Introduction
Background on π L -divisible O L -modules
Ramified Witt vectors, Dieudonné O L -modules and O L -displays
For an OL-algebra R, the ring of branched Witt vectors isWOL(R) (cf. [Haz80]) WOL(R)= RN with the uniqueOL-algebra structure such that 1. In particular, WOL(Fp) is isomorphic with the ring of integers OL˘ of ˘L, where ˘L = L·QLnrp is the completion of the maximum unbranched extension of L. A Dieudonné OL module over k is a free WOL(k) module of finite rank, together with two maps F and V such that F is σ-linear, and V is σ−1-linear, and FV = πL = V F.
There is an equivalence of categories between the category of πL-divisible formal OL-modules over k and the category of Dieudonné OL-modules over k such that V is topologically nilpotent in the πL-adic topology (cf. [Ahs11] , Proposition 2.2.3 and Theorem 5.3.8). Generalizing earlier work by Zink ([Zin02]) and Lau ([Lau08]), Ahsendorf ([Ahs11]) introduced the concept of an Olympic screen. An OL display over an OL algebra R is a quadruple P = (P,Q,F,V−1) with P a finitely generated projective WOL(R) module, Q ⊂ P a submodule and F : P → P andV−1 : Q → P Frobenius linear maps which satisfy some additional conditions.
IfπLis nilpotent inR, then there is an equivalence of categories between the category of nilpotentOL representations overRand the category of formalπL-divisibleOL modules overR (cf. [Ahs11], Theorem 5.3.8).
Isoclinic π L -divisible O L -modules and their endomorphism algebras 10
To show that OLm+n[πm,n] is a discrete valuation ring with uniformizerπm,n, we first note that OLm+n[πm,n] is a subring of the division ring Lm+n(πm,n). ODm,n =End(Gm,n) OLm+n[πm,n], whereπm,is a uniformization of the discrete valuation ringODm,n. where Nrd is the reduced norm of the central division algebra D over L. wherevalis is the normalized valuation on the division algebra D, andvL is the usual normalizedπL-adic valuation on L. b) The result in (a) holds with Replaced by E, andιD replaced byιE. Since $h−1∈Im(θ) ⊆ Eθ, it is clear that this is well defined and is a homomorphism. from which the surjectivity of the above map follows easily.
We begin this section with a useful result that expresses the cohomology of certain rigid analytic spaces in terms of the cohomology of a subspace. Next, we include a result that will be needed later in Section 6.3 to show that the supercuspidal part of the cohomology of the double Lubin-Tate tower realizes the local Langlands and the Jacquet-Langlands correspondences. To study the cohomology of the double Lubin-Tate tower using the external force map Vh−1, it is necessary to first understand how the group actions behave with respect to Vh−1.
Unlike Vh−1, however, the duality map∨ is not WL-equivariant, one of the reasons being that the Tate module of dual af(L/OL)hisOh. However, the duality map∨ can still be used to study the cohomology of Mh∞'s aE××GLh(L) representation, and can therefore be used to show that the supercuspid part of the cohomology of Mh∞ in the intermediate degree realizes the Jacquet- Langland's correspondence up to certain twists. So the action of φ induces an isomorphism from (LTh0)0 to (LTh0)m, and the action of θ(φ) induces an isomorphism from (Mh0)0 to (Mh0)(h−1)m.
Considering the action eOD× on the connected components of (LThn)0, we see that every connected component of (LThn)0 is a finite etalon over (LTh0) of the same degree. Now, considering the action of the groups D× and E×, we see that, for each ∈Z, the map Vh−1 induces an isomorphism from any connected component of (LThn)m to any connected component of (Mhn)(h−1) m. In this section, we will reinterpret the results of the previous section in terms of Lubin-Tate double tower cohomology.
If they are large enough, the determinants of the K elements are distinct mod(1+πmLOL), so each orbit of the induced action of K on the connected LThm components has size d. In particular, we will show that the supercuspid part of the cohomology of the Lubin-Tate dual tower is realized in the intermediate degree by local Langlands and Jacquet-Langlands correspondences (up to the corresponding twists).
Moduli spaces of π L -divisible O L -modules
Basic Rapoport-Zink spaces
Let Cbe the category of schemesSover SpecOL˘ such that it is locally nullpotent on S. Let ¯Sbe the closed subscheme ofSdefined by the ideal sheafpOS. Any twoπL-divisible OL modules with the same Newton polygon are isogenic, so in the definition of Mnd. Let ker(g) be the core of the map g : (L/OL)h → (L/OL)h, and Yrigbe theπL-divisibleOL module αrig(ker(g))Xrig overSrig.
Results on the Lubin-Tate tower
Since the dimensions of D⊗ Lh and Mh(Lh) over Lh are both equal to h2, f is an isomorphism. Let c,d,ebe be as in Lemma 4.1.2 and let eL be the branching degree of L over Qp. a) Im(θ0) is the original image of Im(θ0,n) under the mapping det mod(1+πnLOL), (b) the mapping det mod(1+πnLOL) induces an isomorphism. By Proposition 6.1.1, Vh−1 induces an isomorphism on the formal schemes Vh−1: (LTh)0→ (Mh)0, i.e. an isomorphism on their generic fibers.
Using the fact that K is in the middle of D×, an easy calculation shows that the above is also an isomorphism of representations.
The endomorphism algebras, GL h and representation theory
The endomorphism algebras and GL h
Clearly, this map respects multiplication, so it is in fact a ring homomorphism. If it is unramified overQp, then L contains no non-trivial roots of unity, soc= 0 and µh−1(OL)= µd(OL). To show the inverse inclusion, we need to show that any satisfying det(ϕ) ∈ (OL×)h−1 lies in Im(θ0).
In particular, if =0, then the sea map mod(1+πnLOL) is restricted to a fromker(θ0)toker(θ0,n) isomorphism. To prove (b), we first show that. where L =[L:Qp] and is such that the set of roots of powers p of unity in O×. a) then follows immediately from this.
Representation theory
We're also going to abuse the notation a bit, and whenever a Lubin-Tate tower or Lubin-Tate double tower appears, we actually mean its base change from ˘LtoC. Hazewinkel, twisted Lubin-Tate laws of the formal group, branched Witt vectors and (branched) Artin-Hasse exponents.
The duality map and the exterior power map on moduli spaces
Definitions of the duality map and the exterior power map
In this section we will use the Serre dual and the external effect of a πL-divisible OL module to define maps from LTh∞ toMh∞. In the above, by a slight abuse of notation, the Vh−1α level structure of Vh−1X is given by.
Properties of the exterior power map and group actions
For any n ≥ 0,m,n∈Z, every connected component (LThn)m is isomorphically mapped to some connected component (Mhn)(h−1)m under the mapping Vh−1 :LTh∞ → Mh∞. Using duality, we see that every connected component of (Mhn)0 is also a finite etalon over (Mh0)0 of the same degree. By Theorem 3.2.4, this means that the mapping induced by Vh−1 onto connected components is a bijection.
So K acts trivially on LTh0, and we get an action of Im(θ) on LTh0 induced by the action of D×. To see that we can extend the above action of Im(θ) to an action ofEθ = hO×. D× acts transitively on the connected components of LThn, so for any ϕ < Im(θ), ϕ(Yn) will be disjoint from Yn.
The geometry and cohomology of the dual Lubin-Tate tower
Geometry of the dual Lubin-Tate tower
Cohomology of the dual Lubin-Tate tower
Geometric realization of local correspondences and vanishing results 43
Applying Theorem 3.2.1 and using the fact that JL and rec are compatible with turns, we see that the Hch−1(LTh∞)cusp.