A RELATION BETWEEN TWO MODULI SPACES STUDIED BY V.G. DRINFELD

GERD FALTINGS

Max-Planck-Institut für Mathematik Vivatsgasse 7 53111 Bonn, Germany

1. Introduction

The local Jacquet-Langlands correspondence gives a bijection between supercuspidal representations of the general linear group GL(h) of a local field K and irreducible representations of the multiplicative group of the central division algebra of invariant 1/h. By the local Langlands conjecture these should also be in bijection with the irreducible h-dimensional representations of the Weil-group W_{K}. H. Carayol conjectured two ways to realize these correspondences, namely via the cohomology of either coverings of Drinfeld’s symmetric space, or of the universal deformation of the one dimensional formal group of height h (see [3]). A result along these lines has been shown by Harris and Taylor ([13]), while the author has proven by much more elementary means that the first model incorporates at least the Jacquet-Langlands correspondence ([7]). The purpose of the present paper is to relate the two models via a geometric construction. Namely the corresponding maximal covers (infinite level-structures) become isomorphic as rigid spaces.

A rough idea of what is going one can be given by the associated period-maps. These map the Tate-module into the tensorproduct of crystalline cohomology and Fontaine’s ring of periods A_{crys}. In our case the Tate-module is always trivialised (infinite level-structure), and the crystalline cohomology is (up to torsion) constant. For the one-dimensional formal O_{K}-modules H the period map sends T_{π}(H) = O_{K}^{h} to its crystalline homology W which admits an action by the integers O_{B} of the division-algebra of invariant 1/h. For the Drinfeld symmetric space one classifies formal O_{B}-modules of dimension h, and the period-map sends O_{B} into W^{h} and is O_{B}-linear. Obviously these types of maps correspond one to one, where the Tate-module T_{π}(H) changes sides from étale to crystalline and is replaced by its dual.

One defect of the present paper is that we reduce formal O_{K}-modules to formal groups. It would be more satisfactory to develop an analogue of the classical theory of finite flat group-schemes and p-divisible groups for O_{K}-modules. For example the multiplicative group should be replaced by a Lubin-Tate formal group, Cartierduality by maps into Lubin-Tate, divided powers by π-divided powers ([11]), A_{crys} by an analogue which is naturally an algebra over O_{K}, etc.. This also should take care of positive characteristics. Unfortunately such a theory does not exist, or at least is not known to me, and everything is reduced prosaically to ℚ_{p}. (In the meanwhile such a theory has been constructed.)

Some of the work was done when I visited the Institure for Advanced Study around Easter 2001, so I thank the Institute for its very generous hospitality. It gives me special pleasure to dedicate this work to A.N.Parshin because I profited enormously from his work on diophantine geometry.

This paper has been accepted for publication by the St. Petersburg Mathematical Society.

2. Preliminaries

Throughout the paper K denotes a finite extension of the p-adic numbers ℚ_{p}, that is a non archimedean local field of characteristic zero. Also B denotes the division-algebra of invariant 1/h over K, h an integer. The integers in K or B are denoted by O_{K}, respectively O_{B}. A formal group G of dimension d over a p-adically complete (commutative) ring A is given by a formal group law on the powerseries A[[T_{1}, ...T_{d}]]. It is of finite height h if multiplication by p induces a finite map of degree p^{h} on A[[T_{1}, ...T_{d}]]. In this case there exists a universal vectorextension

0 → F → EG → G → 0,

with F an additive formal group of dimension h−d. Furthermore EG is crystalline, that is if I ⊂ A is a PD-ideal then up to canonical isomorphism EG depends only on the reduction of G modulo I. Also maps (of formal groups) defined modulo I extend canonically to universal vectorextensions (see [15]). For example we can apply this to I = p · A with its canonical PD-structure. Thus if A admits a Frobenius-lift the Frobenius on G modulo p extends to a semilinear endomorphism of EG. Also all this extends to p-divisible groups. We call the Lie-algebra of EG M_{1}(G) (crystalline homology). If V is a complete discrete valuation ring of mixed characteristic, with perfect residue field k, we can form the ring A_{crys} (see [10]). It has a PD-ideal, a Frobeniuslift, and an action of Gal(V̄ /V). If G denotes a p-divisible group over V its universal vectorextension depends up to isogeny only on the fibre of G over k. Thus if G_{0} denotes any lift of G ⊗_{V} k over the Witt-vectors V_{0} = W (k), then M_{1}(G)[1/p] = M_{1}(G_{0}) ⊗_{V_{0}} V [1/p], and this admits a semilinear Frobenius automorphism (the inverse of the map induced by Frobenius on EG_{0}). Furthermore if T_{p}(G) denotes the Tate-module of G any ρ ∈ T_{p}(G) defines over V̄ a homomorphism from ℚ_{p}/ℤ_{p} into G, which lifts over A_{crys}(V) to a map of universal vectorextensions, thus to a map from ℚ_{p} into EG(A_{crys}(V)). Also the induced map on Lie-algebras defines an element of the Lie-algebra of EG, so finally a canonical period-map (as in [8])

T_{p}(G) → M_{1}(G_{0}) ⊗ A_{crys}(V)[1/p].

This period map respects the Galois-action and the image is invariant under Frobenius and lies in the first stage of the product filtration. If we apply this to the multiplicative group (with the canonical generator of M_{1}(𝔾_{m})) we obtain as image of a generator of ℤ_{p}(1) a canonical element t ∈ F^{1}(A_{crys}(V)), well defined up to multiplication by the units ℤ_{p}^{∗}. For general G we consider the determinant of the period-map. For this we choose generators of the determinants of T_{p}(G) and of M_{1}(G_{0}), which are well-determined up to units in ℤ_{p}^{∗}, respectively V_{0}^{∗}. Then the determinant lies in F^{d}(A_{crys}(V).

Lemma 1. The determinant lies in V_{0}^{∗} · t^{d}.

Proof: We may assume that k is algebraically closed. As the Galois-action on the determinant of T_{p}(G) is via the d-th power of the cyclotomic character (see [16]) the determinant must be a K_{0}-multiple of t^{d} (see [10]). We have to show that the factor is a unit. Also choosing a generator of the determinant of M_{1}(G_{0}) which is a Frobenius-eigenvector with eigenvalue p^{−d} we get that the factor is Frobeniusinvariant, that is it lies in ℚ_{p}. (We could have derived this directly from [10] without using the Galois-action, but this result is more delicate).

This assertion is compatible with isogenies and extensions, and also with Cartierduality (the composition of the period map for G with the adjoint of that for G^{dual} is t^{h}). This already implies that the factor is nonzero. Also by a deformation argument we may assume that G = G_{0}: There exists a formal group G over V [[T]] with fibre G_{0} at T = 0 and fibre G at T = π, for a uniformiser π of V. Adjoining the squareroot of π and scaling T by it we may assume that G is constant modulo π. Then the theory of A_{crys} works over V [[T]] (see [8], although we only need a rather elementary case of the general theory essentially covered in [6]). Thus a period of G in F^{d}(A_{crys}(V [[T]])). Using the Galois-action we see that modulo F^{d+1} it is a multiple of t^{d}, with the factor in V [[T]][1/p]. Mapping to the localisation of V [[T]] in π · V [[T]] gives that the factor lies in ℚ_{p}, and by pushout coincides with that for G_{0}, respectively G.

So it suffices to consider G_{0}, which can be an arbitrary lift of its special fibre. Using extensions and isogenies we may assume that G_{0} is a formal group of dimension d and height h, with d and h coprime and 0 < d < h. We also may assume that G_{0} admits as endomorphisms the ring R of integers in the unramified extension ℚ_{p^{h}} of degree h of ℚ_{p}. It then follows that T_{p}(G) is a free R-module of rank one, and that Gal(V̄ /V) acts on it via a character χ_{G} : Gal(V̄ /V) → R^{∗}. It is known that χ_{G} is the product of d conjugates of the Lubin-Tate character χ : Gal(V̄ /V) → R^{∗}. The conjugations are via i-th powers of Frobenius, where 0 ≤ i < d ranges over those numbers for which the corresponding eigenspace of R on Lie(G_{0}) does not vanish.

Furthermore T_{p}(G_{0}) ⊗ R is the direct sum of eigenspaces under R (acting via endomorphims of G_{0}). We index them by integers i, 0 ≤ i < h, such that R acts on i-space via the i-th power of Frobenius. Similarly we have such a decomposition for M_{1}(G_{0}). Now the period-map is integral (no need to invert p), and as it respects endomorphisms it acts (after choice of generators) on the i-space by an element x_{i} ∈ A_{crys}(V). This element lies in F^{1}(A_{crys}(V)) if the corresponding basis-element of M_{1}(G_{0}) does not lie the Lie-algebra of the vectorpart F of EG_{0}, which happens for d indices i. Furthermore the x_{i} are eigenvectors for Gal(V̄ /V), with character the i-th Frobenius conjugate of χ_{G}. Finally x_{i} (and thus also y_{i}) divides t because the period-map has an inverse up to t. The leading term of x_{i} in gr_{j}^{F} (A_{crys}(V)) ≅ ˆ V̄ (j = 0, 1) has a certain valuation v_{i} ∈ ℚ (valuations are normalised such that p has valuation 1). Modulo elements of valuation > v_{i} the Galois-group acts on elements of valuation v_{i} via a tame character with values in k^{∗}. This character is equal to the reduction of the Lubin-Tate character if and only if

v_{i} ≡ p^{−jh}/(p^{h} − 1)mod(ℤ)

for some integer j ≥ 0 (For gr_{0}^{F} there is an easy relation between valuations and tame Galois-actions on leading terms. To check for gr_{1}^{F} one can use that the result is correct for the valuation of t, that is for valuation 1/(p−1), and that it is anyway correct up to a constant shift). For the i-th Frobenius transform one has to multiply the right hand side by p^{i}.

Now let us first study the case d = 1, that is Lubin-Tate groups. It then follows that the valuation v_{i} of x_{i} satisfies either v_{i} > 1 or 0 < v_{i} ≤ p^{i}/(p^{h} −1) (0 ≤ i < h). The first alternative cannot hold because x_{i} divides t which has valuation 1/(p−1).

Also the product of the x_{i} is a Galois-eigenvector for the cyclotomic character and thus a K_{0}^{∗}-multiple of t. Checking valuations one finds that the factor is a unit, and that each x_{i} has valuation v_{i} = p^{i}/(p^{h}−1). Especially the determinant of the period map is the product of the x_{i} and thus up to a unit equal to t, as claimed. Also one checks that for each collection of d < h indices i the product of the corresponding x_{i} has leading term with valuation < 1, thus must generate the corresponding Galois-eigenspace (which is a free cyclic module over V_{0}).

Now for general d < h the corresponding elements x_{i} must be integral multiples of the product of d ”Lubin-Tate-x_{i}’s”. Thus their product is an integral multiple of t^{d}. But this product is the determinant of the period-map, and applying the same procedure to the dual (d replaced by h − d) we obtain that the factor is a unit.

Remark: The result also applies to formal groups of finite height over ˆ V̄, the p-adic completion of the integral closure of V in an algebraic closure K̄. Namely these can be obtained by pushout from a formal group over V [[T]] (use [14] to check that modulo each power of p the formal group is defined over V̄) and the previous deformation-argument applies.

3. valuations of torsion-points

We fix a local field K which is a finite extension of ℚ_{p} and denote by V = O_{K} its ring of integers. p-adic valuations are normalised such that a uniformiser π of O_{K} has valuation 1. The order of the residue-field of V is denoted by q.

In this section we consider formal O_{K}-modules H of dimension 1 and height h, as in [11], over the completion ˆ V̄. In a suitable coordinate T multiplication by π is described by a polynomial

f_{H}(T) = πT + ∑_{i=1}^{h−1} a_{i}T^{q^{i}} + T^{q^{h}},

where the coefficients a_{i} ∈ ˆ V̄ are non-units. They parametrise the (suitably defined) isomorphism-class of H. We say that H satisfies condition (∗) if a_{i} has valuation v(a_{i}) ≥ 1 − i/h. It is shown in [11], Cor.23.26, that any H is isogeneous to one satisfying (∗).

The torsion-points in H form a module isomorphic to (K/O_{K})^{h}. The valuation of such a torsion-point is defined as that of the value of T at that point. The valuations of the π-torsion can be read of from the Newton polygon of f_{H}:

Namely in the (x, y)-plane form the convex hull of all points lying above the points (q^{i}, v(a_{i})), 0 ≤ i ≤ h (a_{0} = π, a_{h} = 1). Then its lower boundary consists of h lines connecting (q^{i−1}, u_{i−1}) to (q^{i}, u_{i}). If v_{i} = (u_{i−1} − u_{i}/(q^{i} − q^{i−1}) denotes the slopes of these lines then H[π] has an 𝔽_{q}-basis x_{i} such that for a linear combination x = ∑_{H} r_{i} · x_{i} (r_{i} ∈ 𝔽_{q}) the valuation of x is equal to the minimal v_{i} for which r_{i} ≠ 0. Furthermore the v_{i} are non-increasing.

Now assume that H satisfies (∗). We want to determine for which isogenies f : H → H′ the target H′ satisfies (∗) as well. We may assume that the isogeny f does not factor over multiplication by π, that is it does not annihilate all πtorsionpoints.

Lemma 2. H′ satisfies (∗) if and only if H′ is the quotient of H under the submodule generated by x_{1}, ..., x_{i}, for some i for which u_{i} = 1 − i/h, and for a suitable choice of x_{i} (There is some ambiguity of several x_{i} have the same valuation).

Proof: First assume that f is of the form above. It is known that for a point x of H the valuation of f(x) is equal to the sum of the valuations v(x−_{H} y), for y ranging over the elements of ker(f) (”−_{H}” means difference in H). We apply this to either x a non-trivial combination of x_{j} with j > i, or to an x with π ·_{H} x a non trivial combination of x_{j} with j ≤ i. The images of these points generate the π-torsion in H′. From the Newton-polygon one checks that for j ≤ i our element has valuation v_{i}/q^{h}, and it follows in all cases that the valuation of x −_{H} y (y ∈ Ker(f)) is equal to v(x). Thus under f the valuation of x is multiplied by the order of f, that is by q^{i}. It follows that if we define x′_{j} = f(x_{j+i}) for j ≤ h − i, and x′_{j} for j > h − i as the image of an x with π ·_{H} x = x_{j+i−h}, then the H′-valuation of x_{j} is either q^{i} · v_{j+i} (j ≤ h − i) or q^{i−h} · v_{j+i−h} (for j > h − i). As v_{1} ≤ q^{h−1} · v_{h} it follows that the sequence of these valuations is non-increasing, with a break at h − i, and that the valuation of any linear combination of the x′_{j} is equal to the infimum of the v′_{j} for which the coefficient is nonzero. One then can read of the parameters u′_{j} = u_{j+i} − u_{i} respectively u′_{j} = u_{j+i−h} + 1 − u_{i}. As u_{i} = 1 − i/n it follows that H′ satisfies (∗).

Conversely assume that this is the case. We can find a basis ρ_{i} for the Tatemodule T_{π}(H) such that the x_{i} are the π-division-points defined by ρ_{i}, and such that the kernel of f is generated by the corresponding π^{n_{i}}-division points, for integers n_{i} ≥ 0 not all strictly positive. That is T_{π}(H′) has basis π^{−n_{i}} · ρ_{i}.

If n denotes the maximum of the n_{i}, choose elements y_{i} with π^{n} ·_{H} y_{i} = x_{i} for those i for which n_{i} = n. Let r be the number of these, and denote them by i_{1} < i_{2} < ... < i_{r}. Any linear combination of the y_{i} not annihilated by π^{n} has valuation ≤ v_{i_{0}}/q^{nh}. As elements y ∈ Ker(f) have valuation ≥ v_{h}/q^{(n−1)h} it follows that under f the valuations of these elements multiply by the degree, that is by q^{∑ n_{i}}.

Now the image is a subgroup of the π-torsion of H′, of rank r. It contains (q −1) elements of valuation v_{i_{1}}/q^{∑(n−n_{i})}, (q^{2} − q) elements of valuation v_{i_{2}}/q^{∑(n−n_{i})}, etc.. The sum of these is equal to

(∑_{j=1}^{r}(q^{j} − q^{j−1})v_{i_{j}})/q^{∑(n−n_{i})}

≤ (∑_{j=1}^{r}(q^{j} − q^{j−1})v_{j})/q^{∑(n−n_{i})}

≤ (∑_{j=1}^{r}(q^{j} − q^{j−1})v_{j})/q^{h−r}

≤ (1 − u_{r})/q^{n−r}.

On the other hand as H′ satisfies (∗) as well the sum must be ≥ r/(hq^{h−r}). Thus u_{r} = 1 − r/h, n = 1, and v_{i_{j}} = v_{j}. Changing the basis x_{i} (it is not quite unique) gives the result.

Remark: The equality u_{i} = 1 − i/h only happens of (q, u_{i}) is a proper vertex of the Newton polygon, that is a change in slopes occurs at that stage. Also this happens if and only if the π-adic valuation of a_{i} is 1 − i/h.

Next we define a valuation (or a π-adic norm) on T_{π}(H)^{dual}. Namely consider the universal O_{K}-vectorextension E_{K}H, as in [11]. The previous universal vectorextension maps to it, and it is the biggest quotient such that O_{K} acts on the Lie-algebra via its natural homomorphism into ˆ V̄ (The Lie-algebra M_{1}(G) of the usual universal vectorextension is a projective ˆ V̄ ⊗ O_{K}-module). An element ρ of T_{π}(H) is given by a sequence x_{n} = π ·_{H} x_{n+1} of torsion-elements. Lift them to E_{K}H and multiply by π^{n}. The result is modulo π^{n} ·F_{K}( ˆV̄) independant of choices, and the limit defines a lift T_{π}(H) → E_{K}H( ˆV̄) (Which is also the lift given by the crystalline nature of universal vector-extensions). In fact this construction even defines a lift on the vectorspace T_{π}(H)[1/π].

The Tate-module T_{π}(H) maps into the vectorpart F_{K}, and by duality we get F_{K}^{dual}( ˆK̄) → T_{π}(H)^{dual} ⊗_{O_{K}} ˆ K̄. The quotient is one dimensional and induces the norm on T_{π}(H)^{dual} (that T_{π}(H)^{dual} injects into it will be seen below). Another way: Lift H to an H over A_{crys}(V). The first crystalline homology M_{1}(H) is independant of the choice involved. Consider the period-map

T_{π}(H) → M_{1}(H).

Further divide by M_{1}(H) ⊗ F^{1}(A_{crys}(V)) and map to the maximal quotient of M_{1}(H) on which O_{K} acts via its embedding into ˆ V̄. The resulting map takes values in F_{K}( ˆV̄) and is the same as before.

It follows by duality that the one-dimensional quotient has the following description: Consider the dual (p-divisible or formal group) H^{dual}. It has height h · [K : ℚ_{p}] and dimension one less than that. It admits an action of O_{K} but is not an O_{K}-module since the action on the Lie-algebra is not correct. Its universal vector-extension has a one-dimensional vectorpart dual to the Lie-algebra of H, on which O_{K} acts by multiplication. The previous construction applied to H^{dual} defines a map

T_{p}(H^{dual}) = T_{p}(H)^{dual}(1) → Lie(H)^{dual}( ˆV̄)

which (up to a factor) is our quotient-map. That it is injective follows from

Lemma 3. Suppose that modulo p H is isogeneous to H_{0} of order p^{n}, that is there exist morphisms in both directions with composition multiplication by p^{n}. Then the induced p-adic valuation on T_{p}(H^{dual}) differs from the standard-valuation by a function taking values between 0 and n + 1.

Proof: Consider an indivisible element ρ ∈ T_{p}(H^{dual}) and suppose its image in Lie(H)^{dual}( ˆV̄) is divisible by p^{n+1}. This means that the corresponding p^{n+1}- torsionpoint x of H^{dual} lifts to a p^{n+1}-torsion-point in EH^{dual}. By the crystalline nature of the universal vectorextension the isogeny modulo p between H and H_{0} lifts to an isogeny over ˆ V̄ between universal vectorextensions, again of order p^{n}. Thus EH_{0}^{dual} must contain a nontrivial torsion-point of exact order p. As we have a Lubin-Tate group all such torsion-points are conjugate under automorphisms, so the period-map for H_{0}^{dual} vanishes modulo p. On the other hand we have already studied the periods of such formal groups, and know that the period divides t with the quotient a non-unit. Thus the period has p-valuation < 1 and is not divisible by p.

Remark: This argument partially extends to families, using the methods from [6]: Suppose G is a formal group over a p-adically complete ring R which is a V -algebra and has toroidal singularities over V (for example is smooth, or has semistable reduction, see [9]). If R̄ denotes its normalisation in the maximale étale cover of R[1/p] then R̄ is almost flat over R, that is all higher T or’s are annihilated by any positive power of p (see [9] for details). Also the relative (logarithmic) differentials Ω_{R̄/R} are almost isomorphic to a direct sum of copies of R̄[1/p]/R, with one direct summand isomorphic to Ω_{V̄ /V} ⊗V̄ R̄. Finally the logarithmic derivatives of roots of unity define an isomorphism

V̄ [1/p]/η^{−1} · V̄ ≅ Ω_{V̄ /V} ,

for a certain η ∈ V (whose valuation is related to the discriminant of V). It then follows from lemma 2 in [6] that the Hodge-tate period map has an inverse up to η (first an almost inverse, but an almost integral element of ˆ R̄[1/p] is integral). Thus the submodule of Lie(G)^{dual} ⊗ ˆ R̄ generated by the image of T_{p}(G^{dual}) contains anything divisible by η. Especially over the field it spans the first stage of the Hodge-filtration.

4. Two Moduli-spaces

V.G.Drinfeld has studied two different moduli-spaces. One of these is his famous symmetric space, namely the complement of all K-rational hyperplanes in ℙ_{K}^{h−1}. It has an integral model X (a formal scheme locally of finite type over the maximal unramified extension of O_{K}) which is the Deligne-scheme: Its R-points (R an artinian local O_{K}-algebra on which p is nilpotent) associated to any lattice Λ ⊂ K^{h} a quotient line Λ ⊗ R ↠ L_{Λ}, invariant under homotheties. Furthermore inclusions of Λ’s should induce maps on the quotient, and for any nontrivial ρ ∈ K^{h} there exists a Λ such that the intersection of Λ and the line spanned by ρ generates L_{Λ}. X classifies formal groupes G of dimension h with multiplication by O_{B}, equipped with a quasiisogeny modulo p (isogeny up to inverting a power of π) of degree 1 to a fixed G_{0}, and such that the integers of the unramified extension of degree h of K act via the regular representation on the Lie-algebra of G (special O_{B}-module in [4]). Choose a uniformiser Π of O_{B} with Π^{h} = π and such that Π normalises the integers of the unramified extension of degree h of K which are contained in B.

Also there exists finite coverings X_{n} → X where X_{n} classifies G’s (as before) together with a generator of its Π^{n}-division points. Here a section g ∈ G is such a generator if Π^{n−1} · g is a generator of the Π-division points, which means that it lies in a certain closed subscheme which is functorial, and which in characteristic zero coincides with the complement of the zero-section in G[Π]. All in all we obtain a proobject in the category of formal schemes, with flat transition-maps. We denote by X_{∞} its projective limit in the category of π-adic formal schemes, that is the projective limit of the topological spaces, endowed with the completion of the inductive limit of sheaves of rings. It admits an action by the subgroup of GL(h, K) × B^{∗} × W_{K} consisting of elements where the sum of π-adic valuations of determinants vanishes (on W_{K} any Frobenius has valuation 1). The action is by changing the quasi-isogeny by isogenies (GL(h, K)), elements of B^{∗} (where one also has to conjugate the O_{B}-action on G), and by composing it with powers of the Frobenius-isogeny on G_{0}. It is shown in [17] 5.48 that the period map into projective space defined by Drinfeld coincides with the crystalline period map.

Similarly in [11] we have the universal deformation-space of the formal O_{K}- module of dimension 1 and height h. Rigid-analytically it corresponds to the open unit disk. It specifying Drinfeld-basis of π^{n}-torsion points one obtain a flat cover which represented by a regular complete local ring of dimension h. The coordinates of the universal division-points give regular parameters (see [13]). In the limit we obtain an infinite covering with group GL(n, O_{K}). On it the bigger group GL(n, K) acts: The scalar π acts trivially. Suppose we are given H with a compatible system of Drinfeld-basis for its division-points, and a g ∈ GL(h, K). Multiply g by a power of π to make it integral, and choose n sufficiently big such that g divides π^{n}. The linear combinations of the g-transforms of the Drinfeld-basis of H[π^{n}] generate a flat subgroup whose underlying scheme is the divisor which is the sum of all linear combinations. Dividing by this subgroup gives a new H with Drinfeld level-structures. Over the closed point the induced isogeny is isomorphic to a power of Π (Π is a uniformiser of O_{B}, and the exponent is the valuation of det(g)), so the special fibre of the quotient is again isomorphic to the fixed H_{0}.

More generally we can find a model over the maximal unramified extension of O_{K}, with an action of the familiar subgroup of GL(h, K) × B^{∗} × W_{K} of elements with sums of valuations 0. Here the first factor acts via the inverse adjoint of the previous, the second and third by quasi-isogenies on H_{0} (the inverse adjoint is needed because the first factor now acts on H instead of H_{0}).

We now construct formal p-adic schemes mapping to these, corresponding to an affinoid covering of the projective system of open unit disks. One such domain is the following: Consider the closed unit disk classifying H’s for which the coefficients a_{i} of the polynomial f_{H} have valuation ≥ 1 − i/h. If we adjoin an h’th root of π it has a smooth formal model, namely the formal affine space in coordinates a_{i}/π^{1−i/h}. There is also a natural model over O_{K}, whose affine ring is the intersection with K[[a_{1}, ..., a_{h−1}]], but it is not smooth (only toroidal), and our affine space is the normalisation of its base-extension. Over these models we have the universal formal group H (where f_{H}(T) has coefficients a_{i}), and a GL(h, O_{K})-covering defined by Drinfeld level structures. This covering is first a pro-object in the category of p-adic formal schemes but we may form its projective limit as a p-adic formal scheme. It is affine with algebra the π-adic completion of the union of the algebras classifying π^{n}-levelstructures.

Next form a disjoint union of these indexed by GL(h, K)/π^{ℤ} · GL(h, O_{K}), and glue them along open subschemes, according to the following rules (dictated by lemma 2):

Everything should by GL(h, K)-equivariant, so it is enough to specify how the copy parametrised by g = 1 is glued (and this has to be GL(h, O_{K}) equivariant).

So for this copy consider the open formal subscheme defined by a_{i}/π^{1−i/h} invertible. Over it divide H by the flat subgroup consisting of π-torsion-points of valuation ≥ i/(h · (q^{i} − 1)). It has order q^{i} and is defined by the vanishing of a suitable factor of f_{H}. The result H′ lies again in our fundamental region, that is a′_{j}/π^{1−j/h} is integral (and invertible for j = h − i). Also Zariski-locally H′ admits a Drinfeld level structure, by composing the one on H with a suitable matrix g which is a product of an element of GL(h, O_{K}) and the inverse of diag(π, π ..., 1, 1) (i π’s). Glue our open subscheme to the corresponding open in the copy defined by g.

The result of this glueing is a π-adic formal scheme Y_{∞}. Over it we have a one-dimensional formal O-module H, together with a quasiisogeny modulo π, of degree 0. If one wants to use pro-objects the result is somehow difficult to describe. It is someting like an increasing union of proobjects in the category of formal πadic schemes. It is easier to form the quotient under a compact open subgroup M ⊂ GL(h, K), which is a π-adic formal scheme Y_{M} over O_{K}, locally of finite type. The Y_{M} form a projective system, with an action of GL(h, K)/π^{ℤ}. Also on everything acts the subgroup of GL(h, K) × B^{∗} × W_{K} of elements of ”unitdeterminant”.

Finally the rigidanalytic space defined by Y_{∞} maps via the a_{i} to the open polydisk. If we restrict to the preimage of a closed polydisk, say defined by the condition that all a_{i} have valuation ≥ ϵ > 0, then the proof of corollary 23.26 in [11] shows that the H involved are isogeneous to an H′ satisfying (∗) such that the degree of the isogeny is bounded in terms of ϵ (First bound the valuations of π-torsionpoints away from 0). Thus the preimage lies in a finite union of transforms of the fundamental region. Hence Y_{∞} defines a model for the infinite rigid covering of the open unit disk defined by an infinite levelstructure on H.

Finally there exists another ”period-mapping”, as in [11], section 23: Rigidanalytically it maps H to the point in P̃ = ℙ^{h−1} which classifies the Hodge-filtration of Lie(EH_{0}) defined by H. After adjoining an h-th root of π P̃ has a model with homogeneous coordinates b_{i}/π^{1−i/h}. The b_{i} are (as the a_{i}) eigenvectors for the units of the unramified extension of degree h of K considered as subgroup of B^{∗}. The uniformiser Π acts by cyclically permuting the b_{i}/π^{1−i/h} (b_{h} = b_{0}), see [11], equ. 22.9. On the fundamental domain (a_{i} valuation ≤ 1−i/h) the period mapping is of the form b_{i} = a_{i} + higherorder. It extends to a regular map Y_{∞} → P̃ which is equivariant under the subgroup of elements with unit determinant in GL(h, K)×B^{∗}, (where the first factor acts trivially on P̃).

We remark that these construction have been done over the ring V obtained by adjoining to O_{K} an h’th root of π. However as the constructions are independant from the choice of such a root we get natural singular models over O_{K}. Fortunately we do not need these: It suffices that our models satisfy a ”weak descent-condition”, namely that over the normalisation of V ⊗_{O_{K}} V the two pullbacks become isomorphic, the isomorphism satisfying transitivity etc.. This ”weak descent” will be sufficient for our applications to vanishing cycles.

5. Relation

We prove that after suitable blowups the two infinite covers X_{∞} and Y_{∞} become isomorphic. We shall construct such an isomorphism over the maximally unramified extension of the extension of O_{K} defined by adjoining an h’th root of π, and it will satisfy the ”weak descent condition” as before, that is nothing really depends on the choice of that root.

We first describe this correspondence on the level of ˆ V̄ -points where the blowups do not show up. We denote by H_{0} a lift of the O-module of dimension 1 and height h, say the one defined by a_{i} = 0. Modulo π it has multiplication by O_{B}, where Π acts as Frobenius. The universal vectorextension EH_{0} is up to isogeny independant of the lift. Also over k there exists a special O_{B}-module G_{0} and an isogeny from H_{0}^{h} into it, of degree q^{h(h−1)/2}. Namely one can choose for G_{0} the product of h copies of H_{0}, with the isogeny on the i-th factor given by Π^{i−1}, and the corresponding conjugation of the O_{B}-action. Thus the Drinfeld halfspace describes O_{B}-modules G with a quasiisogeny module π H_{0}^{h} → G of degree q^{−h(h−1)/2}.

Now assume given such a G defined over ˆ V̄, together with a compatible system of generators of its Π^{n}-division-points. Then EG is isogeneous to EH_{0}^{h}, and the division-points define injections of B into the ˆ V̄ -points of EG and EH_{0}^{h}. The logarithm of 1 ∈ B defines an element x = (x_{1}, ..., x_{h}) ∈ M_{1}(H_{0})^{h} ⊗ ˆ V̄ [1/p]. Its Bspan is a B-stable subvectorspace of codimension h, thus induced from a subspace F_{G} ⊂ ˆ V̄ [1/π]^{h} of codimension 1 which is the classifying point associated to G in the Drinfeld symmetric space. By duality x defines a map ˆ V̄ [1/π]^{h} → M_{1}(H_{0}) ⊗ ˆV̄ [1/π] whose image F_{H} has codimension 1. It defines a point in ℙ̃ and thus an object H ”up to isogeny”. First choose one such H̃ such that modulo π there exists an isogeny H_{0} → H̃, and such that under the lift EH_{0} → EH̃ the elements x_{1}, ..., x_{h} ∈ M_{1}(H_{0}) ⊗ ˆ V̄ [1/π] map to the integral elements in the vector-part of Lie(EH̃) (multiply the isogeny by a high power of π). Then x defines a map O_{K}^{h} → T_{π}(H̃) whose degree can be computed using Lemma 1:

Namely O_{B} · x is the Tate-module of G, and the period-map for G has determinant t^{h}. Up to a factor q^{−h(h−1)/2} this is the determinant of the map from O_{B} into Lie(EH_{0})⊗A_{crys}^{h}[1/p], which in turn is q^{h(h−1)/2} times the h-th power of the determinant of the O_{K}-submodule generated by x_{1}, ..., x_{h} in Lie(EH_{0}) ⊗ A_{crys}(V)[1/p] (see below). Thus this has (up to units) determinant t, and the map above on Tate-modules has degree equal to that of the isogeny (modulo π) H_{0} → H̃. It follows that there is an H over ˆ V̄ with an isogeny H → H̃ such that x_{1}, ..., x_{h} is a basis of T_{π}(H) and such that H has special fibre H_{0}. Furthermore the x_{i} define a Drinfeld level-structure on H.

We have used implicitely that the degree of an isogeny can be read of from Tate-modules (which live in the general fibre) as well as from the crystalline homology (which depends only on the fibre in the residue-field of ˆ V̄). For the remaining assertion above about determinants of O_{B}-stable lattices replace O_{K} by a finite unramified extension which splits B and O_{B} by the h×h-matrices over that extension (this leads to the factor q^{h(h−1)/2}). Then all modules over this matrixring are just sums of h copies of a module over the base, and everything becomes elementary.

Conversely given an H with Drinfeld level-structure, that is a basis x_{1}, ..., x_{h} of its Tate-module, lift them to Lie(EH_{0}) and consider the B-submodule generated by x = (x_{1}, ..., x_{h}) in Lie(EH_{0})^{h}. It is induced from a subspace F_{G} which lies in the Drinfeld halfspace (use lemma 3) and corresponds to a G. Also x determines a level-structre up to a power of Π, and comparing determinants shows that this power is 0. Finally it is clear that these correspondences are inverse to each other. The relevant object is x ∈ Lie(EH_{0})^{h} ⊗ ˆ V̄ [1/π] which determines all the rest.

To see that this bijection on ˆ V̄ -points comes from an isomorphism of formal schemes consider the following blow-ups: Over Y_{∞} the universal period T_{π}(H) → F defines T_{π}(H)^{dual} → Lie(H)^{dual}. Zariski-locally we have that for any indivisible ρ ∈ T_{π}(H)^{dual} its image divides a fixed π-power π^{n}: Namely it suffices to check that over the open affine where H satisfies (∗). There H is modulo p isogeneous to a fixed H_{0}, of degree bounded by p^{n}, and then lemma 3 implies the assertion for all ˆ V̄ -points. If U_{N+1} then denotes the affine classifying H satisfying (∗) and a level π^{N+1}-structure, the image of ρ modulo π^{N+1} is defined over U_{N+1}. Lifting it to a regular section f of Lie(H)^{dual} over U_{N+1} we get from rigid analysis that this section does not vanish on the generic fibre (a rigid space) and thus must divide a power of π. Then π^{N}/f is integral over O_{U_{N+1}}. As the image of ρ coincides with f modulo π^{N+1} it follows that in the extension of O_{U_{∞}} generated by the normalisation of O_{U_{N+1}} the image is of the for u · f with u ≡ 1mod(π). Thus u is a unit, so the image of ρ divides π^{N+1} in this extension. Multiplying with an other π-power which sends the normalisation of O_{U_{N+1}} into O_{U_{N+1}} we get the assertion. Now Ỹ_{∞} is obtained from Y_{∞} by blowing up the ideal generated by the image of Λ, for any lattice Λ ⊂ K^{h}. We claim that it suffices to blow up only finitely many ideals and that they are already defined over Y_{n}: Namely we need only consider Λ’s up to homothety, so we may assume that they lie in T_{π}(H)^{dual} = O_{K}^{h} and that they contain an indivisible ρ as above. But then two Λ’s generate the same ideal if they coincide modulo π^{n}, thus the finiteness of the number of ideals. Also all ideals contain π^{n} and are modulo that generated by elements in O_{Y_{n}}.

For the definition of the appropiate modification X̃_{∞} we note that over X_{∞} the universal torsion-point defines a period-map O_{B} → Lie(G)^{dual} which after inverting π define a map into P̃, and this becomes regular after blow-up of a suitable ideal. By the remark after lemma 3 on sees that the ideal is defined at a finite level.

Finally from Ỹ_{∞} we have a regular map into Drinfeld’s model X_{0}, thus an O_{B}- module G over it. EG is quasiisogeneous to EH_{0}^{h} and EH^{h}, and the lifts of the universal division-points define a map K → EG and thus a B-linear O_{B} → EG → G. Over each ˆ V̄ -point this defines a level-structure, so a level structure in general, and thus a map from Ỹ_{∞} to X_{∞}. This map factors over X̃_{∞} because Ỹ_{∞} maps to P̃, that is the relevant ideals are already invertible in it.

Conversely starting with X̃_{∞} it maps to P̃. Thus over the preimage of any standard affine in P̃ we obtain by [11], Cor.23.15 a H̃, and a map K^{h} → EH̃ → H̃. Its kernel T_{π}(H̃) is an O_{K}-lattice which is Zariski-locally constant, and contained in π^{−n} · O_{K}^{h} with index q^{nh} for sufficiently big n, as can be checked on ˆ V̄ -points. Choosing a basis for T_{π}(H̃) defines a Drinfeld-levelstructure and thus a map into Y_{∞}, which lifts into Ỹ_{∞} because the blowups necessary to map into X_{0} are trivial here. Transforming with a suitable element of GL(h, K) with determinant a unit we obtain the desired H with infinite level-structure (O_{K}^{h} ≅ T_{π}(H)). This gives the inverse X̃_{∞} → Ỹ_{∞}. It is clear that it is compatible with the action of the subgroup of GL(h, K) × B^{∗} × W_{K} of elements with ”unit-determinant”.

6. Consequences for nearby cycles

For an integer r prime to p it is explained in [1] how to define the sheaves of vanishing cycles R^{i}Ψ_{η}(ℤ/(r)) on the special fibres of X̃_{n} and Ỹ_{n}. These coincide with the vanishing cycles defined by any algebraic model. The inductive limit defines vanishing cycles also on the special fibers of X̃_{∞} and Ỹ_{∞}, and these are isomorphic (using [1], Th. 7.1). Furthermore by [2], Cor.2.5 the cohomology with compact support RΓ_{c}(Y_{n,s}, RΨ_{η}(ℤ/(r))) coincides with the local cohomology with support in the closed point RΓ_{x}(RΨ_{η}(ℤ/(r))), for any algebraic O_{K}-scheme whose formal completion in the point x equals the formal scheme which represents deformations H of H_{0} together with a Drinfeld level-π^{n}-structure (This formal scheme is represented by a regular complete local ring). Especially these are finite.

Passing to the limit we obtain the cohomology with compact support of the vanishing cycles on X_{∞}, X̃_{∞}, Y_{∞}, and Ỹ_{∞}. They admit continuous actions of the subgroup of ”elements of determinant a unit” of GL(h, K) × B^{∗} × W_{K}, and the submodule stabilised by any open compact subgroup in GL(h, K) is finite. It is more convenient to consider the induced representation of GL(h, K) × B^{∗} × W_{K} modulo the subgroup generated by (π, 1, 1), as in [7].

Finally for l-adic cohomology we form the projective limit of cohomologies mod l^{s}, and in it the elements stabilised by an open compact subgroup in GL(h, K) which is a pro-p-group. We get the same result if we first form the projective limit at some finite level (that is form the l-adic cohomology of some Y_{n} which is finite over ℤ_{l} as seen above, not for general reasons), and then the direct limit over all levels. Finally inverting l gives ℚ_{l}-adic cohomology.

It is shown in [13] that the result realises the Jacquet-Langlands correspondence as well as the local Langlands-conjecture: Namely a given supercuspidal representation σ of GL(h, K) occurs with finite multiplicity and the alternating sum of the multiplicities is (−1)^{h−1} · h. Furthermore in the Grothendieck-group of representations the eigenspace for the dual representation σ^{dual} is equivalent to the tensorproduct of the associated (local Jacquet-Langlands) representation JL(σ) of B^{∗}, and the h-dimensional representation of W_{K} predicted by the local Langlandscorrespondence. Our main result now implies that similar assertions hold as well for the cohomology of (the coverings of) the Drinfeld upper halfspace (no more duality because the two actions of GL(h, K) differ by the outer automorphism ^{t}g^{−1} which sends σ to its dual, as can be checked on traces using that ^{t}g is conjugate to g).

Remark: Without too much effort one can improve a little bit on [13], namely one shows that the σ-eigenspace occurs only in cohomology in degree h − 1. For the Drinfeld symmetric space this has been done by M.Harris in [12], for the other space it is at least known to the authors of [13]. One possible argument goes as follows:

[13] constructs a Shimura-variety (of dimension h − 1) whose completion at a closed point is the deformation-space for H_{0}. For our purpose we fix suitable levelstructures away from π and denote by Sh_{n} the quotient associated to a Drinfeld level-π^{n}-structure. It classifies abelian varieties A with certain endomorphisms and polarisations, and their local deformations are classified by a certain onedimensional π-divisible direct summand H ⊂ A[π^{∞}] (the latter is essentially the sum of H and its dual). The special fibers S̄h_{n} have stratifications according to the rank of the maximal étale quotient of H. Let S̄h_{n}^{i} denote the locally closed reduced subscheme where this rank is h − i. It has pure codimension i, and our deformation-spaces for H_{0} arise as formal completions at points in S̄h_{n}^{h−1}. For simplicity we denote by Sh the pro-scheme given by the projective system Sh_{n}. It admits an action of GL(h, K).

Over the formal completion of Sh_{n} along S̄h_{n}^{i} the π-divisible group H becomes an extension

H^{inf} → H → H^{et},

with H^{inf} formal of dimension 1 and height i, and H^{et} étale of height h−i. There is an open and closed subscheme S̄h_{n}^{i,0} ⊆ S̄h_{n}^{i} classifying level-structures for which the first i generators lie in H^{inf}.

For the projective system we obtain similarly S̄h^{i} and S̄h^{i,0}. They are stable under GL(h, K) respectively P_{i}(K), where P ⊂ GL(h, K) denotes the parabolic stabilising a subspace of dimension i. In fact the formal completion along S̄h^{i} is the formal GL(h, K)-scheme induced (or better produced) from the formal P_{i}(K)- scheme ”completion along S̄h^{i,0}”. It follows that the cohomology of S̄h^{i} with values in the vanishing cycles of S̄h is induced from the P_{i}(K)-representation on the cohomology of S̄H^{i.0}. If we show that the unipotent radical N_{i}(K) of P_{i}(K) acts trivially on any irreducible subquotient in this cohomology the induced representation has (for i > 0) no supercuspidal subspace.

So suppose given an u ∈ N_{i}(K). It has some denominator π^{m}, and induces maps Sh_{n+m} → Sh_{n}. We first show that it acts trivially (that is the same way as u = 0) on closed points of S̄h_{n+m}^{i,0}), and then also trivially on stalks of vanishing cycles.

An R-point of the formal completion of Sh_{n+m} along S̄h_{n+m}^{i,0} is given by an abelian variety A over R, with certain additional data which include a π-divisible extension

H^{inf} → H → H^{et},

and π^{n+m}-division points e_{1}, ..., e_{i} ∈ H^{inf}, f_{i+1}, ..., f_{h} ∈ H which form Drinfeldbasis for H^{inf} and H^{et}, respectively. u maps π^{m} · f_{j} to elements f̃_{j} which are the sum of π^{m} · f_{j} and a linear combination of the e’s. The u-image is defined by dividing H by the submodule (of order q^{mh} generated by the π^{m} · e_{j} and f̃_{j}. As usual the scheme structure is that of the divisor which is the sum of all points. Also the level-structure on the quotient is defined by the images of e_{j} and π^{−m} ·f̃_{j} (these are welldefined modulo the previous). Of course all this translates into welldefined operations on the level of abelian varieties.

Now over any algebraically field k of characteristic p all e_{i} = 0, and u operates trivially on k-points. For the action on stalks of vanishing cycles it depends by [2], Th.3.1 only on the formal completion along the relevant point. This formal completion describes deformations of H as above, and can be analysed as follows:

The formal π-divisible module H_{inf} with π^{n+m}-levelstructure is parametrised by a local ring of relative dimension i − 1, as before. The extension above is given by chosing h − i sections f̃_{j} ∈ H^{inf}, and then H is the quotient of H^{inf} × K^{h−i} under the O_{K}-submodule generated by π^{n} · (f̃_{j}, 1). Finally the images f_{j} of (f̃_{j}, 1) define the levelstructure.

It follows that the formal completion is formally smooth over the moduli-space classifying H^{inf} with its levelstructure. Thus the vanishing cycles are induced from this, and as u respects the projection it acts trivially on them.

Now coming back to the Shimura-variety we see that the cohomology with compact support of S̄h − S̄h^{h−1} has no supercuspidal subquotient, and by duality the same holds for the cohomology without support. Thus for a given supercuspidal σ the σ component in the cohomology of Sh (that is the cohomology of S̄h with values in the vanising cycles) coincides with the σ-component in either the restriction of the vanishing cycles to S̄h^{h−1} or in the cohomology with support in that subscheme. From the first description we know that it occurs only in cohomological degrees ≤ h − 1, from the second in degrees ≥ h − 1 (by duality). Combining we get the result.

Thus in the cohomology of X_{∞} or Y_{∞} supercuspidal representations occur only in the middle degree h − 1. The corresponding space is the product of the associated representation JL(σ^{dual}) or JL(σ) and an h-dimensional representation of W_{K}. The hard content of [13] is that this representation realises the local Langlandsconjecture.

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