Mod p Langlands correspondences via

arithmetic geometry

Notes from a Minicourse at KIAS, August 2016 ∗

Judith Ludwig

January 5, 2017

Lecture 1: Introduction to mod p Langlands correspon-dences and background on adic spaces

1.1 Mod p Langlands correspondences - Motivation

Let L ∼= C or L ∼= Ql. Let p 6= l be a prime number and F/Qp be a finite extensionwith ring of integers OF , uniformizer $ and residue field Fq. Let GF := Gal(F/F )denote the absolute Galois group of F . The classical local Langlands correspondencefor GLn(F ) is an injection

continuous representations

ρ : GF → GLn(L)

/∼=−→

irreducible smooth

L− representations

of GLn(F )

/∼=ρ 7−→ π(ρ)

which is characterised by certain identities of L-and ε-factors. Enlarging the left sideto include all Fobenius-semisimple Weil–Deligne representations, this can be madeinto a bijection.

We recall some basic definitions.

Definition 1.1. Let k be a field. A representation π : GLn(F ) → Autk(V ) on ak-vector space V is called smooth if for every v ∈ V , the stabilizer

StabGLn(F )(v) ⊂ GLn(F )

∗Last modified January 5, 2017. Thanks to Jaclyn Lang and Sug Woo Shin for helpful comments.

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is open. A smooth representation π is called admissible if for every compact opensubgroup K ⊂ GLn(F ) the subspace

πK := v ∈ V : π(k)v = v ∀k ∈ K

of vectors fixed by K is finite-dimensional.

Let D/F be a division algebra with center F and invariant 1/n and let D∗ bethe group of units in D.

The classical local Jacquet–Langlands correspondence is an injectionirreducible smooth

L− representations

of D∗

/∼=−→

irreducible smooth

L− representations

of GLn(F )

/∼=π 7−→ JL(π)

which is characterized by a certain equality of traces. There is a maximal compactopen subgroup O∗D ⊂ D∗, which has the property that O∗DF ∗ = D∗, where F ∗ isidentified with the center. This implies that the objects on the left side are finite-dimensional.

Remark 1.2. Both correspondences can be realized simultanously via geometry;more precisely they can both be realized in the `-adic cohomology of the Lubin–Tatetower, which is a tower of deformation spaces of p-divisible groups. We we willintroduce them below.

Question: Do we have similar correspondences when L is replaced by Fp, Qp? Isit possible to understand the mod p and the p-adic cohomology of the Lubin–Tatetower?

In this course we well study mod p correspondences.1 In [8], Scholze constructeda candidate for the mod p local Langlands and the mod p Jacquet–Langlands cor-respondence via geometry.

Goal of these lectures: Explain Scholze’s construction and study an example.

Before we start with geometry let us remark, that there are two main differencesbetween the theory of smooth admissible representations in characteristic zero andin characterstic p.

1A brief remark on the p-adic case: When L = Qp, there are a lot more Galois representations

as the topologies of Gal(Qp/Qp) and GLn(Qp) are more compatible. The category of smoothrepresentations is too small to see all p-adic Galois representations.

2

• Every non-zero smooth mod p representation of a pro-p group H has non-zeroH-fixed vectors.

• In characteristic zero we have the Haar measure as a tool at hand and thereforecan use harmonic analysis to study representations. This fails mod p.2

Any Fp-valued Haar measure on GLn(F ) is zero.

Let’s prove this: Let K(1) := ker(GLn(OF ) → GLn(Fq)). This is a com-pact open subgroup of GLn(F ). The subgroup K(2) := ker(GLn(OF ) →GLn(OF /$2OF )) has index qn

2inK(1) (the quotient isK(1)/K(2) ∼= Mn(Fq)).

Any Haar measure µ on GLn(F ) is translation invariant and in particularµ(K(1)) = [K(1) : K(2)]µ(K(2)) ∈ Fp, therefore µ(K(1)) = 0, but K(1) isopen, and so µ is zero.

1.2 Crash course on adic spaces

The category - where our geometric objects (like the Lubin–Tate spaces) live - is thecategory of adic spaces. In this section we briefly review the most important defi-nitions and constructions. One motivation to study adic spaces is that the categoryof adic spaces encompasses both, formal schemes and rigid spaces.

Definition 1.3. Let R be a topological ring. A subset S ⊂ R is called bounded iffor any open neighbourhood U of zero there exists an open neighbourhood V of zerowith V · S ⊂ U . An element x ∈ R is called power-bounded if xnn∈N ⊂ R isbounded.

Definition 1.4. • A topological ring R is called Huber if it contains an opensubring R0 ⊂ R whose topology is generated by a finitely generated ideal I ⊂R0. We call R0 a ring of definition and I an ideal of definition.

• A Huber pair is a pair (R,R+) where R is a Huber ring and R+ ⊂ R is anopen and integrally closed subring consisting of power-bounded elements.

• A morphism of Huber pairs (R,R+)→ (S, S+) is a continuous homomorphismϕ : R→ S such that ϕ(R+) ⊂ S+.

Examples of Huber rings:

• Any ring R with the discrete topology (R0 = R, I = 0).

2In particular we cannot characterize correspondences in characteristic p as we do in character-istic zero, i.e., via traces.

3

• Recall that a topological ring R is called adic if there exists an ideal I ⊂ R,called ideal of definition, s.t. (In)n is a basis of open neighbourhoods of 0 ∈ R.Any adic ring R with a finitely generated ideal of definition is a Huber ring.

• For F/Qp a complete extension with ring of integers OF consider the Tatealgebra

F 〈X1, · · · , Xn〉 :=∑

ai1,··· ,inXi11 · · ·X

inn | ai1,··· ,in ∈ F, ai1,··· ,in → 0

.

Any quotient of F 〈X1, · · · , Xn〉 is a Huber ring. For R = F 〈X1, · · · , Xn〉, wecan take R0 = OF 〈X1, · · · , Xn〉 and I = (p).

Definition 1.5. A continuous valuation on a topological ring R is a map

| · | : R→ Γ|·| ∪ 0,

where Γ|·| is a multiplicative totally ordered abelian group such that

• |f · g| = |f | · |g|, ∀f, g ∈ R,

• |1| = 1 ∈ Γ|·|, |0| = 0,

• |f + g| ≤ max|f |, |g|,∀f, g ∈ R and

• ∀γ ∈ Γ|·|, f ∈ R : |f | ≤ γ ⊆ R is open.

We say two continuous valuations | · | and | · |′ are equivalent if

|a| ≤ |b| ⇐⇒ |a|′ ≤ |b|′

for all a, b ∈ R.

Let (R,R+) be a Huber pair. Then we define the topological space X =Spa(R,R+) as the set

X = Spa(R,R+) := equivalence classes of cts. valuations x on R,

s.t. |f(x)| := x(f) ≤ 1 ∀f ∈ R+

equipped with the topology generated by so called rational subsets

X

(T1

s1, · · · , Tn

sn

):= x ∈ X : ∀i, |fi(x)| ≤ |si(x)| 6= 0 ∀fi ∈ Ti ,

where for i = 1, . . . , n, si ∈ R, and Ti ⊂ R is a finite subset which generates an openideal.

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We put a natural structure presheaf OX on X as follows.

We equip the ring R[

1s1, · · · , 1

sn

]with a topology making R0

[T1s1, · · · , Tnsn

]open

equipped with the J := I ·R0

[T1s1, · · · , Tnsn

]-adic topology. This defines a ring topol-

ogy on R[

1s1, · · · , 1

sn

]and turns it into a Huber ring. Define

R

⟨T1

s1, · · · , Tn

sn

⟩:= J-adic completion of R

[T1

s1, · · · , Tn

sn

].

This construction allows us to define structure presheaves OX and O+X on X =

Spa(R,R+) by defining them on the basis of rational subsets U := X(T1s1, · · · , Tnsn

)as

OX(U) := R

⟨T1

s1, · · · , Tn

sn

⟩and O+

X(U) as the completion of the integral closure of R+[T1s1, · · · , Tnsn

]in OX(U).

For each x ∈ X, the valuation f 7→ |f(x)| on R extends to a valuation | · |x onthe stalk OX,x.

Warning: The presheaf OX is not necessarily a sheaf.

Definition 1.6. A Huber pair (R,R+) is called sheafy, if OX is a sheaf.

For a Huber ring R, let R ⊂ R denote the subring of power-bounded elements.

Example: If R = F 〈X1, · · · , Xn〉/a is a quotient of a Tate algebra, and R+ = R,then (R,R+) is sheafy.

Let V be the category whose objects are triples

(X,OX , (| · |x : x ∈ X)),

where X is a topological space, OX is a sheaf of complete topological rings onX and | · |x is an equivalence class of valuations on the stalk OX,x. A morphismfrom (X,OX , (| · |x : x ∈ X)) to (Y,OY , (| · |y : y ∈ Y )) is a pair (f, ϕ) consistingof a continuous map f : X → Y and a morphism ϕ : OY → f∗OX of sheavesof topological rings such that, for every x ∈ X, the induced ring homomorphismϕx : OY,f(x) → OX,x is compatible with the valuations | · |x and | · |f(x).

Note that any sheafy Huber pair (R,R+) gives rise to an object(Spa(R,R+),OSpa(R,R+), (| · |x : x ∈ Spa(R,R+))

)∈ V.

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Definition 1.7. • An affinoid adic space is an object of V which is isomorphicto the triple associated with a sheafy Huber pair.

• An adic space is an object (X,OX , (| · |x : x ∈ X)) in V which is locally anaffinoid adic space, i.e., every point x ∈ X has neighbourhood U ⊂ X suchthat (U, (OX)|U , (| · |x : x ∈ U)) is an affinoid adic space.

• A morphism X → Y of adic spaces is a morphism in V.

Remark 1.8. Below, we often denote an adic space simply by X and we use |X| torefer to its underlying topological space.

Let K be a complete non-archimedean field.

Theorem 1.9. There is a fully faithful functor

r : rigid analytic varieties/K −→ adic spaces /Spa(K,K)Sp(R) 7−→ Spa(R,R)

Theorem 1.10. There is a fully faithful functor

t : locally noetherian formal schemes −→ adic spaces Spf(R) 7−→ Spa(R,R)

X 7−→ Xad

There is an important construction, the so called generic fibre construction thatwill come up later, so we review it briefly.

Let K be a non-achimedean field, complete with respect to a discrete valuation| · |K . Let OK be the ring of integers, $ a uniformizer and let k be the residue field.Then Spa(OK ,OK) has two points, given by

| · |K : OK → R≥0, a 7→ |a|K

and

| · |k : OK → OK/$|·|triv→ R≥0.

The point | · |K corresponds to a morphism

η : Spa(K,OK)→ Spa(OK ,OK).

The adic generic fibre of a formal scheme Xad/OK is defined as

Xadη := Xad ×Spa(OK ,OK) Spa(K,OK).

Example: We describe the points of the closed unit disc X = Spa(Cp〈T 〉,OCp〈T 〉).Fix a norm | · | : Cp → R≥0. The topological space X has five types of points:

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• Points of type (1), the classical points: Any x ∈ OCp gives a map

Cp〈T 〉 −→ Cp|·|−→ R≥0

f 7→ f(x) 7−→ |f(x)|.

• Points of type (2) and (3): Let 0 ≤ r ≤ 1 be a real number, x ∈ OCp . Then

| · |r,x : f =∑

an(T − x)n 7→ supn|an|rn = sup

y∈OCp|y−x|≤r

|f(y)| ∈ R≥0

defines a continuous valuation. We say |·|r,x is of type (2) if r ∈ |C∗p|, otherwisewe say | · |r,x is of type (3). Note that when r = 0, this gives back the classicalpoint x. The valuation |·|r,x depends only on D(x, r) = y ∈ OCp : |y−x| ≤ r.In particular for r = 1, | · |1,x is independent of x ∈ OCp ; the valuation | · |1 iscalled the Gaußpoint.

• The field Cp is not spherically complete. Therefore there exists a sequence ofdiscs

Cp ⊃ D1 ⊃ D2 ⊃ · · ·with

⋂Di = ∅. Fix such a sequence, then

f 7→ infi

supx∈Di

|f(x)| ∈ R≥0

defines a continuous valuation. We call a point corresponding to such a valu-ation a point of type (4).

• Finally there are some valuations of rank 2. For that let x ∈ OCp , 0 < r ≤ 1and choose ∗ ∈ <,>. Let

Γ∗,r := R>0 × γZ

be the totally ordered abelian group, where r′ < γ < r for all r′ < r if ∗ =<and r′ > γ > r for all r′ > r if ∗ =>. Then

f =∑n

an(T − x)n 7→ max |an|γn ∈ R≥0 × γZ

defines a rank 2 valuation on Cp〈T 〉. One can check that if r /∈ |C∗p|, then allthese are equivalent to the corresponding point of type (3). But if r ∈ |C∗p|this construction gives new points. They are called points of type (5) and theydepend only on D(x,< r) := y ∈ OCp : |y − x| < r if ∗ =< and on D(x, r)if ∗ =>.

One checks that all these valuations are indeed ≤ 1 on OCp〈T 〉. All points exceptthose of type (2) are closed. A point of type (2) has points of type (5) in its closure.

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Lecture 2: Perfectoid spaces and Scholze’s functor

2.1 Scholze’s functor

Let F/Qp be a finite extension. Let D be a division algebra over F with center Fand invariant 1/n. In [8], Scholze constructs a functor

S :

smooth admissible

Fp-representations

of GLn(F )

−→

smooth admissible

Fp-representations

of D∗

π 7−→ S(π).

The space S(π) also carries an action of Gal(F/F ).

Goal of this lecture: Explain his construction.

2.2 Perfectoid spaces

As the construction of S involves the Lubin–Tate space at infinite level, which is aperfectiod space, I will give the definition of a perfectoid space next. For a Huberring R, recall that R denotes the subset of power-bounded elements.

Definition 2.1. A perfectoid field is a non-archimedean field K of residue charac-teristic p > 0, complete with respect to a rank 1 non-discrete valuation such that

Frob : K/$ → K/$, x 7→ xp

is surjective, where $ ∈ K∗ is a topologically nilpotent unit.

Examples: Cp, Qp(ζp∞) and Fp((t1/p∞

)) are examples of perfectoid fields. Non-examples are Qp and Fp.

Fix a perfectoid field K.

Definition 2.2. A perfectoid K-algebra is a Banach algebra R such that R ⊂ Ris open and bounded and

Frob : R/$ → R/$, x 7→ xp

is surjective.

Note that by definition the subring R of a perfectoid algebra R is open andbounded. Therefore R can serve as a ring of definition.

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Definition 2.3. An affinoid perfectoid K-algebra is a Huber pair (R,R+) such thatR is a perfectoid K-algebra.

For an affinoid perfectoid K-algebra, the presheaves OX and O+X are always

sheaves.3 The reason for this is the boundedness of R in R and the presence of atopologically nilpotent unit. Here’s the general picture:

Definition 2.4. 1. A Huber ring is called Tate if it contains a topologically nilpo-tent unit.

2. A Huber ring R is called uniform, if R ⊂ R is bounded.

3. A Huber pair (R,R+) is called stably uniform, if OX(U) is uniform for allrational subsets U ⊆ X = Spa(R,R+).

Exercise: Show that Qp[T ]/(T 2) is not uniform.

Theorem 2.5 ([1, Theorem 7]). If (R,R+) is a stably uniform Huber pair, with RTate, then OSpa(R,R+) is a sheaf.

Theorem 2.6 ([6, Theorem 6.3]). If (R,R+) is affinoid perfectoid, U ⊂ X =Spa(R,R+) rational, then (OX(U),O+

X(U)) is affinoid perfectoid. In particular OXis a sheaf, so X = Spa(R,R+) is an adic space.

As before fix a perfectoid field K.

Definition 2.7. An affinoid perfectoid space over K is an affinoid adic spaceSpa(R,R+) over K with R a perfectoid algebra. A perfectoid space is an adicspace X over a K that has a cover by affinoid perfectoid spaces. Morphisms betweenperfectoid spaces are the morphisms of adic spaces.

One word of caution: If X/K is a perfectoid space and Spa(R,R+) ⊂ X is anaffinoid open, then we don’t know whether R is a perfectoid K-algebra.

2.3 The Lubin–Tate tower

As we recall in this section, the Lubin–Tate tower is a tower Mn of rigid analyticvarieties parametrizing deformations of a p-divisible group together with some levelstructure. The inverse limit lim←−Mn can be equipped with the structure of a perfec-toid space.

Let F = Qp, k = Fp and fix a connected p-divisible group H/k of dimension oneand height n. Then D = End(H) is a division algebra with center Qp and invariant1/n.

3It suffices to check the sheaf property for OX , as if OX is a sheaf, then this implies that O+X is

also a sheaf.

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Definition 2.8. Let NilpW (k) be the category of W (k)-algebras R in which p isnilpotent. A deformation of H to R ∈ NilpW (k) is a pair (G, ρ) where G is a p-divisible group over R and

ρ : H ⊗k R/p→ G⊗R R/p

is a quasi-isogeny.

Let DefH be the functor

DefH : NilpW (k) → Sets

R 7→ (G, ρ) : deformation of H to R/∼=.

Theorem 2.9 ([5]). The functor DefH is representable by a formal scheme M/W (k).We have a decomposition

M ∼=⊔i∈Z

M(i)

according to the height i of the quasi-isogeny and non-canonically

M(i) ∼= Spf(W (k)[[t1, . . . , tn−1]]).

Let M0 := Madη ×Qp Cp be the (base change to Cp of the) adic generic fibre of

M. One can introduce level structures to get spaces Mm for any integer m ≥ 0, aswell as finite etale maps Mm →M0.

The tower (Mm)m can be equipped with the structure of a perfectoid space. Inorder to express this we need the following notion.

Definition 2.10. [cf. Def. 2.4.1 in [9]] Let (Xm)m∈I be a cofiltered inverse systemof adic spaces with finite etale transition maps. Let X be an adic space. Write

X ∼ lim←−Xm

if there exists a compatible family of maps X → Xm such that

• the map of underlying topological spaces |X| → lim←−|Xm| is a homeomorphismand

• there is an open cover of X by affinoid subsets Spa(R,R+) ⊂ X such that themap

lim−→Spa(Rm,R

+m)⊂Xm

Rm → R

has dense image, where the direct limit runs over all open affinoids Spa(Rm, R+m) ⊂

Xm over which Spa(R,R+) ⊂ X → Xm factors.

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Theorem 2.11 ([9, 6.3.4]). There exists a unique up to unique isomorphism per-fectoid space M∞ over Cp such that

M∞ ∼ lim←−Mm.

The infinite-level Lubin–Tate space M∞ represents the functor from completeaffinoid (W (k)[1

p ],W (k))-algebras to Sets which sends (R,R+) to the set of triples

(G, ρ, α) where (G, ρ) ∈M0(R,R+) and

α : Z2p → T (G)adη (R,R+)

is a morphism of Zp-modules such that for all points x = Spa(K,K+) ∈ Spa(R,R+),the induced map

α(x) : Z2p → T (G)adη (K,K+)

is an isomorphism, cf. [9, Section 6.3]. Here T (G) is the integral Tate module,i.e., the sheaf on NilpopR defined as T (G)(S) = lim←−G[pn](S), and T (G)adη is the adicgeneric fibre (over Spa(Qp,Zp)).

The infinite-level Lubin–Tate spaceM∞ has commuting actions of the group D∗

(which acts via its action on H), of the group GLn(Qp) and of the Weil group WQp .We have a decomposition into perfectoid spaces

M∞ ∼=⊔i∈ZM(i)∞

at infinite level and for each i ∈ Z the stabilizer StabGLn(Qp)(M(i)∞ ) is given by the

subgroupG′ := g ∈ GLn(Qp)| det(g) ∈ Z∗p.

If F/Qp is a finite extension we have similar deformation spaces of p-divisiblegroups with anOF -action, and in the limit again a perfectoid spaceM∞ with actionsof GLn(F ), D∗/F and WF .

Remark 2.12. The classical local Langlands correspondence and the classical Jacquet–Langlands correspondence can be realized in the l-adic cohomology of the Lubin–Tatetower. As we would like to understand mod p correspondences, it seems that it couldbe a good idea to study the mod p cohomology ofM∞. Unfortunately the cohomologygroups H i

et(M∞,Fp) are not at all well behaved, e.g., they depend on the choice of

a complete algebraic closure of F and they are not admissible (see [2]).

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In order to get a candidate for a mod p local Langlands correspondence and amod p Jacquet–Langlands correspondence, Scholze uses the Gross–Hopkins periodmorphism

πGH :M∞ → Pn−1Cp .

• This is a surjective map.

• It is GLn(F ) equivariant (where the action on Pn−1Cp is the trivial action).

• It is also equivariant for the D∗ action (via the natural action of D∗(F ) ∼=GLn(F ) on Pn−1

F×F C∗p)

• It factors through a corresponding map at all finite levels πGH,m :Mm → Pn−1Cp

and all these are etale covering maps.

Note that the existence of πGH implies that the analytic projective space Pn−1Cp is

very far from being simply connected.

2.4 Definition of Scholze’s functor

We now have all the prerequisites to define Scholze’s functor S.Let π be an admissible smooth representation of GLn(F ) on an Fp-vector space.

For an etale map U → Pn−1Cp define

Fπ(U) := Mapcont,GLn(F )(|U ×Pn−1 M∞|, π).

Here one turns the right action of GLn(F ) on M∞ into a left action and then thesubscript GLn(F ) means GLn(F )-equivariant maps.

By [8, Proposition 3.1], this defines a sheaf Fπ on (Pn−1Cp )et. Moreover the functor

that sends π to Fπ is an exact functor.

Theorem 2.13 ([8, Theorem 1.1]). For any admissible smooth representation π ofGLn(F ), the cohomology groups

Si(π) := H iet(P

n−1Cp ,Fπ), i ≥ 0,

are admissible D∗-representations. They carry an action of Gal(F/F ) and theyvanish for all i > 2(n− 1).

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Lecture 3: The case of a principal series representationof GL2(Qp)

In this lecture we study Scholze’s functor in the case where F = Qp, n = 2 and π isa principal series representation.

Recall that we have an exact functorsmooth admissible

Fp-representations

of GL2(Qp)

−→

sheaves on (P1Cp)et

π 7−→ Fπ.

The cohomology groups Si(π) := H iet(P1

Cp ,Fπ) are admissible D∗-representations

which carry a continuous action of Gal(Qp/Qp). Furthermore Si(π) = 0 for alli > 2.

One can always easily compute S0(π). For a representation π of GL2(Qp) letπSL2(Qp) denote the space of vectors that are invariant under SL2(Qp). This is asubrepresentation of π.

Proposition 3.1 ([8, Prop. 4.7]). The natural map

H0et(P1

Cp ,FπSL2(Qp)) → H0et(P1

Cp ,Fπ)

is an isomorphism.

Note that if π is irreducible and πSL2(Qp) 6= 0, then π is finite-dimensional. Butthe only irreducible finite-dimensional smooth representations of GL2(Qp) are theone-dimensional representations, so we see that

S0(π) = 0,

whenever π is irreducible and not one-dimensional.

We now want to study S2(π) when π is a principal series representation. Forthat let B(Qp) ⊂ GL2(Qp) denote the Borel subgroup of upper triangular matrices.Let q = pm for some m ≥ 1. Let χi : Q∗p → F∗q , i = 1, 2 be two smooth characters.They give rise to a character χ = (χ1, χ2) of the Borel subgroup via

b =

(b1 ∗

b2

)7→ χ1(b1)χ2(b2).

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The principal series representation IndGL2(Qp)B(Qp) (χ) of GL2(Qp) is defined as

IndGL2(Qp)B(Qp) (χ) :=

f : GL2(Qp)→ Fq :

f locally constant andf(bg) = χ(b)f(g)∀b ∈ B(Qp), g ∈ GL2(Qp)

,

where GL2(Qp) acts via right translation. The representation IndGL2(Qp)B(Qp) (χ) is irre-

ducible if and only if χ1 6= χ2.In the rest of this lecture we’ll explain some ideas behind the proof of the following

theorem.

Theorem 3.2 ([4, Theorem 4.6]). Let π := IndGL2(Qp)B(Qp) (χ) be a principal series

representation, where χ = (χ1, χ2), and χi : Q∗p → F∗q are smooth characters. ThenS2(π) = 0.

Remark 3.3. The theorem and Proposition 3.1 imply that, for χ1 6= χ2, the coho-

mology Si(IndGL2(Qp)B(Qp) (χ)) is concentrated in the middle degree.

The proof of the theorem can be divided into two steps.

• Step 1: One constructs a quotient M∞/B(Qp) in the category of perfectoidspaces over Cp.

• Step 2: The Gross–Hopkins period morphism πGH factors through the quo-tient M∞/B(Qp). Let πGH : M∞/B(Qp) → P1

Cp be the induced map. Thismap has many nice properties, in particular, it is quasi-compact. One canshow that Fπ ∼= πGH,∗Fχ for a finite rank local system Fχ on M∞/B(Qp),which is defined in terms of χ. Then one uses that M∞/B(Qp) is perfectoidand the nice properties of πGH to deduce vanishing of S2(π).

Below, we will focus on explaining the ideas behind the construction of thequotient in Step 1.

For Step 2 let us just remark that once one has constructed the perfectoid spaceM∞/B(Qp) one defines the local system Fχ on M∞/B(Qp) as follows.

Let χ = (χ1, χ2), χi : Q∗p → F∗q , be a pair of smooth characters considered as a

representation of the Borel subgroup B(Qp). For Uet→M∞/B(Qp) let

Fχ(U) := Mapcont,B(Qp)(|U ×M∞/B(Qp)M∞, χ).

This defines an etale local system of rank 1 over Fq on M∞/B(Qp).The Gross–Hopkins period morphism πGH : M∞ → P1

Cp factors through the

quotient M∞/B(Qp) and we get an induced map

πGH :M∞/B(Qp)→ P1Cp ,

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which turns out to be quasi-compact.One can show that πGH,∗Fχ = Fπ and that the higher direct images all vanish,

RiπGH,∗Fχ = 0, ∀i > 0.

HenceSi(

IndGL2(Qp)B(Qp) (χ)

)= H i

et(M∞/B(Qp),Fχ).

We see that at the expense of a more complicated space, the sheaf has simplifieda great deal. And although the space is more complicated, it is perfectoid. Thisfact and the good properties of πGH make it possible to show vanishing of, well, notdirectly of H2

et(M∞/B(Qp),Fχ), but instead of H2et(M∞/B(Qp),Fχ ⊗ O+/p). It

turns out that this suffices to deduce vanishing of S2(

IndGL2(Qp)B(Qp) (χ)

). For details

regading these cohomological calculations we refer to [4].

3.1 The quotient M∞/B(Qp)

First, let us mention a general result on quotients.

Proposition 3.4. Let X be a rigid analytic space (resp. a perfectoid space definedover a perfectoid field of characteristic zero). Let G be a finite group which acts on X.Assume that X has a cover by G-stable affinoid open subsets Spa(R,R+). Then X/Gexists as a rigid space (resp. a perfectoid space). The quotient Spa(R,R+)/G is

affinoid and given by Spa(RG, R+G).

Proof. See Theorem 1.1, Theorem 1.2 and Theorem 1.4 in [3].

The spaceM∞/B(Qp) is constructed in two steps. Recall thatM∞ =⊔iM

(i)∞ .

First one constructs the quotient M(0)∞ /B(Zp). In a second step one constructs

M∞/B(Qp) from M(0)∞ /B(Zp).

Theorem 3.5 ([4, Theorem 3.4]). There exists a perfectoid space M(0)∞ /B(Zp) to-

gether with compatible maps

M(0)∞ /B(Zp)→M(0)

m /B(Z/pmZ)

such thatM(0)∞ /B(Zp) ∼ lim←−

m

M(0)m /B(Z/pmZ).

The spaceM(0)∞ /B(Zp) is the quotient ofM(0)

∞ by B(Zp) in the category of perfectoidspaces.

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The space M(0)∞ /B(Zp) is constructed using ∞-level modular curves. In [7],

Scholze has shown that the ∞-level modular curve

X ∗Γ(p∞) ∼ lim←−m

X ∗Γ(pm)

is a perfectoid space. It has an action of GL2(Qp) and there is a GL2(Qp)-equivariantperiod map, the so called Hodge–Tate period morphism

πHT : X ∗Γ(p∞) → P1Cp

where the action of GL2(Qp) on P1Cp is the usual left action turned into a right

action, i.e., it is given by

[x0 : x1]

(a bc d

)= [dx0 − bx1 : −cx0 + ax1].

We have a decomposition

X ∗Γ(p∞) = X ordΓ(p∞) t X

ssΓ(p∞)

= π−1HT(P1(Qp)) t π−1

HT(Ω2),

where Ω2 denotes the Drinfeld upper half plane Ω2 = P1\P1(Qp).The Lubin–Tate tower sits inside the ∞-level modular curve, in fact

X ssΓ(p∞) =⊔

finite

M(0)∞ .

Fix an embedding ι :M(0)∞ → X ssΓ(p∞).

Furthermore “part of the tower(X ∗Γ0(pm)

)m

” is perfectoid. More precisely there

is a family of open subspaces4

X ∗Γ0(pm)(ε)a ⊂ X∗Γ0(pm), 0 ≤ ε < 1/2,

that in the limit give a perfectoid space

X ∗Γ0(p∞)(ε)a ∼ lim←−X∗Γ0(pm)(ε)a.

There is a corresponding open X ∗Γ(p∞)(ε)a ⊂ X∗Γ(p∞) and an open map

φ : X ∗Γ(p∞)(ε)a → X∗Γ0(p∞)(ε)a. (1)

4Roughly speaking: X ∗Γ0(pm)(ε)a is the locus where the subgroup is anticanonical and the Hasse

invariant satisfies |Ha| ≥ |p|p−mε.

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In fact (and very importantly for us) the spaces X ∗Γ(p∞)(ε)a and X ∗Γ0(p∞)(ε)a areaffinoid perfectoid. Finally

M∞ ⊂∞⋃i=0

X ∗Γ(p∞)(ε)a · γi,

where γ =

(p

p−1

)∈ GL2(Qp). We can build a candidate for the quotient

M(0)∞ /B(Zp) using the open map φ from (1).Before we explain this, let us mention a technical annoyance regarding inverse

limits of adic spaces: When X is an adic space, which is a limit of a cofiltered system(Xm)m by which we mean that

X ∼ lim←−Xm,

then unfortunately we do not know whether an arbitrary affinoid open

U = Spa(R,R+) ⊂ X

is the preimage p−1m (Um) of an open affinoid subset Um = Spa(Rm, R

+m) ⊂ Xm for

sufficiently large m and if it satisfies R+ ∼= (lim−→R+m)∧. For this reason some of the

arguments below become slightly technical and to facilitate notation we introducethe following notion.

Definition 3.6. Let (K,OK) be a non-archimedean field. Let

(Xm = Spa(Rm, R+m))m∈I

be a cofiltered inverse system of affinoid adic spaces over Spa(K,OK) with finiteetale transition maps. Assume X = Spa(R,R+) is an affinoid adic space overSpa(K,OK) with a compatible family of maps pm : X → Xm. We write

X ≈ lim←−Xm

if R+ is the $-adic completion of lim−→mR+m.

Obviously if X ≈ lim←−Xm then also X ∼ lim←−Xm.

In order to construct the quotientM(0)∞ /B(Zp) we first show that every point in

M(0)∞ has a nice affinoid perfectoid neighbourhood. For that we use the geometry of

the Hodge–Tate period morphism. Let D1 ⊂ P1 be the closed unit disc embeddedin P1 via x 7→ [x : 1] in usual homogeneous coordinates. Then it is shown in [7] thatthe preimage π−1

HT(D1) is affinoid perfectoid and that there exists 0 < ε < 1/2 suchthat X ∗Γ(p∞)(ε)a ⊂ π

−1HT(D1). Fix such an ε and define Y := X ∗Γ(p∞)(ε)a.

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Proposition 3.7. Let i ≥ 0 be an integer and let x ∈ |M(0)∞ ∩Y γi| be a point. Then

there exists an open neighbourhood U of x in M(0)∞ , such that

• U is affinoid perfectoid,

• U ⊂ Y γi,

• U is invariant under γ−iB(Zp)γi,

• U ≈ lim←−Um for affinoid open subsets Um ⊂M(0)m and m large enough.

Proof. We sketch the main ideas of the proof; for details see [4, Prop. 3.7]. Foran integer n ≥ 1, let z1, . . . , zpn ∈ Zp be a set of representatives of Zp/pnZp andconsider the rational subset Xn ⊂ D1 defined by

Xn(Cp) = x ∈ D1(Cp) : |x− zi| ≥ p−n for all i = 1, . . . , pn= x ∈ D1(Cp) : |x− z| ≥ p−n for all z ∈ Zp.

It is well known that (Xn)n∈N is a cover of Ω2∩D1. For any n ≥ 0, the affinoid openXn ⊂ P1 is stable under B(Zp). Indeed let g =

(a bd

)∈ B(Zp) and x ∈ Xn(Cp), then

|x− z| ≥ p−n, ∀ z ∈ Zp

and therefore

|xγ − z| = |(dx− b)/a− z|= |dx− b− az| = |x− (az − b)/d|≥ p−n.

Therefore π−1HT(Xn) ⊂ π−1

HT(D1) is also B(Zp)-stable. By the results of [7] it is alsoaffinoid perfectoid. Then the intersection

U := π−1HT(Xn) ∩ Y ∩M(0)

∞

is still affinoid perfectoid and gives the neighbourhood we want in the case i = 0,i.e., x ∈ |Y |.

For x ∈ |Y γi|, i 6= 0, one easily checks that the translates Xnγi are invariant

under the group

γ−iB(Zp)γi :=

(a b

d

)∈ B(Qp) : a, d ∈ Z∗p, b ∈ p−2iZp

.

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The property of being affinoid perfectoid is GL2(Qp)-invariant. One can then checkthat

U := π−1HT(Xnγ

i) ∩ Y γi ∩M(0)∞

gives an affinoid perfectoid neighbourhood of x ∈ Y γi, invariant under γ−iB(Zp)γi.For the “tower property”, i.e., the last bullet point we refer to the proof of Propo-sition 3.7 in [4].

Sketch of proof of Theorem 3.5.Let x ∈ |M∞| be a point. Assume it is contained in |Y |. Choose U as in the

proposition. Recall that we have the open map φ : Y → X ∗Γ0(p∞)(ε)a. Define

U/B(Zp) := φ(U) ⊂ X ∗Γ0(p∞)(ε)a.

This is a perfectoid space and indeed the quotient of U by B(Zp). If x ∈ |Y γi|,and U ≈ lim←−Um is a good neighbourhood of x as in the last proposition, consider

φ(Uγ−i). We may assume Uγ−i ≈ lim←−Wm for Wm ⊂ M(0)m and m large enough.

One then shows that φ(Uγ−i) ∼ lim←−Wm/B(Z/pmZ). This is not quite the space wewant though.

Define

Hi := γiB(Zp)γ−i =

(a b

d

)∈ B(Zp) | b ∈ p2iZp

⊂ B(Zp)

and letHi,m ⊂ B(Z/pmZ)

be the image of Hi under the natural reduction map B(Zp)→ B(Z/pmZ). Considerthe tower (Wm/Hi,m)m≥2i of affinoid adic spaces

Wm/Hi,m = Spa(Rm, R+m).

Then the natural maps

Wm/Hi,m →Wm/B(Z/pmZ)

are finite etale maps and the degree does not change for m large enough as

B(Zp)/Hi → B(Z/pmZ)/Hi,m

is a bijection. In fact the diagram

Wm+1/Hi,m+1

// Wm+1/B(Z/pm+1Z)

Wm/Hi,m// Wm/B(Z/pmZ)

19

is cartesian. The pullback

W/Hi := W2i/Hi,2i ×W2i/B(Z/p2iZ) φ(Uγ−i)→ φ(Uγ−i)

is finite etale, therefore affinoid perfectoid say Uγ−i/Hi = Spa(R,R+). Define

U/B(Zp) := Spa(R,R+).

One now verifiesU/B(Zp) ≈ lim←−

m

Um/B(Z/pmZ)

by checking that the towers (Um/B(Z/pmZ))m≥4i and (Wm/Hi,m)m≥4i are equiva-

lent, i.e., that the systems agree in (M(0)0 )proet.

Finally one glues the spaces U/B(Zp) and verifies that the resulting space is

indeed the quotient M(0)∞ /B(Zp) in the category of perfectoid spaces.

To get from M(0)∞ /B(Zp) to a perfectoid space M∞/B(Qp) we use the decom-

position M∞ =⊔M(i)∞ to first reduce to the construction of

M(0)∞ /B′, B′ = b ∈ B(Qp)| det(b) ∈ Z∗p.

This latter space one can construct using the Gross–Hopkins period map at levelzero, see [4, Section 3.6].

References

[1] Kevin Buzzard and Alain Verberkmoes. Stably uniform affinoids are sheafy.arXiv: 1404.7020v2, 2015.

[2] Przemys law Chojecki. On mod p non-abelian Lubin-Tate theory for GL2(Qp).Compos. Math., 151(8):1433–1461, 2015.

[3] David Hansen. Quotients of adic spaces. to appear in Math. Res. Letters, 2016.http://www.math.columbia.edu/~hansen/overcon.pdf.

[4] Judith Ludwig. A quotient of the Lubin–Tate tower. arXiv:1611.03685v1, 2016.

[5] M. Rapoport and Th. Zink. Period spaces for p-divisible groups, volume 141of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ,1996.

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[6] Peter Scholze. Perfectoid spaces. Publ. Math. Inst. Hautes Etudes Sci., 116:245–313, 2012.

[7] Peter Scholze. On torsion in the cohomology of locally symmetric spaces.arXiv:1306.2070, 2013.

[8] Peter Scholze. On the p-adic cohomology of the Lubin–Tate tower.arXiv:1506.04022, 2015.

[9] Peter Scholze and Jared Weinstein. Moduli of p-divisible groups. Camb. J.Math., 1(2):145–237, 2013.

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