Multiplicities of cohomological automorphic forms on $\mathrm{GL}_2$ and mod $p$ representations of $\mathrm{GL}_2(\mathbb{Q}_p)$

We prove a new upper bound for the dimension of the space of cohomological automorphic forms of fixed level and growing parallel weight on GL2 over a number field which is not totally real, improving the one obtained in [19]. The main tool of the proof is the mod p representation theory of GL2(Qp) as started by Barthel-Livné and Breuil, and developed by Paškūnas.


Introduction
Let F be a finite extension of Q of degree r, and r 1 (resp. 2r 2 ) be the number of real (resp. complex) embeddings. Let F ∞ = F ⊗ Q R, so that GL 2 (F ∞ ) = GL 2 (R) r1 × GL 2 (C) r2 . Let Z ∞ be the centre of GL 2 (F ∞ ), K f be a compact open subgroup of GL 2 (A f ) and let If d = (d 1 , ..., d r1+r2 ) is an (r 1 + r 2 )-tuple of positive even integers, we let S d (K f ) denote the space of cusp forms on X which are of cohomological type with weight d.
In this paper, we are interested in understanding the asymptotic behavior of the dimension of S d (K f ) as d varies and K f fixed. Define When F is totally real, Shimizu [27] proved that for some constant C independent of d. However, if F is not totally real, the actual growth rate of dim C S d (K f ) is still a mystery; see the discussion below when F is quadratic imaginary.
The main result of this paper is the following (see Theorem 6.1 for a slightly general statement). Theorem 1.1. If F is not totally real and d = (d, ..., d) is a parallel weight, then for any fixed K f , we have To compare our result with the previous ones, let us restrict to the case when F is imaginary quadratic. In [13], Finis, Grunewald and Tirao has proven the bounds using base change and the trace formula respectively (the lower bound is conditional on K f ).
In [19], Marshall has improved the upper bound to be while our Theorem 1.1 gives dim S d (K f ) ≪ ǫ d 3/2+ǫ , hence a saving by a power d 1/6 . It worths to point out that such a power saving is quite rare for tempered automorphic forms. Indeed, purely analytic methods, such as the trace formula, only allow to strengthen the trivial bound by a power of log, cf. [13]. We refer to the introduction of [19] for discussion on this point and a collection of known results.
Finally, let us mention that the experimental data of [13] (when F is quadratic imaginary) suggests that the actual growth rate of dim C S d (K f ) is probably d. We hope to return to this problem in future work.
Let us first explain Marshall's proof of the bound (1.1). It consists of two main steps, the first of which is to convert the problem to bounding the dimension of certain group cohomology of Emerton's completed cohomology spaces H j (in mod p coefficients) and the second one is to establish this bound. For the first step, he used the Eichler-Shimura isomorphism, Shapiro's lemma and a fundamental spectral sequence due to Emerton. For the second, he actually proved a bound in a more general setting which applies typically to H j . To make this precise, let us mention a key intermediate result in this step (stated in the simplest version). Let K 1 = 1 + pZ p pZ p pZ p 1 + pZ p , T 1 (p n ) = 1 + pZ p p n Z p p n Z p 1 + pZ p and Z 1 ∼ = 1 + pZ p be the center of K 1 . Also let F be a sufficiently large finite extension of F p . By a careful and involved analysis of the structure of finitely generated torsion modules over the Iwasawa algebra Λ := F[[K 1 /Z 1 ]], Marshall proved the following ( [19,Prop. 5]): if Π is a smooth admissible F-representation of K 1 /Z 1 which is cotorsion 1 , then for any i ≥ 0, Our proof of Theorem 1.1 follows closely the above strategy. Indeed, the first step is identical to Marshall's. Our main innovation is in the second step by improving the bound (1.2). The key observation is that Emerton's completed cohomology is not just an admissible representation of K 1 , but also carries naturally a compatible action of GL 2 (Q p ), which largely narrows the possible shape of H j . Indeed, this is already observed in [19] and used once 2 when deriving (1.1) from (1.2). However, the mod p representation theory of GL 2 (Q p ) developed by , Breuil [4] and Paškūnas [24,25], allows us to make the most of this fact and prove the following result. Theorem 1.2. Let Π be a smooth admissible F-representation of GL 2 (Q p ) with a central character. Assume that Π is admissible and cotorsion. Then for any i ≥ 0, We obtain the bound by using numerous results of the mod p representation theory of GL 2 (Q p ). First, the classification theorems of [2] and [4] allow us to control the dimension of invariants for irreducible π, in which case we prove (1.3) dim F H i (T 1 (p n )/Z 1 , π) ≪ n.
In fact, to do this we also need more refined structure theorems due to Morra [21,22]. Second, the theory of Paškūnas [24] allows us to pass to general admissible cotorsion representations.
To explain this, let us assume moreover that all the Jordan-Hölder factors of Π are isomorphic to a given supersingular irreducible representation π. Paškūnas [24] studied the universal deformation of π ∨ and showed that the universal deformation space is three dimensional. We show that the admissibility and cotorsion condition imposed on Π forces that Π ∨ is a deformation of π ∨ over a one-dimensional space. Knowing this, we deduce easily Theorem 1.2 from (1.3).
We point out that to prove Theorem 1.2 for i ≥ 1 and to generalize it to a finite product of GL 2 (Q p ), we need to solve several complications caused by the additional requirement of carrying an action of GL 2 (Q p ). In doing so, we prove some results which might be of independent interest. We explain these in more detail below.
The first complication comes when we try to prove Theorem 1.2 for higher cohomology degrees. To apply the standard dimension-shifting argument, we need also consider admissible representations Π which are not necessarily cotorsion, that is, the Pontryagin dual Π ∨ has a positive rank over Λ. Using the bound in the torsion case, one is reduced to consider torsionfree Π ∨ . The usual argument (as in [19, §3.2]) uses the existence of morphisms Λ s → Π ∨ and Π ∨ → Λ s with torsion cokernels, where s is the Λ-rank of Π ∨ . However, these are only morphisms of Λ-modules, so the bound for torsion modules does not apply to these cokernels. To solve this issue, we prove that under certain conditions a torsion-free Λ-module which carries a compatible action of GL 2 (Q p ) is actually free. The proof of this fact uses crucially a result of Kohlhaase [18].
To explain the second, we recall the following interesting result of Breuil-Paškūnas [6]: if Π is a smooth admissible F-representation of GL 2 (Q p ) with a central character, then there exists a GL 2 (Q p )-equivariant embedding Π ֒→ Ω, where Ω| GL 2 (Zp) is an injective envelope of Π| GL 2 (Zp) in the category of smooth F-representations of GL 2 (Z p ) with the (fixed) central character. Although this construction works for the group GL 2 (F ) for any local field F , it does not generalize (at least not obviously) to a finite product, say G = GL 2 (Q p ) × · · · × GL 2 (Q p ). This causes an obstacle in generalizing Theorem 1.2 to G. To overcome this we prove, using the theory of Serre weights, a weaker replacement of the construction of Breuil-Paškūnas. Roughly, it says that we may always embed Π into some Ω which, although not necessarily an injective envelope of Π| GL2(Zp) , is an injective object. This statement generalizes to G.
Notation. Throughout the paper, we fix a prime p and a finite extension F over F p taken to be sufficiently large.
Acknowledgement. Our debt to the work of Vytautas Paškūnas and Simon Marshall will be obvious to the reader. We also thank Marshall for his comments on an earlier draft.

Non-commutative Iwasawa algebras
Let G be a p-adic analytic group of dimension d and G 0 be an open compact subgroup of G. We assume G 0 is uniform and pro-p. Let be the Iwasawa algebra of G 0 over F. A finitely generated Λ-module is said to have codimension c if Ext i Λ (M, Λ) = 0 for all i < c and is non-zero for i = c; the codimension of the zero module is defined to be ∞. We denote the codimension by j Λ (M ). If M is non-zero, then j Λ (M ) ≤ d. For n ≥ 0, define inductively G n+1 := G p n [G n , G 0 ] which are normal subgroups of G 0 ; the decreasing chain G 0 ⊇ G 1 ⊇ · · · is called the lower p-series of G 0 , see [1, §2.4]. We have |G n : G n+1 | = p d . With this notation, the utility of the above theorem is the following result (see [9,Thm. 2.3]).
Corollary 2.2. Let M be a finitely generated Λ-module of codimension c. Then for some rational number λ(M ) > 0.
Proof. We assume first that φ is nilpotent, say φ k0 = 0 for some k 0 ≥ 1. Then M admits a finite filtration by φ k (M ) (for k ≤ k 0 ). Since each of the graded pieces is a quotient of M/φ(M ), the assertion follows from (2.1). Now assume that φ is not nilpotent, so by Lemma 2.4 below φ induces an injection φ k0 (M ) → φ k0 (M ) for some k 0 ≫ 1 and the RHS of (2.3) does not depend on the choice of k 0 . The above argument shows that Hence, by (2.1) applied to the short exact sequence 0 That is, by replacing M by φ k0 (M ), we may assume φ is injective and need to show δ Λ (M ) = δ Λ (M/φ(M )) + 1. Indeed, this follows from [14,Lem. A.15].
Lemma 2.4. Let M be a finitely generated Λ-module. Let φ ∈ End Λ (M ) be such that n≥1 φ n (M ) = 0. Then one of the following holds: (i) φ is nilpotent; 3 It would be more natural to impose the condition φ(M ) ⊂ JM . We consider the present one for the following reasons. On the one hand, in practice we do need consider φ such that n≥1 φ n (M ) = 0 but φ(M ) JM . On the other hand, since M is finitely generated, the condition n≥1 φ n (M ) = 0 implies φ n (M ) ⊂ JM for n ≫ 1, see the proof of Lemma 4.15.
Proof. For any k ≥ 1, φ induces a surjective morphism Since Λ is noetherian and M is finitely generated, any ascending chain of submodules of M/φ(M ) is stable, so there exists k 0 ≫ 0 such that Recall that the projective dimension, denoted by pd Λ (M ), is defined to be the length of a minimal projective resolution of M . It is proved in [28,Cor. 6 On the other hand, we also have pd Λ (M ) = pd Λ (M/φ(M )) − 1.
2.1. Torsion vs torsion free. Assume now G 0 is a uniform and pro-p. Then Λ is a noetherian integral domain. Let L be the field of fractions of Λ. If M is a finitely generated Λ-module, then M ⊗ Λ L is a finite dimensional L-vector space, and we define the rank of M to be the dimension of this vector space. We see that rank is additive in short exact sequences and that M has rank 0 if and only if M is torsion.
They are both integral domains. Let L Qp be the field of fractions of Λ Qp . If M is a finitely generated module over Λ Qp , we define its rank as above and the analogous facts hold. For n ≥ 1, let K n = 1+p n Zp p n Zp p n Zp 1+p n Zp . Also let Z 1 := K 1 ∩Z. Since Z 1 is pro-p, any smooth character χ : Z → F × is trivial on Z 1 , so any F-representation of G (resp. K) with a central character can be viewed as a representation of G/Z 1 (resp. K/Z 1 ). Set Since K 1 /Z 1 is uniform (as p > 2) and pro-p, the results in §2 apply to Λ. Note that dim(K 1 /Z 1 ) = 3. To simplify the notation, we write j(·) = j Λ (·), δ(·) = δ Λ (·) and pd(·) = pd Λ (·).
If H is a closed subgroup of G and σ is a smooth representation of H, we denote by Ind G H σ the usual smooth induction. When H is moreover open, we let c-Ind G H σ denote the compact induction, meaning the subspace of Ind G H σ consisting of functions whose support is compact modulo H.
Let ω : Q × p → F × be the mod p cyclotomic character. If H is any group, we write 1 H for the trivial representation of H (over F).
For 0 ≤ r ≤ p−1, let Sym r F 2 denote the standard symmetric power representation of GL 2 (F p ). Up to twist by det m with 0 ≤ m ≤ p − 1, any absolutely irreducible F-representation of GL 2 (F p ) is isomorphic to Sym r F 2 . Inflating to K and letting p 0 0 p act trivially, we may view Sym r F 2 as a representation of KZ. Let I(Sym r F 2 ) := c-Ind G KZ Sym r F 2 denote the compact induction to G. It is well-known that End G (I(Sym r F 2 )) is isomorphic to F[T ] for a certain Hecke operator T ( [2]). For λ ∈ F we define π(r, λ) := I(Sym r F 2 )/(T − λ).
Proof. (i) The first assertion is clear. For the second, we may assume Π is absolutely irreducible. Corollary 2.2 allows us to translate the problem to computing the growth of dim F Π Kn . If Π is non-supersingular, then it is easy, see [22,Prop. 5.3] for a proof. If Π is supersingular, this is first done in [23, Thm. 1.2] and later in [22,Cor. 4.15] (of course, both proofs are based on [4]).
Assume δ(Π ∨ ) = 1. Consider the G-socle filtration with graded pieces soc i Π (i ≥ 1) given by soc 1 Π = soc G Π, soc 2 (Π) = soc G (Π/soc 1 Π), etc. Since Π is admissible, soc i Π is non-zero and of finite length. Since there is no non-trivial extension between two characters, for any two successive pieces soc i Π, soc i+1 Π, (at least) one of them contains an infinite dimensional irreducible representation of G. On the other hand, using Corollary 2.2 and the additivity of λ(·) with respect to short exact sequences, we deduce that the number of irreducible subquotients of Π ∨ which have Gelfand-Kirillov dimension 1 is finite. 5 Putting these together, we see that the socle filtration of Π is finite, hence Π has finite length.
Recall the following result of Kohlhaase.
Recall that a block in Rep F (G) is an equivalence class of irreducible objects in Rep F (G), where τ ∼ π if and only if there exists a series of irreducible representations τ = τ 0 , τ 1 , . . . , τ n = π such that Ext 1 G (τ i , τ i+1 ) = 0 or Ext 1 G (τ i+1 , τ i ) = 0 for each i. 5 Strictly speaking, we also need to know that λ(·) is uniformly bounded below for any infinite dimensional irreducible representation. This can be seen by the result of Morra recalled in (i), or by the general theory of Hilbert polynomials. Proof. See [24,Prop. 5.34].
The following theorem describes the blocks (when p ≥ 5 as we are assuming).
Theorem 3.4. Let π ∈ Rep F (G) be absolutely irreducible and let B be the block in which π lies. Then one of the following holds: Proof. See [24,Prop. 5.42].
Convention: By [24, Lem. 5.10], any smooth irreducible F p -representation of G with a central character is defined over a finite extension of F p . Theorem 3.4 then implies that for a given block B, there is a common field F such that irreducible objects in B are absolutely irreducible. Hereafter, given a finite set of blocks, we take F to be sufficiently large such that irreducible objects in these blocks are absolutely irreducible.
3.2. Projective envelopes. Fix π ∈ Rep F (G) irreducible and let B be the block in which π lies. Let Inj G π be an injective envelope of π in Rep l,fin F (G); the existence is guaranteed by [24,Cor. 2.3]. Let P = P π ∨ := (Inj G π) ∨ ∈ C and E = E π ∨ := End C (P ). Then P is a projective envelope of π ∨ in C and is naturally a left E-module. Since P is indecomposable, Proposition 3.3 implies that (the dual of) every irreducible subquotient of P lies in B. Also, E is a local F-algebra (with residue field F). Paškūnas has computed E and showed in particular that E is commutative, except when B is of type (III) listed in Theorem 3.4; in any case, we denote by R = Z(E) the center of E. Hence E = R except for blocks of type (III). (i) R is naturally isomorphic to the Bernstein center of C B . In particular, R acts on any object in C B and any morphism in C B is R-equivariant.
However, if B is of type (IV), P is not flat over E. This causes quite a bit of complication in the proof of our main result. To solve this, we determine in §3.7 all the Tor-groups Tor E i (F, P ). We state a result which will be used there.
Lemma 3.6. For i ≥ 1, we have Proof. Choose a resolution of F by finite free E-modules: F • → F → 0. Then the homology of F • ⊗ E P computes Tor E i (F, P ). It is clear that Since Hom C (P, −) is exact, this implies as required.
Proposition 3.7. (i) F ⊗ E P (resp. F ⊗ R P ) has finite length in C.
Proof. (i) By definition, F ⊗ E P is characterized as the maximal quotient of P which contains π ∨ with multiplicity one. This object is denoted by Q in [24, §3] and can be described explicitly. If B is of type (I) or (III), Q is just π ∨ . If B is of type (II), it has finite length by [25,Prop. 6.1]. If B is of type (IV), it follows from Proposition 3.30 below in §3.7 where the explicit structure of F ⊗ E P is determined.
To see that F ⊗ R P has finite length, we may assume B is of type (III). Then E is a free R-module of rank 4, so that (ii) the result follows from the explicit description of F ⊗ E P , using Theorem 3.2 and Proposition 3.34 in the case π = 1 G .
3.3. Serre weights. We keep the notation in the previous subsection. Let π ∈ Rep F (G) be irreducible. By a Serre weight of π we mean an isomorphism class of (absolutely) irreducible F-representations of K, say σ, such that Hom K (σ, π) = 0. Denote by D(π) the set of Serre weights of π. The description of D(π) can be deduced from [2] and [4]; see [25, Rem. 6.2] for a summary.
Proof. The statement is trivial if B is of type (I) or (III). For type (II) or type (IV), it is a direct check (using the assumption p ≥ 5), see [25,Rem. 6.2].
Then π ∞ (r, λ, χ) is a locally finite smooth F-representation of G and we have an exact sequence Proposition 3.9. Assume λ = 0. The following statements hold.
(ii) The morphism θ identifies π ∞ (r, λ, χ) with the largest G-stable subspace of Inj G π(r, λ, χ) which is generated by its is the pro-p Iwahori subgroup. In particular, the image of θ does not depend on the choice of θ.
(i) Let σ ∈ Rep F (K) be irreducible. Whenever non-zero, Hom K (P, σ ∨ ) ∨ is a cyclic Emodule. If J σ denotes the annihilator, there exists x / ∈ J σ such that (ii) Let σ = ⊕ σ σ where the sum is taken over all σ such that Hom K (P, σ ∨ ) = 0. Then Proof. (i) If B is not of type (I), the result is a reformulation of Proposition 3.9(iv). If B is of type (I), it is proved in [25, Thm. 6.6, (38)].
(ii) Remark that although E is non-commutative when B is of type (III), E/J σ is commutative by Proposition 3.9, where J σ denotes the annihilator of Hom K (P, σ ∨ ) ∨ . So it makes sense to talk about the Cohen-Macaulayness. That being said, if B is not of type (I), the result follows from Proposition 3.9(iv). If B is of type (I), it is a special case of [ We record a result in the context of commutative algebra which will be used in Section 5.
Lemma 3.12. Let σ ∈ Rep F (K) be irreducible such that Hom K (P, σ ∨ ) is non-zero. View Hom K (P, σ ∨ ) ∨ as an R-module and let J ′ σ be the annihilator. There exist g, h ∈ J ′ σ such that J ′ σ /(g, h) has finite length.
Proof. First note that R = E and J ′ σ = J σ except when B is of type (III). If B is of type (I) or (II), R is isomorphic to a power series ring over F in three variables, so we may even choose g, h such that J σ = (g, h). If B is of type (III), R = Z(E) is isomorphic to a power series ring in three variables and Proposition 3.28 proved in §3.6 below implies that the image of and it is proved in [16,Lem. 3.9] that J σ = (y, z, w) with a suitable choice of variables. It suffices to take g = y − z and h = w.
Then f is injective and P /f P is projective in C(K).

Proof. Consider the exact sequence
Denote by Im(f ) the image of f : P → P . Then f * factors as with α surjective. Since f * is injective by assumption, β is also injective and α is an isomorphism.
Since P is projective in C, it remains projective in C(K) by [12]. Applying Hom K (−, σ ∨ ) ∨ to 0 → Im(f ) → P → P /f P → 0, we get an exact sequence of E-modules: The injectivity of β implies Ext 1 K ( P /f P , σ ∨ ) ∨ = 0. This being true for every irreducible σ ∈ Rep F (K), we deduce that P /f P is projective in C(K).

As a consequence, Im(f ) is also projective in
Since α is an isomorphism, we obtain Hom K (N, σ ∨ ) ∨ = 0. This being true for any σ, we finally obtain N = 0, so f : P → P is injective.
The following result complements [25, Thm. 5.2]. 7 Corollary 3.14. If π ∈ {1 G , Sp}, there exists f ∈ E such that f : P → P is injective and P/f P isomorphic to a projective envelope of Sym 0 F 2 ⊕ Sym p−1 F 2 in C(K). 7 Although our result is stated for mod p coefficients, the p-adic case can be deduced from this by the proof of ] as E-modules. Let f ∈ E be any lifting of S, then we obtain In particular, P/f P is coadmissible and we conclude by Proposition 3.13.

Principal series and deformations.
Recall that T denotes the diagonal torus of G.
If η : T → F × is a smooth character, set π η = Ind G B η (possibly reducible). Let Inj T η be an injective envelope of η in Rep F (T ) and set Π η = Ind G B Inj T η. Then Π η is a locally finite smooth representation of G. It is easy to see that soc G Π η = soc G π η , which we denote by π. So there is a G-equivariant embedding Π η ֒→ Inj G π and by [24,Prop. 7.1] the image does not depend on the choice of the embedding.
By [24, §3.2], (Inj T η) ∨ is isomorphic to the universal deformation of the T -representation η ∨ (with fixed central character), with E η ∨ being the universal deformation ring. In particular, it is flat over E η ∨ . The result follows from this and the definition of M η ∨ .
Proof. For the first surjection, see [24,Prop. 7.1]. Since End G (π(r, λ, χ)) = F, the dual version of (3.1) implies End C (π ∞ (r, λ, χ) Proof. Since M is coadmissible while M η ∨ is not by Corollary 3.16, the kernel of M η ∨ ։ M is non-zero; denote it by N . We claim that Hom C (M η ∨ , N ) = 0. For this it suffices to prove Hom C (P, N ) = 0, because any morphism P → N must factor through P ։ M η ∨ → N , see [24,Prop. 7.1(iii)]. Assume Hom C (P, N ) = 0 for a contradiction. Then π ∨ (recall π := soc G π η ) does not occur in N . This is impossible unless π η is reducible, i.e. π η ∼ = π(p − 1, 1) up to twist. Assuming this, we have π = 1 G and all irreducible subquotients of N are isomorphic to Sp ∨ . In particular, we obtain Hom K (N, (Sym 0 F 2 ) ∨ ) = 0. However, this would imply an isomorphism 3.5. Coadmissible quotients. Keep the notation in the previous subsection. Let M ∈ C be a coadmissible quotient of P = P π ∨ . We set m(M ) := Hom C (P, M ) which is a finitely generated E-module. There is a natural morphism Proposition 3.20. Let M ∈ C be a coadmissible quotient of P = P π ∨ . The following statements hold.
Proof. Let Ker be the kernel of (3.5). By [24, Lem. 2.9] we have so Hom C (P, Ker) = 0 because P is projective in C. This implies that Ker does not admit π ∨ as a subquotient. In particular, if B is of type (I) and (III) of Theorem 3.4, then Ker = 0 and ev is an isomorphism, so both the assertions are trivial. In the rest of the proof, we assume B is of type (II) or (IV).
we have a natural isomorphism of compact E-modules: Therefore it is enough to consider those σ such that Hom K (P, σ ∨ ) = 0. By Corollary 3.10, these are exactly the weights in D(π) if π / ∈ {1 G , Sp} up to twist, and are Assume π / ∈ {1 G , Sp} up to twist. Lemma 3.8 implies that Hom K (Ker, σ ∨ ) = 0 for σ ∈ D(π) because π ∨ does not occur in Ker. Hence, we obtain an isomorphism Since M is coadmissible, they are finite dimensional.
Assume π = 1 G . The above argument (using Lemma 3.8) shows We are left to treat the case σ = Sym p−1 F 2 . However, Proposition 3.9(iv) implies that the E-modules Hom K (P, σ ∨ ) ∨ , with σ ∈ {Sym 0 F 2 , Sym p−1 F 2 }, are naturally isomorphic. So we deduce the result from (3.6). The proof in the case π = Sp is similar.
(ii) It is equivalent to show that Ker is a torsion Λ-module. Since the case of type (II) is similar and simpler, we assume in the rest that B is of type (IV), so that B consists of three irreducible objects and we let π 1 , π 2 be the two other than π. Since Ker is coadmissible by (i) and does not admit π ∨ as a subquotient, we can find s 1 , s 2 ≥ 0 and a surjection P ⊕s1 Let Q 1 (resp. Q 2 ) be the maximal quotient of P π ∨ 1 (resp. P π ∨ 2 ) none of whose irreducible subquotients is isomorphic to π ∨ . Then the above surjection must factor through Q ⊕s1 Hence, it is enough to show that any coadmissible quotient of Q 1 (resp. Q 2 ) is torsion. This follows from the results in [24, §10] as we explain below. Up to twist we may Let us first assume π = π α , so that up to order π 1 = 1 G and π 2 = Sp. We have the following exact sequences (233) in loc. cit. Combining this with Proposition 3.19 implies the assertion.
If π = 1 G , then (up to order) π 1 = Sp and π 2 = π α . The assertion for Q 1 follows from the exact sequence (see [24, (179) The assertion for Q 2 follows from (3.9) together with the following one [24, (235)]. A similar argument works in the case π = Sp. Remark 3.21. (i) The above proof shows that in any case Q i has a finite filtration (in fact of length ≤ 2) with graded pieces being subquotients of M η ∨ .
(ii) If π = 1 G and if we set π 1 = Sp, π 2 = π α , then we have the following description of We record the following consequence of the above proof.
Theorem 3.23. Let π ∈ Rep F (G) be irreducible and M ∈ C be a coadmissible quotient of P = P π ∨ . There exists f ∈ R such that f annihilates M and P/f P is a finite free Λ-module.
Proof. By Proposition 3.20, we may assume M = m(M ) ⊗ E P . The quotient map P ։ M induces a surjective map E ։ m(M ), that is m(M ) is a cyclic E-module. Let a denote the annihilator.
Let σ = ⊕ σ σ where the sum is taken for all the irreducible σ ∈ Rep F (K) such that Since Hom K (P, σ ∨ ) ∨ is a Cohen-Macaulay E-module of dimension 1 by Corollary 3.11 and If B is of type (III), the above argument only gives an element f ∈ E while we need an element in R. However, Corollary 3.27 below shows that f f * ∈ R and verifies the required condition.
3.6. Blocks of type (III). In this subsection, we assume B is of type (III), that is, . Then E is non-commutative. Let R = Z(E) be the center of E. After twisting we assume π ∼ = Ind G B 1 ⊗ ω −1 and that the central character of P is ω (being the one of π ∨ ).
The goal of this subsection is to explain how to pass from E to R, hence complete the proof of Theorem 3.23. To this aim, we need pass to Galois side via a functor of Colmez. We first introduce some notation.
Here µ p ∞ is the group of p-power order roots of unity in Q p and Q ur p is the maximal unramified extension of Q p . We choose a pair of generatorsγ,δ of G ab such that γ → (1 + p, 0) andδ → (1, 1). Then G is a free pro-p group generated by 2 elements γ, δ which lift respectivelyγ,δ. See [26, §2] for details.
• Let R ps,1 denote the universal deformation ring over O (recall O := W (F)) that parameterizes all two-dimensional pseudo-characters of G lifting the trace of the trivial F-representation and having determinant equal to 1. For our purpose, we only need to consider R ps,1 := R ps,1 ⊗ O F. Let T : G → R ps,1 be the associated universal pseudo-character. • Colmez [11] has defined an exact and covariant functor V from the category of smooth, finite length representations of G on F-vector spaces with a central character to the category of continuous finite length representations of G Qp on F-vector spaces. We will use a modified version as in [24, §5.7], denoted byV, which applies to objects in C.
Following [24, (145)], we let (note that in loc. cit. the ring is defined over O and is denoted by R) where J is the closure of the ideal generated by g 2 − T (g)g + 1 for all g ∈ G Qp . One may show that the center of R ′ is equal to R ps,1 and the natural morphism By [24,Cor. 9.25], R ′ is a free R ps,1 -module of rank 4 with a basis given by {1, u, v, t ′ } where t ′ := uv − vu. Using [24, (160)], one checks that Proof. It follows from Lemma [24, Lem. 9.3] and the proof of [24,Cor. 9.27].
We finally obtain the following result which completes the proof of Theorem 3.23.  If P/f P is coadmissible, so are P/(f * )P and P/(f f * )P . In particular, f f * = 0.
Proof. Let σ ∈ Rep F (K) be a weight such that Hom K (P, σ ∨ ) = 0. By Corollary 3.10, this implies σ ∈ D(π). The exact sequence P → P → P/f P → 0 induces Since P/f P is coadmissible by assumption, Hom K (P/f P, σ ∨ ) is finite dimensional. By identifying End E ( Via the isomorphism ϕ : E ∼ = R ′op , we are reduced to determine the image of

3.7.
Blocks of type (IV). In this subsection, we complement some results in the work of Paškūnas [24,25] when B is of type (IV). The notation here are the same as in the previous subsections. In particular, π ∈ Rep F (G) is irreducible of type (IV), and P π ∨ is a projective envelope of π ∨ in C and E π ∨ = End C (P π ∨ ). Note that the rings E π ∨ are naturally isomorphic (to F[[x, y, z, w]]/(xw − yz)) for any π ∈ B (see [24, §10]), so the subscript will be omitted in the rest (while the one of P π ∨ will be kept). Up to twist, we may assume B = {1 G , Sp, π α }.
Proposition 3.30. Let π ∈ B and set Q π ∨ = F ⊗ E P π ∨ . In the following statements, the existence of the extensions is guaranteed by Lemma 3.29.

Tor
where Inj T η denotes an injective envelope of η in Rep F (T ). In the rest we only consider η ∈ {1 T , α}. By Lemma 3.18 there is a natural surjection q : , we may choose the variables such that q : E ։ E η ∨ is given by modulo (z, w).
Proof. First, via Colmez's functor we may identity E with the special fiber of a certain universal Galois pseudo-deformation ring over O := W (F), see [24,Thm. 10.71]. This ring is denoted by R ψ in loc. cit. and we write R ψ for its special fiber. Let r denote the reducible locus of R ψ (see [24, Cor. B.6] for its definition) and r its image in Lemma 3.32. We have Proof. By Lemma 3.31, we have a resolution of E η ∨ by free E-modules: We deduce that Proof. We first observe the following facts: (a) SL 2 (Q p ) acts trivially on Tor E i (F, P π ∨ α ) for i ≥ 1. Indeed, [24,Cor. 10.43] states this for i = 1 but the proof works for all i ≥ 1. This implies that Tor E i (F, P π ∨ α ) is isomorphic to a finite direct sum of 1 ∨ G .
this is a special case of Lemma 3.6. Recall the following exact sequences (3.10). From (3.20) and Lemma 3.32, we obtain a long exact sequence More generally, if N ⊂ m ⊗ E P 1 ∨ G is a subobject in C which is a finite direct sum of Sp ∨ , then (m ⊗ E P 1 ∨ G )/N is a Cohen-Macaulay Λ-module of codimension 2.

Similarly, using Lemma 3.32 the sequence (3.21) induces
Proof. (i) We prove this by induction on the length of m. We may enlarge F so that irreducible subquotients of m (as E-modules) are isomorphic to F. If m = F, we need to prove Q 1 ∨ G is Cohen-Macaulay of codimension 2. By Proposition 3.30(i) and Theorem 3.2, it is enough to prove κ ∨ is Cohen-Macaulay. This follows from [18,Prop. 5.7], which says that κ ∨ is isomorphic to Ext 2 Λ (Sp ∨ , Λ) up to twist (the latter module is naturally equipped with an action of G). Alternatively, we may apply [24, Lem. 10.23] which says that κ ∨ has projective dimension 2, hence is Cohen-Macaulay.
If the length of m is ≥ 2 then let m 1 m be a submodule of length 1 and let m 2 := m/m 1 . We then obtain a long exact sequence (ii) Let N ⊂ m ⊗ E P 1 ∨ G be as in the statement and M be the corresponding quotient. If m = F, then M has the shape as in (3.24), and we conclude in the same way. The general case is proved by induction in a similar way as in (i).

Main result
We keep the notation of Section 3. For n ≥ 1, let The main result of this section is as follows.
Theorem 4.1. Let Π ∈ Rep F (G). Assume that Π is admissible and that Π ∨ is torsion as a Λ-module. Then for any i ≥ 0, we have (for a general finitely generated torsion Λ-module which need not carry a compatible action of G). In Section 6, we use Theorem 4.1 to improve some results in [19].
The rest of the whole section is devoted to the proof of Theorem 4.1 (and its extension to SL 2 (Q p )). The proof is divided into several steps, the first of which is the following. Lemma 4.3. In Theorem 4.1, we may assume that Π is indecomposable and has an irreducible G-socle.
Proof. Let S be the G-socle of Π. Since Π is admissible, S decomposes as a finite direct sum ⊕ r i=1 π i with π i irreducible. For each i, we let Inj G π i be an injective envelope of π i in Rep F (G). The inclusion π 1 ֒→ Π extends to a G-equivariant morphism α 1 : Π → Inj G π 1 . It is clear that Ker(α 1 ) has G-socle isomorphic to ⊕ r i=2 π i and Im(α 1 ) has G-socle π 1 . Continuing this with Ker(π 1 ), we get a finite filtration of Π such that each graded piece, say gr i (Π), has an irreducible G-socle. Since Π ∨ is torsion as a Λ-module if and only if each (gr i (Π)) ∨ is, we obtain the result.
The plan of the proof of Theorem 4.1 is as follows: in §4.1, we prove a bound of the dimension of Π T1(p n ) for Π of finite length; in §4.2 we prove a lemma which allows to control the dimension of invariants from that of representations of lower canonical dimension; we combine these results to prove the theorem for i = 0 in §4.3 and for i ≥ 1 in §4.4.

Irreducible representations.
The following control theorem will play a key role in the proof of Theorem 4.1.
It is clear that we may assume Π is irreducible in Theorem 4.4. Further, by the recall in §3.1, up to twist it is enough to prove the following Theorem 4.5. For any 0 ≤ r ≤ p − 1 and λ ∈ F, we have dim F π(r, λ, 1) T1(p n ) ≪ n.
Proof. Let A = a b c d ∈ K. We have the following facts: (i) if A ∈ K 0 (p), i.e. c ∈ pZ p , we have two subcases: H .
It is easy to check that this is a disjoint union, hence the cardinality of K 0 (p n )\K 0 (p)/H is 1 + (n − 1)(p − 1).
Combining (i) and (ii) the cardinality of K 0 (p n )\K/H is equal to Proposition 4.7. Let n ≥ 1 and σ be a smooth F-representation of K 0 (p n ) of finite dimension d. Let V be a quotient K-representation of Ind K K0(p n ) σ, then dim F V H ≤ 2dpn.
Proof. Let W be the corresponding kernel so that we have an exact sequence hence an equality of dimensions Now note that H ∼ = 1 + pZ p ∼ = Z p is a pro-p group of cohomological dimension 1, so by Lemma 4.8 below we have We are thus reduced to prove the proposition in the special case V = Ind K K0(p n ) σ. Using [2, Lemma 3], it is easy to see that any irreducible smooth F-representation of K 0 (p n ) is onedimensional, so there exists a filtration of σ by sub-representations, of length d, such that all graded pieces are one-dimensional. Hence, we may assume d = 1, in which case the result follows from Lemma 4.6.
Lemma 4.8. Let W be a finite dimensional F-representation of Z p , then Proof. This is clear if dim F W = 1 because then W must be the trivial representation of Z p so that H 1 (Z p , W ) ∼ = Hom(Z p , F) is of dimension 1. The general case is proved by induction on dim F W using the fact that H 2 (Z p , * ) = 0 and that W always contains a one-dimensional sub-representation. Remark 4.9. In the proof of Proposition 4.7, we crucially used the fact that H has cohomological dimension 1. This fact, very special to the group GL 2 (Q p ), is also used in [4] and [23] (but for the unipotent subgroup of B(Z p )).
Moreover, the following properties hold (see [5, §4]): (i) c-Ind G KZ σ| K ∼ = ⊕ n≥0 R n ; (ii) the Hecke operator T | Rn : R n → R n+1 ⊕ R n−1 is the sum of a K-equivariant injection T + : R n ֒→ R n+1 and (for n ≥ 1) a K-equivariant surjection T − : R n ։ R n−1 . (iii) we have an isomorphism of K-representations Denote by π 0 and π 1 the two direct summands of π in (4.5). For all n ≥ 0, we let R n denote the image of R n → π(r, 0, 1). Then R n ⊂ π 0 if n is even, and R n ⊂ π 1 if n is odd.
Lemma 4.10. For all n ≥ 0, we have R n ⊂ R n+2 and dim F R n = (r + 1)p n .
Proof. The inclusion R n ⊂ R n+2 follows from (4.5). Moreover, (4.5) shows that if n is even, then and if n is odd, The result then follows from (4.4).
At this point, we need the following result of Morra. Recall that π = π 0 ⊕ π 1 as Krepresentations.
Moreover, π i is (nearly) uniserial in the following sense: if W 1 , W 2 are two K-stable subspaces of π i such that Proof. See [22,Cor. 4.14,4.15] for the dimension formula. Note that the formula in loc. cit. is for the dimension of π Kn 0 ⊕ π Kn 1 . The second statement follows from [21, Thm. 1.1] which describes the K-socle filtration of π i . To explain this, fix i ∈ {0, 1}. By [21, Thm. 1.1], π i admits a filtration Fil k π i , k ≥ 0 such that for suitable characters χ k : B(F p ) → F × . In particular, the graded pieces have dimension p + 1 except for the first. Moreover, the filtration satisfies the property that for any K-stable subspace W ⊂ π i and any k ≤ k ′ , the condition dim Now, for the given W 1 let k 1 be the smallest index such that The assumption then implies that W 2 contains Fil k1 π i , proving the result.
Proof. We have assumed r ≥ 1, so by Lemma 4.10 we get for n ≥ 1: By the (nearly) uniserial property of π i , this implies π Kn 0 ⊂ R n if n is even, while π Kn 1 ⊂ R n if n is odd. Putting them together, we obtain the result.
Proof of Theorem 4.5 when λ = 0. Since T 1 (p n ) contains K n , we have an inclusion π T1(p n ) ⊂ π Kn , so Corollary 4.12 implies Noting that dim F σ ≤ p, we obtain by Proposition 4.7 hence the result.

4.1.2.
Non-supersingular case. Assume from now on λ = 0. We define the subspaces R n (n ≥ 0) of c-Ind G KZ σ as above. We still have the properties (i) and (ii) recalled there. The only difference, also the key difference with the supersingular case, is that the induced morphisms R n → π(r, λ, 1) are all injective (because λ = 0). Moreover, if we write R n for the image of R n in π(r, λ, 1), then R n ⊂ R n+1 and π(r, λ, 1) = lim − → n≥0 R n .
We then conclude as in the supersingular case.
Proof of Theorem 4.5 when λ = 0. Since T 1 (p n ) contains K n , we obtain by Proposition 4.13 The result then follows from Proposition 4.7.
We close this subsection by the following consequence.
In any case, we have Proof.
That is, by replacing M by φ k0 (M ), we may assume φ is injective in the rest.
If I is another (two-sided) ideal of Λ containing J k+1 , then we obtain by modulo I again: Since dim F M/(J + I)M is bounded by c 0 := dim F M/JM which depends only on M , and since Q k is a successive extension of Q (k times), we obtain the following inequality: We specialize the above inequality to our situation. Recall that Λ is topologically generated by three elements, say z 1 , z 2 , z 3 , such that every element of Λ can be uniquely expressed as a sum over multi-indices α = (α 1 , α 2 , α 3 ) ∈ N 3 : Moreover, z α z β = z α+β up to terms of degree > |α| + |β|, see [19,Thm. 10]. The ideal J is simply spanned by the set of elements z α with |α| > 0. Let I n denote the two-sided ideal of Λ generated by the maximal ideal of F[[T 1 (p n )/Z 1 ]]. Then it is easy to see that J 3p n is contained in I n . Applying (4.6) to I = I n , we obtain giving the result.
Remark 4.16. In the proof of Lemma 4.15, it is crucial that we are working with T 1 (p n ) instead of K 1 (p 2n ) (this group will show up in §6 for application), although they are (up to finite order) conjugate to each other in GL 2 (Q p ). We have learnt this trick of "averaging" from [19] (used in a different manner there).

4.3.
The proof in degree 0.
Proof of Theorem 4.1 for i = 0. Let M := Π ∨ ∈ C. Then the Pontryagin dual induces natural isomorphisms for i ≥ 0 So we could instead work with M .
By Lemma 4.3, we may assume M is a quotient of P π ∨ for some irreducible π ∈ Rep F (G). Write P = P π ∨ , E = E π ∨ and R = Z(E) in the following.
By Theorem 3.23 there exists a regular element f ∈ m R (where m R denotes the maximal ideal of R) such that: (a) P/f P is a finitely generated free Λ-module; (b) f annihilates M , i.e. M is a quotient of P/f P .
Since R is a Cohen-Macaulay integral domain of Krull dimension 3, we can find g, h ∈ m R such that f, g, h form a regular sequence. As a consequence, f, g, h is a system of parameters for R. We deduce from Proposition 3.7 that P/(f, g, h)P is of finite length in C, hence so is M/(g, h)M (noting that f annihilates M ). Theorem 4.4 then implies that

4.4.
Higher homological degrees. In [19] or [8], once a bound is obtained for torsion Λmodules, the extension to higher homological degrees is rather easy using the trivial dimension formula for free Λ-modules. However, it is much subtler in our situation, because to be able to apply Theorem 4.1 we need guarantee in each step that the module M ∈ C in consideration can be split into two parts in C, not just as Λ-modules. To overcome this difficulty, we prove the following result.
Proposition 4.17. Fix π as above. Let M ∈ C be a non-zero coadmissible quotient of P . Assume that M is torsion-free as a Λ-module.
(i) If B is not of type (IV), then M is a free Λ-module.
(ii) If π ∼ = χ • det, then M is a free Λ-module and the kernel of the evaluation morphism (3.5) has finite length.
Since Ker 1 ∼ = (Sp ∨ ) m for some m ≥ 0 and since m/(g, h) has finite length, we conclude by Proposition 3.34 as in (i).
The next result implies in particular Theorem 4.1.
Theorem 4.18. Let M ∈ C and assume it is finitely generated of rank s as a Λ-module.
Then for some constant c > 0 depending on M , Proof. (i) The proof of (4.9) is similar to the proof of (26) in [19, §3.2] but using Proposition 4.17 as the main input. Since the rank function is additive with respect to short exact sequences, using Lemma 4.3 we may assume M is a quotient of P π ∨ with π ∨ ∈ C irreducible. The proof proceeds by induction on s. If s = 0, then it follows from Theorem 4.1. Assume s ≥ 1 and (4.9) holds for objects of rank ≤ s − 1.
Let M tor be the torsion part of M (as a Λ-module) and M tf be the quotient M/M tor . Then M tor is stable under the action of G, so that Theorem 4.1 applies to M tor (it is also finitely generated over Λ because M is and Λ is noetherian). So by Theorem 4.1, we may assume M is torsion-free of rank s.
If up to twist π / ∈ {Sp, π α }, then M is already a free Λ-module by Proposition 4.17, so the result is obvious. Assume π = Sp. We must have Hom C (P 1 ∨ G , M ) = 0, otherwise M would be torsion by (3.9). Choose a non-zero morphism P 1 ∨ G → M with M ′ being its image and M ′′ := M/M ′ . Since M is torsion-free, so is M ′ , hence M ′ is actually a free Λ-module by Proposition 4.17. Since the rank of M ′′ is ≤ s − 1, we conclude by induction. The case π = π α is proven in a similar way.
(ii) To prove (4.10), again we may assume M is a quotient of P π ∨ for some irreducible π ∨ ∈ C, and further a quotient of P where P is a finite free Λ-module by Theorem 3. 23. 9 Letting N be the kernel, we have a short exact sequence 0 → N → P → M → 0 which induces From (4.11) and using (4.9) we obtain for some constant c > 0 where s ′ denotes the Λ-rank of N . Since P is free of rank s + s ′ , we have proving the result for i = 1. Finally, the estimation for higher i follows from (4.12) by induction.
4.5. GL 2 (Q p ) vs SL 2 (Q p ). For the application in Section 6, we need to consider smooth admissible F-representations of SL 2 (Q p ) and their Pontryagin duals. It is easy to translate the results above to SL 2 (Q p ) case. If H is a subgroup of GL 2 (Q p ), we denote by H ′ the intersection H ∩ SL 2 (Q p ).
Lemma 4.19. Let Π be a smooth admissible F-representation of SL 2 (Q p ). Then for all i ≥ 0, Proof. Write Id for the identity matrix of SL 2 (Q p ). The center of SL 2 (Q p ) is {±Id} and we have an isomorphism SL 2 (Q p )/{±Id} ∼ = GL 2 (Q p )/Z. Depending on the action of −Id, Π decomposes as Π + ⊕ Π − , where −Id acts on Π ± via ±1. Up to twist, we only need treat Π + . But then we may view Π + as a representation of GL 2 (Q p ) with a trivial central character and conclude by Theorem 4.1.

Generalization
For application in §6, we need generalize Theorem 4.1 to representations of a finite product of GL 2 (Q p ). Although the main result (Theorem 5.1) we will prove below is similar to [19,Prop. 4], the proof is quite different. The reason is that to carry out the inductive step as in [19, §3.1], one need a stronger statement than Theorem 4.1; cf. [19,Prop. 7]. Instead, we use a direct generalization of the proof of Theorem 4.1, at the cost of obtaining a weaker result. 10 See also Remark 6.2.
We let G = GL 2 (Q p ), K = GL 2 (Z p ) and other subgroups of G are defined as in the previous sections. Given r ≥ 1, we let That is, G is a product of r copies of G, and so on. If n = (n 1 , ..., n r ) ∈ (Z ≥1 ) r , let The aim of this section is to prove the following result.
Theorem 5.1. Let M ∈ C(G) and assume it is finitely generated of rank s as a Λ-module. Then for some constant c > 0 depending on M , where κ(n) := max i {n i }.
In §5.1 we establish some results generalizing the case r = 1. We give the proof of Theorem 5.1 in §5.2, and in §5.3 translate it into a form adapted for application.
Lemma 5.2. (i) The tensor product π 1 ⊗ · · · ⊗ π r is an irreducible admissible representation of G and each irreducible admissible representation of G is of this form.
(iii) Let π = ⊗ r i=1 π i and π ′ = ⊗ r i=1 π ′ i be irreducible representations of G. Then π ∼ π ′ (i.e. in the same block) if and only if π i ∼ π ′ i for all i.
(ii) is a direct consequence of Theorems 3.1 and 3.2.
where ⊗ denotes the completed tensor product (always over F). Let R = Z(E) be the center of E. (ii) If π i / ∈ {Sp, π α } (up to twist) for any i, then F⊗ E P (resp. F⊗ R P ) is Cohen-Macaulay.
Proof. Both assertions follow from Proposition 3.7.
Proof. Since any irreducible object in C(G) has canonical dimension ≤ r and π ∨ occurs as the G-cosocle of M , the "only if" part is obvious. The next result is a weaker generalization of Theorem 3.1.
Proof. We may assume M is irreducible so M ∼ = ⊗ r i=1 π ∨ i with π i irreducible. The result then follows from the case r = 1, see Theorem 4.4. 5.1.2. Serre weights. Similar to Lemma 5.2, an irreducible representation of K is of the form σ = ⊗ r i=1 σ i with each σ i ∈ Rep F (K) irreducible. We have the obvious notion of Serre weights for π = ⊗ r i=1 π i . Clearly, σ ∈ D(π) if and only if σ i ∈ D(π i ) for each i. The following lemma is a direct generalization of Corollary 3.11.
are two weights such that σ i = σ ′ i whenever π i is supersingular, then Hom K (P, σ ∨ ) ∨ ∼ = Hom K (P, σ ′∨ ) ∨ as E-modules when they are both non-zero. As in the proof of Proposition 3.20, we have Hom C(G) (P, Ker) = 0, that is π ∨ does not occur in Ker.
We need to show that Hom K (m⊗ E P, σ ∨ ) is finite dimensional for any irreducible σ ∈ Rep F (K). Since Hom K (m⊗ E P, σ ∨ ) ∨ ∼ = m⊗ E Hom K (P, σ ∨ ) ∨ it suffices to consider those σ such that Hom K (P, σ ∨ ) ∨ = 0, or equivalently Hom K (P π ∨ i , σ ∨ i ) ∨ = 0 for all i. The rest of the proof is identical to the proof of Proposition 3.20, using Lemma 5.8.
Proof. By Lemma 5.9(i), we may assume M = m⊗ E P , where m = Hom C(G) (P, M ). The projectivity of P implies that m is a cyclic E-module; let a be the annihilator.
Let σ = ⊕ σ σ where the sum is taken for all the irreducible σ ∈ Rep F (K) such that Hom K (P, σ ∨ ) = 0. Then we have By Lemma 5.8, Hom K (P, σ ∨ ) ∨ is a Cohen-Macaulay E-module of dimension s. Since M is coadmissible, Hom K (M, σ ∨ ) ∨ is finite dimensional. So we may find f 1 , ..., f r ∈ a which form a regular sequence for Hom K (P, σ ∨ ) ∨ . As in the proof of Theorem 3.23 we may modify f i so that they all lie in a ∩ R. Using repeatedly Proposition 3.13 we obtain the condition (iii) and that f 1 , ..., f r is a P -sequence.
We are left to check that f 1 , ..., f r is an R-sequence. When r = 1, this is trivial because f 1 = 0 and R is an integral domain. In general case, we first observe that for any 1 ≤ i ≤ r there is a natural isomorphism ..., f i )).
Proof. Since M is coadmissible, the proof of Lemma 4.3 implies that it admits a finite filtration such that each of the graded pieces is a (coadmissible) quotient of P π ∨ for some irreducible π ∈ Rep F (G). By the horseshoe lemma in homological algebra, we may assume M is just a coadmissible quotient of P π ∨ , in which case we conclude (inductively) by Proposition 5.10.
Remark 5.12. The above result can be viewed as a generalization (in a weak form) of the construction of Breuil-Paškūnas [6]. Note that when r ≥ 2 it is not clear (to the author) how to generalize the construction to G. Assume that n≥1 φ n (M ) = 0.
Proof. The proof is identical to that of Lemma 4.15, except that we need k ≥ 3p κ(n) to guarantee that J k is contained in the two-sided ideal of Λ generated by the maximal ideal of Here J denotes the maximal ideal of Λ.

Principal series.
Definition 5.14. Let S be a subset of {1, ..., r}. Given irreducible π i ∈ Rep F (G) for each i / ∈ S and a character η i ∈ Rep F (T ) for each i ∈ S, we define P (π i , η i , S) ∈ C(G) by Denote by E S the ring End C(G) (P (π i , η i , S)). Then If i ∈ S, let π i := soc G π ηi and let P π ∨ i be a projective envelope of π i so that M η ∨ i becomes naturally a quotient of P π ∨ i . Set . Then E S is naturally a quotient of E. Let R S be the image of R in E S (note that R S might be smaller that the center of E S .) Then R S is Cohen-Macaulay of Krull dimension 3r − |S|. Moreover, if σ = ⊗ i σ i ∈ D(⊗ i π i ), then which is a cyclic E S -module and a Cohen-Macaulay E S -module of Krull dimension r by Corollary 3.11. Lemma 5.15. With the above notation, if f 1 , ..., f r ∈ R is a regular sequence for Hom K (P, σ ∨ ) ∨ , then they are R S -regular. Proof. Although it is possible to prove the result using a similar argument as in Proposition 5.10, we instead do it via a commutative algebra argument based on the following observation. Let J ′ σ ⊂ R S be the annihilator of Hom K (P, σ ∨ ) ∨ (viewed as an R S -module). Then J ′ σ is a prime ideal of height 2r − |S|. If we can find a sequence of elements g 1 , ..., g 2r−|S| ∈ J ′ σ such that J ′ σ /(g 1 , ..., g 2r−|S| ) has finite length, then we are done. Indeed, the latter condition implies that (we still use f i to denote its image in R S ) g 1 , ..., g 2r−|S| , f 1 , ..., f r form a system of parameters of R S . Since R S is Cohen-Macaulay of dimension 3r − |S|, the sequence is in particular R S -regular, hence f 1 , ..., f r is also R S -regular. Finally, it is easy to construct such elements g i by Lemma 3.12.
Proposition 5.16. Let M be a coadmissible quotient of P (π i , η i , S). Then M is torsion over Λ and Proof. We may view M as a coadmissible quotient of P , hence by Proposition 5.10 we find an R-sequence f 1 , ..., f r which annihilate M . As in the proof of Lemma 5.15, we may complete f 1 , ..., f r by g 1 , ..., g 2r−|S| to obtain a system of parameters of R S . Since M/(g 1 , ..., g 2r−|S| ) has finite length, we conclude by Lemma 5.13 and Lemma 5.7. As in the proof of Proposition 3.20(ii), it is enough to show the following: Claim: For i ∈ S, let π ′ i ∈ B i be distinct with π i and let Q ′ i be the maximal quotient of P π ′∨ i none of whose irreducible subquotients is isomorphic to π ∨ i . Then By Remark 3.21, each Q ′ i has a finite filtration with graded pieces being subquotients of M η ∨ . The claim then follows from Proposition 5.16. 5.1.5. Torsion free vs free. We fix an irreducible representation π = ⊗ i π i ∈ Rep F (G) and let B (resp. B i ) be the block of π (resp. π i ). Let In the rest, we make the following assumption: such that each of the graded pieces is of the shape ⊗ r i=1 λ i , where λ i ∈ C(G) Bi and -either λ i is irreducible, not of type (IV), -or λ i ∈ {Sp ∨ , κ ∨ }, where we recall that κ is defined in (3.15).
By definition, if M is CM-type, then M = lim ← −k M k with M k being quotients of finite length and CM-type. Lemma 5.19. Let λ = ⊗ r i=1 λ i be a graded piece as in Definition 5.18. Let S κ ⊂ {1, ..., r} be the set of indices i such that λ i = κ ∨ . Then the G-socle (resp. G-cosocle) of λ is Proof. This is clear from the definition of κ.
Lemma 5.20. Let M ∈ C(G) B be non-zero, of finite length and CM-type. Then as a Λ-module M is Cohen-Macaulay of projective dimension 2r.
Proof. We may assume M is of the form ⊗ r i=1 λ i with notation in Definition 5.18, in which case the result follows from Theorem 3.2 and the fact that κ ∨ is Cohen-Macaulay, see the proof of Proposition 3.34.
Lemma 5.23. Let π ∈ Rep F (G) be irreducible satisfying (H) and m be a finitely generated E-module. Then for any i ≥ 0, Tor E i (m, P π ∨ ) is CM-type.
Proof. We first treat the case when m is of finite length. By induction on the length of m and using Lemma 5.22, it suffices to show Tor E i (F, P ) are CM-type. This follows from the Künneth formula and Propositions 3.30 and 3.33 (under the assumption (H)).
For general m, we note that m ∼ = lim ← −k m/m k E where m E is the maximal ideal of E. Therefore, m ⊗ E P is isomorphic to lim ← −k (m/m k E ) ⊗ E P , hence is CM-type. The result for i ≥ 1 follows from a dimension-shifting argument. with δ Λ (Ker i ) ≤ δ Λ (Ker i−1 ) − 1. However, since Ker is torsion, we eventually arrive at some i such that g i acts nilpotently on Ker i−1 and conclude as in (1).

The proof.
Theorem 5.25. Let π = ⊗ r i=1 π i ∈ Rep F (G) be irreducible with δ Λ (π ∨ ) = r. Let P = P π ∨ and M ∈ C(G) be a non-zero coadmissible quotient of P . Then Proof. Being non-zero, M admits π ∨ as its G-cosocle, so we have δ Λ (M ) ≥ r. We do induction on δ Λ (M ). If δ Λ (M ) = r, then Lemma 5.5 implies that M has finite length and the result follows from Lemma 5.7. Assume the result is proven for all M of canonical dimension ≤ t and treat the case δ Λ (M ) = t + 1 below.
We know that R is a Cohen-Macaulay local ring of Krull dimension 3r by Theorem 3.5. Since M is coadmissible, there exist f 1 , ..., f r ∈ m R as in Proposition 5.10. We complete them by g 1 , ..., g 2r ∈ m R to obtain a system of parameters.
so it suffices to prove the result for M 1 . For this reason we assume, without loss of generality, that we are in the following setting: there exist g 1 , ..., g r ′ ∈ R (with 1 ≤ r ′ ≤ 2r) such that M/(g 1 , ..., g r ′ )M ∈ C(G) is of finite length and that g 1 is not nilpotent on M .
If δ Λ (M 1 ) = t, it follows from the inductive hypothesis. If δ Λ (M 1 ) = t + 1, we consider the action of g 2 , ..., g r ′ on M 1 and conclude by an induction on r ′ (i.e. the length of the sequence).
We are ready to prove Theorem 5.1 whose statement we recall.
Proof. (i) We first assume M is torsion. Since dim G = 3r, we get δ Λ (M ) ≤ 3r − 1. As in Lemma 4.3, we may assume M is a (coadmissible) quotient of P = P π ∨ for some irreducible π ∈ Rep F (G). If δ Λ (π ∨ ) = r, then the result is a special case of Theorem 5.25.
Assume in the rest δ Λ (π ∨ ) < r. This amounts to saying that the subset S ⊂ {1, ..., r} of indices i such that π i is one-dimensional is non-empty; indeed, we have δ Λ (π ∨ ) = r − |S| by Lemma 5.2(ii). Up to twist by a suitable central character, we may assume π i = 1 G for i ∈ S. We do induction on |S|. Assume 1 ∈ S without loss of generality. Define P ′ = ⊗ i P ′ i by P ′ 1 = P π ∨ α and P ′ i = P π ∨ i , i = 1. Then we have a natural inclusion P ′ ֒→ P by (3.7), with quotient isomorphic to P (π i , α, {1}), see Definition 5.14. This allows to divide M into two parts: a submodule M ′ being image of P ′ and the corresponding quotient M ′′ = M/M ′ . We obtain the required bound by Theorem 5.25, Proposition 5.16 and the inductive hypothesis. Now we prove (5.4) in general. We may assume M is torsion-free. Let S ′ ⊂ Σ be the set of indices i such that π i ∈ {Sp, π α }. We do induction on the quantity s + |S ′ |. The case when S ′ = ∅ follows from Proposition 5.24 which implies that M is a free Λ-module. If S ′ = ∅, say 1 ∈ S ′ without loss of generality, then we may divide M into 0 → M ′ → M → M ′′ → 0 in such a way that -M ′ is a quotient of P ′ = ⊗ i P ′ i where P ′ 1 = P 1 ∨ G and P ′ i = P π ∨ i , i = 1, and the inductive hypothesis implies that (5.4) holds for M ′ ; -M ′′ has rank < s, hence also verifies (5.4) by induction (remark that M ′′ is not necessarily torsion-free, in which case we split it as 0 → M ′′ tor → M ′′ → M ′′ tf → 0 and apply the inductive hypothesis to M ′′ tf ). This completes the proof of (5.4).
(ii) Finally, using Corollary 5.11, (5.5) is proved by the same argument as in Theorem 4.18.

5.3.
Change of groups. We keep the notation in the previous subsection. For n ≥ 1, let K 1 (p n ) := K 1 ∩ K 0 (p n K 1 (p n )) = 1 + pZ p pZ p p n Z p 1 + pZ p .
On the other hand, using (essentially) the fact K 1 /K 1 (p n ) ∼ = pZ/p n Z, Marshall proved the following interesting result ([19, Cor. 14]). where α(i) is a non-increasing sequence of non-negative integers. Then there exists a filtration 0 = L 0 ⊂ · · · ⊂ L l = L of L by submodules L i such that L i /L i−1 ∼ = F[K 1 /K 1 (p α(i)+1 )].
Proof. The proof goes as in that of [19,Lem. 19]. We explain this briefly. First, if L = F[K 1 /K 1 (p m )] for some m ∈ (Z ≥1 ) r , we apply Shapiro's lemma to obtain (5.7) Using a diagonal element of G, precisely For general L, Lemma 5.28 provides a filtration of L L = F 0 ⊃ F 1 ⊃ · · · such that every quotient F i /F i+1 is isomorphic to F[K 1 /K 1 (p m )] for some m ≤ n and each isomorphism class of quotient occurs at most p r times. We then deduce from the first case that dim ≪ κ(n) 2r p (r− 1 2 )κ(n) . Here we use the fact that the cardinality of the set {m : m ≤ n} is r i=1 n i , hence bounded by κ(n) r .

Application
Let F be a number field of degree r, and r 1 (resp. 2r 2 ) be the number of real (resp. complex) embeddings. Let F ∞ = F ⊗ Q R, so that SL 2 (F ∞ ) = SL 2 (R) r1 × SL 2 (C) r2 . Let K ∞ be the standard maximal compact subgroup of SL 2 (F ∞ ).
Let {σ 1 , ..., σ r } be the set of complex embeddings of F and let d ∈ (Z ≥1 ) r be an r-tuple indexed by the σ i such that d i = d j when σ i and σ j are complex conjugates. Let W d be the representation of SL 2 (F ∞ ) obtained by forming the tensor product σi real Theorem 6.1. Let Y = SL 2 (F )\SL 2 (A)/K f K ∞ for some compact open subgroup K f ⊂ SL 2 (A f ). If F is not totally real and d = (d 1 , ..., d r ) as above, then Proof. The proof follows closely the one presented in [19]. We content ourselves with briefly explaining the main ingredients. Below we abuse the notation by letting the same letters to denote subgroups of SL 2 obtained by intersection from GL 2 .
(1) By [19,Lem. 18], there exists a p-adic local system V d defined over O = W (F), such that . This need to choose a bijection between the set of complex places and p-adic places of F .
where H j is the j-th completed homology of Emerton with (trivial) coefficients in O, and H j,Qp = H j ⊗ Zp Q p . Note that H j is a coadmissible module over O[[K 1 ]] and carries a natural compatible action of r i=1 SL 2 (Q p ). (3) Let n = (n 1 , ..., n r ) where n i is the smallest integer such that p ni ≥ d i (resp. p ni ≥ d i /2) if σ i is real (resp. complex). By [19,Lem. 17] we may choose lattice V di ⊂ V di such that V di /p ⊂ F[[K 1 /K 1 (p ki )]]. Let L d be the reduction mod p of ⊗ r i=1 V di . (4) Let M j be the reduction modulo p of the image of H j → H j,Qp . We then have (5) Because SL 2 (C) does not admit discrete series, the assumption that F is not totally real implies that H j,Qp is a torsion O[[K 1 ]] ⊗ Zp Q p -module, see [8,Thm. 3.4]. So by Lemma 2.6, M j is a torsion Λ-module. Therefore our Theorem 5.29 applies, via Lemma 4.19, and shows that dim F H i (K 1 , M j ⊗ L d ) ≪ κ(n) 2r p (r− 1 2 )κ(n) ≪ ǫ κ(d) r− 1 2 +ǫ .
Remark 6.2. In [19], Marshall considered a more general setting, allowing a subset of the weights d i to be fixed and letting the others vary. We have restricted ourselves for two reasons. On the one hand, Theorem 6.1 provides interesting bounds only when all weights vary in a uniform way (because of the appearance of κ(d)), for example, when d is parallel. On the other hand, this already includes the most interesting cases: for example, when F is imaginary quadratic, we do have d 1 = d 2 . Now we change our notation. Let Z ∞ be the centre of GL 2 (F ∞ ), K f be a compact open subgroup of GL 2 (A f ) and let X = GL 2 (F )\GL 2 (A)/K f Z ∞ .
If d = (d 1 , ..., d r1+r2 ) is an (r 1 + r 2 )-tuple of positive even integers, let S d (K f ) denote the space of cusp forms on X which are of cohomological type with weight d. Then using the Eichler-Shimura isomorphism, see [19, §2.1], Theorem 6.1 can be restated as follows. Theorem 6.3. If F is not totally real then for any fixed K f and d = (d 1 , ..., d r1+r2 ) as above, we have dim C S d (K f ) ≪ ǫ κ(d) r−1/2+ǫ .