Affinoids in the Lubin-Tate perfectoid space and simple supercuspidal representations II: wild case

We construct a family of affinoids in the Lubin-Tate perfectoid space and their formal models such that the middle cohomology of their reductions realizes the local Langlands correspondence and the local Jacquet-Langlands correspondence for the simple supercuspidal representations. The reductions of the formal models are isomorphic to the perfections of some Artin-Schreier varieties, whose cohomology realizes primitive Galois representations. We show also the Tate conjecture for Artin-Schreier varieties associated to quadratic forms.


Introduction
Let K be a non-archimedean local field with residue field k. Let p be the characteristic of k. We write O K for the ring of integers of K, and p for the maximal ideal of O K . We fix an algebraic closure k ac of k. The Lubin-Tate spaces are deformation spaces of the one-dimensional formal O K -module over k ac of height n with level structures. We take a prime number ℓ that is different from p. The local Langlands correspondence (LLC) and the local Jacquet-Langlands correspondence (LJLC) for supercuspidal representations of GL n are realized in the ℓ-adic cohomology of Lubin-Tate spaces. This is proved in [Boy99] and [HT01] by global automorphic arguments. On the other hand, the relation between these correspondences and the geometry of Lubin-Tate spaces is not well understood.
In this direction, Yoshida constructs a semi-stable model of the Lubin-Tate space with a full level p-structure, and studies its relation with the LLC in [Yos10]. In this case, the Deligne-Lusztig varieties appear as open subschemes in the reductions of the semi-stable models, and their cohomology realizes the LLC for depth zero supercuspidal representations. In [BW16], Boyarchenko-Weinstein construct a family of affinoids in the Lubin-Tate perfectoid space and their formal models so that the cohomology of the reductions realizes the LLC and the LJLC for some representations which are related to unramified extensions of K (cf. [Wei10] for some special case at a finite level). It generalizes a part of the result in [Yos10] to higher conductor cases. In the Lubin-Tate perfectoid setting, the authors study the case for the essentially tame simple supercuspidal representations in [IT13], where simple supercuspidal means that the exponential Swan conductor is equal to one. See [BH05] for the notion of essentially tame representations. The result in [IT13] is generalized to some higher conductor essentially tame cases by Tokimoto in [Tok20] (cf. [IT17a] for some special case at a finite level).
In all the above cases, Langlands parameters are of the form Ind W K W L χ for a finite separable extension L over K and a character χ of W L , where W K and W L denote the Weil groups of K and L respectively. Further, the construction of affinoids directly involves CM points which have multiplication by L. In this paper, we study the case for simple supercuspidal representations which are not essentially tame. In this case, the Langlands parameters can not be written as inductions of characters. Hence, we have no canonical candidate of CM points which may be used for constructions of affinoids.
We will explain our main result. All the representation are essentially tame if n is prime to p. Hence, we assume that p divides n. We say that a representation of GL n (K) is essentially simple supercuspidal if it is a character twist of a simple supercuspidal representation. Let q be the number of the elements of k and D be the central division algebra over K of invariant 1/n. We write q = p f and n = p e n ′ , where n ′ is prime to p. We put m = gcd(e, f ). The main theorem is the following: Theorem. For r ∈ µ q−1 (K), there is an affinoid X r in the Lubin-Tate perfectoid space and its formal model X r such that • the special fiber X r of X r is isomorphic to the perfection of the affine smooth variety defined by • the stabilizer H r ⊂ GL n (K) × D × × W K of X r naturally acts on X r , and • c-Ind GLn(K)×D × ×W K Hr H n−1 c (X r , Q ℓ ) realizes the LLC and the LJLC for essentially simple supercuspidal representations.
See Theorem 2.6 and Theorem 6.5 for precise statements. As we mentioned, we have no candidate of CM points for the construction of affinoids. First, we consider a CM point ξ which has multiplication by a field extension of K obtained by adding an n-th root of a uniformizer of K. If we imitate the construction of affinoids in [IT13] using the CM point ξ, we can get a nontrivial affinoid and its model, but the reduction degenerates in some sense, and the cohomology of the reduction does not give a supercuspidal representation. What we will do in this paper is to modify the CM point ξ using information of field extensions which appear in the study of our simple supercuspidal Langlands parameter. The modified point, which is constructed in Proposition 2.2, is no longer a CM point, but we can use this point for a construction of a desired affinoid. Since the modification comes from the study of the Langlands parameter, we expect that such constructions work also for other Langlands parameters.
In the above mentioned preceding researches, the Langlands parameters are inductions of characters, and realized from commutative group actions on varieties. In the case for Deligne-Lusztig varieties, they come from the natural action of tori. In our simple supercuspidal case, they come from non-commutative group actions. For example, the restriction to the inertia subgroup of a simple supercuspidal Langlands parameter factors through a semidirect product of a cyclic group with a Heisenberg type group, which acts on our Artin-Schreier variety in a very non-trivial way.
In the following, we briefly explain the content of each section. In Section 1, we collect known results on the Lubin-Tate perfectoid space, its formal model and group action on it.
In Section 2, we construct a family of affinoids and their formal models. Further we determine the reductions of them. The reduction is isomorphic to the perfection of some Artin-Schreier variety.
In Section 3, we describe the group action on the reductions. In Section 4, we show that the Tate conjecture holds for Artin-Schreier varieties of associated to quadratic forms. Further, we study the action of some special element on cycle classes in the etale cohomology of the Artin-Schreier variety. This becomes a key ingredient for the proof of the main theorem.
In Section 5, we give an explicit description of the LLC and the LJLC for essentially simple supercuspidal representations, which follows from results in [IT15] and [IT18b]. In Section 6, we give a geometric realization of the LLC and the LJLC in the cohomology of our reduction.

Acknowledgements
The authors would like to thank a referee for helpful comments and suggestions. This work was supported by JSPS KAKENHI Grant Numbers 26707003, 15K17506, 18H01109, 20K03529.

Notation
For a non-archimedean valuation field F , its valuation ring is denoted by O F . For a nonarchimedean valuation field F and an element a ∈ O F , its image in the residue field is denoted byā. For a ∈ Q and elements f , g with valuation v that takes values in Q, For a topological field extension E over F , let Gal(E/F ) denote the group of the continuous automorphisms of E over F . For an ideal I of a topological ring, let I − denote the closure of I.
1 Lubin-Tate perfectoid space 1.1 Lubin-Tate perfectoid space and its formal model Let K be a non-archimedean local field with residue field k of characteristic p. Let q be the number of the elements of k. We write p for the maximal ideal of O K . We fix an algebraic closure K ac of K. Let k ac be the residue field of K ac .
Let n be a positive integer. We take a one-dimensional formal O K -module G 0 over k ac of height n, which is unique up to isomorphism. Let K ur be the maximal unramified extension of K in K ac . We write K ur for the completion of K ur . Let {Spf A m } m≥0 be the tower of Lubin-Tate formal schemes defined by Drinfeld level p m -structure as explained in [IT13, §1.1]. Note that the generic fibers of these formal schemes are connected components of usual Lubin-Tate spaces. Let I the ideal of lim − → A m generated by the maximal ideal of A 0 . Let A be the I-adic completion of lim − → A m . We put M G 0 ,∞ = Spf A. Let K ab be the maximal abelian extension of K in K ac . We write K ab for the completion of K ab . Let ∧G 0 denote the one-dimensional formal O K -module over k ac of height one. Then we have M ∧G 0 ,∞ ≃ Spf O K ab by the Lubin-Tate theory. We have a determinant morphism by [Wei16, 2.5 and 2.7] (cf. [Hed10]). Then, we have the ring homomorphism O K ab → A determined by (1.1). We fix a uniformizer ̟ of K. Let M ∞ be the open adic subspace of Spa(A, A) defined by |̟(x)| = 0 (cf. [Hub94,2]). We regard M ∞ as an adic space over K ur . Let C be the completion of K ac . For a deformation G of G 0 over O C , we put We put η = Spa( K ab , O K ab ). By the ring homomorphism O K ab → A, we can regard M ∞ as an adic space over η, for which we write M ∞,η . We putη = Spa(C, O C ) and M ∞,η = M ∞,η × η η. Then, M ∞,η is a perfectoid space over C in the sense of [Sch12, Definition 6.15] by [Wei16, Lemma 2.32]. We call M ∞,η the Lubin-Tate perfectoid space.
In the following, we recall an explicit description of . Let G 0 be the formal O K -module over k ac obtained as the reduction of G 0 . We put O D = End G 0 and D = O D ⊗ O K K, which is the central division algebra over K of invariant 1/n. Let [ · ] denote the action of O D on G 0 . Let ϕ be the element of D such that [ϕ](X) = X q . Let K n be the unramified extension of K of degree n. We consider the K-algebra embedding of K n into D determined by [ζ](X) =ζX for ζ ∈ µ q n −1 (K n ).
Then we have ϕ n = ̟ and ϕζ = ζ q ϕ for ζ ∈ µ q n −1 (K n ). Let ∧G 0 be the one-dimensional We choose a compatible system {t m } m≥1 such that Let v be the normalized valuation of K such that v(̟) = 1. The valuation v naturally extends to a valuation on C, for which we again write v. Note that v(t) = 1/(q − 1). For an integer i ≥ 0, we put Let W K be the Weil group of K. Let Art K : K × ∼ − → W ab K be the Artin reciprocity map normalized such that a uniformizer is sent to a lift of the geometric Frobenius element. We use similar normalizations also for the Artin reciprocity maps for other non-archimedean local fields. Let σ ∈ W K . Let n σ be the image of σ under the composite Let a K : W K → O × K be the homomorphism given by the action of W K on {t m } m≥1 . It induces an isomorphism a K : • the symbol ∧G 0 denotes the sum under the additive operation of ∧G 0 , • we take the sum over n-tuples (m 1 , . . . , m n ) of integers such that m 1 +· · ·+m n = n(n−1)/2 and m i ≡ m j mod n for i = j, • sgn(m 1 , . . . , m n ) is the sign of the permutation on Z/nZ defined by i → m i+1 .
We put For l ≥ 1, we put Then, we have an isomorphism . Let + G 0 and + ∧G 0 be the additive operations for G 0 and ∧G 0 respectively.
For q-th power compatible systems X = (X q −j ) j≥0 and Y = (Y q −j ) j≥0 that take values in O C , we define q-th power compatible systems X + Y , X − Y and XY by the requirement that their j-th components for j ≥ 0 are where we take the sum in the above sense and the index set is the same as (1.3).

Group action on the formal model
We define a group action on the formal scheme M (0) ∞,O C , which is compatible with usual group actions on Lubin-Tate spaces with finite level (cf. [BW16, 2.11]). We put Let G 0 denote the kernel of the following homomorphism: Then, the formal scheme M (0) ∞,O C admits a right action of G 0 . We write down the action. In the sequel, we use the following notation: for a positive integer m, and we simply write a also for the q-th power compatible system (a q −m ) m≥0 .
The symbol G 0 denotes this summation for q-th power compatible systems. First, we define a left action of GL n (K) × D × on the ring for 1 ≤ i ≤ n. Let g ∈ GL n (K). We write g = (a i,j ) 1≤i,j≤n . Then, let g act on the ring B n by (1.7) Now, we give a right action of G 0 on M (0) ∞,O C using (1.6) and (1.7). Let (g, d, 1) ∈ G 0 . We set where α ∈ O × K . This is well-defined, because the equation Remark 1.4. For a ∈ K × , the action of (a, a, 1) ∈ G 0 on M ∞,O C is trivial by the definition.

CM points
We recall the notion of CM points from [BW16, 3.1]. Let L be a finite extension of K of degree n inside C.
We say that a point of M ∞ (C) has CM by L if the corresponding deformation over O C has CM by L.
Let ξ ∈ M ∞ (C) be a point that has CM by L. Let (G, φ, ι) be the triple corresponding to ξ. Then we have embeddings i M,ξ : L → M n (K) and i D,ξ : L → D characterized by the commutative diagrams 2 Good reduction of affinoids

Construction of affinoids
We take a uniformizer ̟ of K. Let r ∈ µ q−1 (K). We put ̟ r = r̟. We take ϕ r ∈ C such that ϕ n r = ̟ r . We apply results in Section 1 replacing ̟ with ̟ r . We put L r = K(ϕ r ). By the ∞,η (C), we write (ξ 1 , . . . , ξ n ) for the coordinate of ξ with respect to (X 1 , . . . , for 1 ≤ i ≤ n and j ≥ 0. Further, we have the following: (1) ξ r has CM by L r .
∞,η,1 as a subspace of D n,perf C by (1.4). We put η r = ξ q−1 r,1 and write η r = (η q −j r ) j≥0 . Note that v(η r ) = 1/n. We write n = p e n ′ with gcd(p, n ′ ) = 1. We assume that e ≥ 1 in the sequel, since the case where e = 0 is already studied in [IT13]. We put We take q-th power compatible systems θ r = (θ q −j r ) j≥0 and λ r = (λ q −j r ) j≥0 in C satisfying We define ξ ′ r ∈ D n,perf .
Hence, we see the claim by Newton's method.
Remark 2.3. The key ingredients for the construction of ξ 0 r are the elements θ r and λ r defined by (2.1). Up to some difference of normalizations, these elements are analogues of β ζ and γ ζ in [IT15, §2.2], which are generators of a field extension used in a construction of a Langlands parameter there.
We take ξ 0 r as in Proposition 2.2. We put (2. 2) The definition of X r is independent of the choice of θ r and λ r . We define B r ⊂ D n,perf C by the same condition (2.2).

Formal models of affinoids
We simply write f (X) for f (X 1 , . . . , X n ), and f (ξ r ) for f (ξ r,1 , . . . , ξ r,n ). We will use the similar notations also for other functions. We put Proof. We put .
We note that (X 1 , . . . , X n ) satisfies the assumption of Lemma 1.3 by the definition of B r and Lemma 2.1 (4). Then we see that using Lemma 1.3 and the definition of δ ′ 0 . The claims follow from this, because We put s i = (x i /x i+1 ) q i (q−1) for 1 ≤ i ≤ n − 1, and s i s −1 n−1 = 1 + Y i for 1 ≤ i ≤ n − 2, s n−1 = 1 + Y n−1 . (2.6) We put m = gcd(e, f ) and (2.7) We put f = m 0 and e = m 1 . We define m 2 , . . . , m N +1 by the Euclidean algorithm as follows: We have We put and define T 2 , . . . , T N by Then we see that inductively by (2.7). We see also that (2.10) Then we have Y ≡ Y n−1 mod > 1 n(p e + 1) (2.11) by (2.9) and (2.10). We define a subaffinoid B ′ r ⊂ B r by v(z) ≥ 0. We choose a square root η 1/2 r = (η q −j /2 r ) j≥0 and a (p e + 1)-st root η 1/(p e +1) r = (η q −j /(p e +1) r ) j≥0 of η r compatibly. We set , . . . , y 1/q ∞ n−2 , z 1/q ∞ . The parameters y, y 1 , . . . , y n−2 , z give the morphism Θ : B ′ r → B. We simply say an analytic function on B for a q-th power compatible system of analytic functions on B. We put Lemma 2.5. The morphism Θ is an isomorphism.
Proof. We will construct the inverse morphism of Θ. We can write Y n−1 and S as analytic functions on B by (2.8), (2.9), (2.10) and (2.12). Then we can write x i /x i+1 as an analytic function on B by (2.6). By (2.4) and (2.5), we have By this equation, we can write x n as an analytic functions on B. Hence, we have the inverse morphism of Θ.
We put δ B (y, y 1 , . . . , y n−2 , z) = (δ| B ′ r ) • Θ −1 equipped with its q j -th root δ q −j B for j ≥ 0. We put Let X r denote the special fiber of X r .
Theorem 2.6. The formal scheme X r is a formal model of X r , and X r is isomorphic to the perfection of the affine smooth variety defined by Proof. Let (X 1 , . . . , X n ) be the coordinate of B r . By Lemma 2.4, we have v(f (X)) ≥ q − 1 nq and v(S) > q − 1 nq . (2.14) We have We have X q n n X 1 by (2.15). We put (2.17) Then we have v(R(X)) > 1/n by Lemma 2.4 and (2.14). The equation δ(X) = δ(ξ 0 r ) is equivalent to (2.19) The equation (2.19) is equivalent to (2.20) We put Then we have v(R 1 (X)) > 1/n. The equation (2.20) is equivalent to The equation (2.21) is equivalent to As a result, δ(X) = δ(ξ L ) is equivalent to (2.22) on B r . By Lemma 2.4 and (2.22), we have v(z) ≥ 0 on X r . This implies X r ⊂ B ′ r . We have the first claim by Lemma 2.5 and the construction of X r . The second claim follows from (2.11) and (2.22).
Remark 2.7. If n = p = 2, then the smooth compactification of the curve over k defined by (2.13) is the supersingular elliptic curve, which appears as an irreducible component of a semi-stable reduction of a one-dimensional Lubin-Tate space in [IT17b] and [IT12].   Let (g, d, 1) ∈ G 0 . We take the integer l such that dϕ −l D,r ∈ O × D . Let (X 1 , . . . , X n ) be the coordinate of X r . Assume v((g, d) · X i ) = v(X i ) for 1 ≤ i ≤ n at some point of X r . Then we have (g, d) ∈ (ϕ M,r , ϕ D,r ) l (I × × O × D ). Proof. This is proved in the same way as [IT13, Lemma 3.1].
We put g r = (ϕ M,r , ϕ D,r , 1) ∈ G. For a ∈ k ac , we simply write a also for the q-th power compatible system (a q −j ) j≥0 .
Let P be the Jacobson radical of the order I, and p D be the maximal ideal of O D . We put Proof. Assume that (g, d) ∈ GL n (K) × D × stabilizes X r . Then we have det(g) = Nrd D/K (d).
We will show that for some integer l by Lemma 3.1, since (g, d) stabilizes X r and we have v(X i ) = 1/(nq i−1 (q − 1)) for 1 ≤ i ≤ n at any point of X r by Lemma 2.1 (4) and (2.2). Further, we may assume that (g, d) ∈ I × × O × D , since we already know that (ϕ M,r , ϕ D,r ) stabilizes X r by Proposition 3.2 (1).
Action of the Weil group We put ϕ ′ r = ϕ p e r and E r = K(ϕ ′ r ). Let σ ∈ W Er in this paragraph. We put a σ = Art −1 Er (σ) and u σ = a σ ϕ ′ r −nσ ∈ O × Er . We take b σ ∈ µ q−1 (K) such thatb p e σ =ū σ ∈ k. We put I be the element defined by a i,i = b σ for 1 ≤ i ≤ n − 1, a n,n = b σ c σ and a i,j = 0 if i = j. We put Then g σ stabilizes each component in (1.5). We choose elements α r , β r , γ r ∈ K ac such that (3.16) For σ ∈ W Er , we set Then we have a r,σ , b r,σ , c r,σ ∈ O C and by (2.1) and (3.16). Let be the group whose multiplication is given by Let Q ⋊ Z be the semidirect product, where l ∈ Z acts on Q by g(a, b, c) → g(a q −l , b q −l , c q −l ). Let (g(a, b, c), l) ∈ Q ⋊ Z act on X r by , (a(y + b p e )) q l , (a p e +1 2 y q l i ) 1≤i≤n−2 . (3.18) We have the surjective homomorphism Proposition 3.4. Let σ ∈ W Er . Then, g σ ∈ G stabilizes X r , and induces the automorphism of X r given by Θ r (σ).
Proof. This follows from [BW16, Lemma 3.1.3]. Note that our action of W K is inverse to that in [BW16].
Lemma 3.7. The action of S r on M ∞,η stabilizes X r , and induces the action on X r .
The group S r normalizes i ξr (L × r ) · (U 1 I × U 1 D ) 1 by Proposition 3.3. We put Then H r acts on X r by Lemma 3.7 and the proof of Proposition 3.3.

Artin-Schreier variety
as F p m -representations. By Poincaré duality, we have an isomorphism ψ as F p m -representations. Let ψ ′ : F p ֒→ Q × ℓ be any non-trivial character. Then, for each ψ ∈ F ∨ p m \ {1}, there exists a unique element ζ ∈ F × p m such that ψ = ψ ′ • p ζ . Hence, we know that as F p m -representations. Therefore, the required assertion follows.
Consider the fibration Let 0 denote the origin of A n 0 . The inverse image π −1 ζ (0) has p connected components. For a ∈ F p , we define Z a ζ to be the connected component of π −1 ζ (0) defined by z ζ = a. We know that each Z a ζ is isomorphic to the affine space of dimension n 0 . Let be the cycle class map.
(2) The cohomology group H N (X N,ζ , Q ℓ (n 0 )) is generated by the cycle classes cl([Z a ζ ]) for a ∈ F p with the relation a∈Fp cl([Z a ζ ]) = 0.
Proof. For 1 ≤ i ≤ n 0 , let U i be the open subscheme of A n 0 defined by the condition that the i-th coordinate is not zero. Then {U i } 1≤i≤n 0 is a covering of A n 0 \ {0}. We can see that π ζ is a trivial affine bundle on each U i by (4.1). Hence the first claim follows. We set U = π −1 ζ (A n 0 \ {0}). We have the long exact sequence and H N (U, Q ℓ ) ≃ H N (A n 0 \ {0}, Q ℓ ) = 0, which follows from the first claim. Therefore, H N (X N,ζ , Q ℓ (n 0 )) is generated by the cycle classes cl([Z a ζ ]) for a ∈ F p . On the other hand, we have a∈Fp cl([Z a ζ ]) = 0, since a∈Fp [Z a ζ ] = 0 in CH n 0 (X N,ζ ). Since dim H N (X N,ζ , Q ℓ (n 0 )) = p − 1 by Lemma 4.2, we obtain the claim. Then the q-th geometric Frobenius in Gal(F q /F q ) acts on H N (X N , Q ℓ ) by q n 0 . Hence [Jan90,7.13 Conjecture (C)] for X N,Fq also follows.

Action on cohomology
In this section, we assume that p = 2. Let n ≥ 4 be an even integer. Let m = gcd(e, f ) as in Subsection 2.2. We consider the affine smooth variety X of dimension n − 2 defined by We take ζ 3 ∈ F q \ {1} such that ζ 3 3 = 1. Then, we define u 1 , . . . , u n−2 by Then the variety X is isomorphic to the affine variety X n−2 defined by where n 0 = (n − 2)/2. For ζ ∈ F × 2 m , we simply write X ζ for the variety X n−2,ζ , which is defined in Subsection 4.1 where N = n − 2. Recall that X ζ has the defining equation For a ∈ F 2 , we consider the other n 0 -dimensional cycle Z ′a ζ in X ζ defined by where ε 1 is defined at (3.2).
Using the above, we can check that Therefore, we obtain in H n−2 (X ζ , Q ℓ (n 0 )) using Lemma 4.3 and Proposition 4.5. Hence, the claim follows from Lemma 4.2 and Lemma 4.3.
Definition 5.4. We say that a smooth irreducible representation of GL n (K) is essentially simple supercuspidal if it is a character twist of a simple supercuspidal representation.
Proof. This follows from Theorem 5.3, because LL and JL are compatible with character twists.