conditions. Then the action of J naturally translates to the right-hand side of (12.9) and upgrades it to our desired map.
Cohomological correspondences between Shimura varieties
Preliminaries
Mainaig dagitoy iti kinaperpekto dagiti lokal a Shtukas babaen ti mod A fiber.
Main Theorem
Here the three vertical maps are the natural quotients of the finite group 𝐾ℓ/𝐾𝑛. 2)!(Z/ℓ𝑛Zh𝑑2i)𝑛 be a cohomological correspondence in CorrSh𝜈. As discussed in loc.cit, the action of J on H∗𝑐(Sh𝜇𝑖,L𝑊,Qℓh𝑑𝑖i) is expected to coincide with the usual Hecke algebra action, which can be understood as the Shimura manifold analog of V. Since the Shimura manifold we consider is zero-dimensional, it follows from [XZ17, A.2.3(5)] that the cohomological correspondence loc★𝑝(SO(𝑓)) can be identified with a Zℓ-valued function on Sh𝜇|𝜇.
Non-Vanishing of the Geometric Jacquet-Langlands Transfer
