AN IRREDUCIBILITY CRITERION FOR SUPERSINGULAR mod p REPRESENTATIONS OF GL 2 (F ) FOR TOTALLY RAMIFIED EXTENSIONS F OF Q p

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1 AN IRREDUCIBILITY CRITERION FOR SUPERSINGULAR mod p REPRESENTATIONS OF GL 2 F FOR TOTALLY RAMIFIED EXTENSIONS F OF Q p MICHAEL M. SCHEIN Abstract. Let F be a totally ramified extension of Q p. We consider supersingular representations of GL 2 F whose socles as GL 2 O F -modules are of a certain form that is expected to appear in the mod p local Langlands correspondence and establish a condition under which they are irreducible.. Introduction Let F be a finite extension of Q p with valuation ring O. Choose a uniformizer π O and denote the residue field by k = O/π. A question of immediate relevance to the emerging mod p local Langlands correspondence is to construct smooth mod p representations of the group G = GL 2 F. If K = GL 2 O and Z is the center of G, then any irreducible F p -representation σ of the finite group GL 2 k may be viewed naturally as a representation of KZ. We may then consider the compact induction ind G KZσ; a precise definition is given below. Barthel and Livné proved [BL], Prop. 8 that the endomorphism algebra End G ind G KZσ is isomorphic to a polynomial ring F p [T ] for an explicitly defined generator T. Moreover, they showed [BL], Theorem 33 that any irreducible mod p representation V of G is, up to twist by an unramified character, a quotient of ind G KZσ/T λind G KZσ for some σ as above and some λ F p. If λ 0, then Barthel and Livné classified these quotients completely. On the other hand, quotients of ind G KZσ/T ind G KZσ are called supersingular and are still very poorly understood. In this paper we prove an irreducibility criterion for certain quotients of ind G KZσ/T ind G KZσ when F/Q p is totally ramified. Given a tamely ramified continuous irreducible Galois representation ρ : GalF /F GL 2 F p, for any finite extension F/Q p, Serre s weight conjecture and its generalizations associate to ρ a set Dρ of irreducible F p -representations of GL 2 k; these are called the modular weights of ρ. These conjectures were formulated by Serre for F = Q p, by Buzzard, Diamond, and Jarvis [BDJ] for F unramified over Q p, and by the author [Sch] in general; the reader is referred to those articles and to the beginning of the last section of this paper for more details. These conjectures may be seen as describing the socle of the smooth representation πρ of GL 2 F associated to ρ by the mod p local Langlands correspondence: generically, one expects soc K πρ = σ Dρ σ. In particular, this implies that a surjection ind G KZσ πρ exists if and only if σ Dρ. Let F/Q p be totally ramified of degree e. Consider the F p -representation σ = det w Sym r F 2 p of GL 2 F p, where 0 < r < p 2. Let f σ ind G KZσ be a non-zero function supported on the single coset KZ that satisfies f σ id σ I. Here I K is the upper triangular pro-p-iwahori subgroup. Observe that f σ generates an irreducible K-submodule isomorphic to σ. The following lemma is proved by computation. Lemma.. Let 0 < r < p 2 and let σ and f σ be as above. Date: August 4, 2009; revised November, Mathematics Subject Classification. S37, F80. Key words and phrases. Supersingular representations, mod p local Langlands correspondence, Galois representations.

2 2 MICHAEL M. SCHEIN a The image of π 0 f σ in ind G KZσ/T ind G KZσ is invariant under the action of I and generates a K-submodule that is irreducible and isomorphic to det w+r Sym p r F 2 p. b The image of µ 0,µ F p µ r+ π 2 [µ 0 ] + π[µ ] f σ in ind G KZσ/T ind G KZσ is invariant under the action of I and generates a K-submodule that is irreducible and isomorphic to det w+r+ Sym p r 3 F 2 p. Here [µ] O is the canonical Teichmüller lift of µ F p. Proof. The first statement is Lemma 3.6. The second follows from the case n = of Lemma 3. and Proposition 3.3. Now let 0 < r p 2e, and consider the set D = {σ 0,..., σ e } {σ 0,..., σ e } of F p -representations of GL 2 F p, where σ i = det i Sym r+2i F 2 p, σ i = det r+i Sym p r 2i F 2 p This D arises as Dρ for a suitable Galois representation ρ, and it consists of 2e distinct regular weights. Let β =. We now define e explicit elements of ind G π 0 KZσ 0 /T ind G KZσ 0 as follows. For i e define f i = β µ r+2i π 2 [µ 0 ] + π[µ ] f i µ 0,µ F p π 2 [µ z i = 0 ] + π[µ ] βf i. µ p r 2i µ 0,µ F p Proposition.2. Let 0 < r p 2e, and let the set D of weights be defined as above. Let τ : ind G KZσ 0 /T ind G KZσ 0 W be a quotient. Suppose that W has no non-supersingular subrepresentations, that soc K W σ D σ, and that τf e W is non-zero. Then for each 0 i e the K-submodules of W generated by the elements τf i resp. τβf i are irreducible and isomorphic to σ i resp. σ i. Proof. This is Proposition 3.8 below. Admitting these two propositions, we can immediately establish the following irreducibility criterion, which is the main result of this paper. See Remark 3.9 for a variation. Theorem.3. Let 0 < r p 2e, and let the set D of weights be defined as above. Let τ : ind G KZσ 0 W be a quotient. Suppose that W has no non-supersingular subrepresentations, that soc K W σ D σ, and that τf e W is non-zero. Suppose also that τz i 0 for all i e. Then W is an irreducible G-module. Proof. Let U W be an irreducible G-submodule. Since f 0 generates ind G KZσ 0 as a G-module, to conclude U = W it suffices to show that τf 0 U. Note that if τβf i U, then also τf i U. By our assumption on the K-socle of W and the previous proposition, it then follows that any irreducible K-submodule of W must contain one of the elements τf 0,..., τf e. Let 0 l e be the smallest number such that τf l U, and suppose that l > 0. By Frobenius duality there is a non-zero map ψ l : ind G KZσ l W such that ψ lf σ l = τβf l. Since Hom G ind G KZσ l, W is a one-dimensional space, every non-zero element is an eigenvector for the action of the commutative algebra End G ind G KZσ l. Therefore, ψ l must factor through a quotient ind G KZσ l /T λindg KZσ l for some λ F p. We must have λ = 0, since otherwise the image of ψ l in W would have a non-supersingular subrepresentation. By assumption ψ l µ 0,µ F p µ p r 2l π 2 [µ 0 ] + π[µ ] f σ = τz l l is a non-zero element of W. The second part of Lemma. then implies that τz l generates an irreducible K-submodule

3 AN IRREDUCIBILITY CRITERION 3 of W that is isomorphic to σ l. But since each irreducible submodule in soc K W appears with multiplicity one, it follows that τz l = cτf l for a suitable non-zero scalar c F p, contradicting the minimality of l. It follows that l = 0, and hence U = W. We briefly discuss previous work to place this theorem in context. A first result towards studying the supersingular representations of GL 2 F was attained by Breuil, who showed in [Bre] that if F = Q p then ind G KZσ/T ind G KZσ is irreducible for all σ. He proved this by explicitly computing the I-invariants of ind G KZσ/T ind G KZσ and observing that every non-zero I- invariant generates ind G KZσ/T ind G KZσ as a G-module. Since I is a pro-p group, any irreducible submodule of ind G KZσ/T ind G KZσ must have non-trivial I-invariants, and the result follows. A more conceptual version of this argument was given by Ollivier in [Oll], and other proofs were found by Emerton [Eme], Theorem 5. and Vignéras unpublished. Moreover, ind G KZσ/T ind G KZσ has the expected socle, and an explicit correspondence between irreducible Galois representations and supersingular representations of GL 2 Q p was stated in [Bre]. The smooth representation theory of GL 2 F for F Q p is much more complicated, since ind G KZσ/T ind G KZσ is of infinite length and there are many more supersingular representations of GL 2 F than there are Galois representations to pair them with. When F/Q p is unramified, Breuil and Paskunas [BP] have applied Paskunas method of diagrams to prove the existence of many supersingular representations with socle σ D σ. These were again shown to be irreducible by an argument on I-invariants, although the argument relies on the combinatorics of D and is considerably more complicated than in the case F = Q p. In fact, their method of construction essentially works for arbitrary extensions F/Q p. Alternatively, Hu [Hu] associated a canonical diagram to any supersingular representation not necessarily irreducible of GL 2 F for arbitrary F. In general it has been difficult to show that the representation of GL 2 F associated to a given diagram is irreducible, since the method of Breuil and Paskunas for proving irreducibility fails in this case. We note that the Breuil-Paskunas construction applied to totally ramified F/Q p yields representations with no non-supersingular subrepresentations and with K-socle σ D σ. However, neither these representations nor Hu s canonical diagrams are understood explicitly enough at present to verify the non-vanishing of τf e and τz i and establish irreducibility by means of Theorem.3 in any example. The second section of the paper is rather technical. It uses the methods of Breuil s original paper [Bre] to prove Corollary 2., which will provide information about the I-invariants of certain quotients of ind G KZσ 0 /T ind G KZσ 0. Lemma. and Proposition.2 are proved in the third section. In fact, we obtain more precise information about V e, which is used when constructing irreducible supersingular representations of GL 2 F. This work will appear in a separate article. We note that the constructions and results of this paper may be generalized to arbitrary extensions F/Q p, although the presence of an unramified subextension complicates the computations. The author is grateful to Christophe Breuil, Yongquan Hu, and Vytautas Paskunas for enlightening conversations and for comments on an earlier version of this paper. He thanks the referee for suggesting a number of improvements to the exposition... Notations and background results. In this section we establish notation and recall some results that we will need. Let p be an odd prime. Recall that F is totally ramified extension of Q p with valuation ring O and π O is a uniformizer. Then O/π = F p. Let e = [F : Q p ] be the ramification index. We assume that e > ; note that in the case F = Q p the questions we investigate have been resolved completely by Breuil. Let G = GL 2 F. Then K = GL 2 O G is a maximal compact subgroup. Let B GL 2 F p be the subgroup of upper triangular matrices. Fix the Iwahori subgroup I = ω B K, where ω : K GL 2 F p is the natural projection. Let I be the pro-p Sylow subgroup of I. Write Z for the center of G and K for the kernel of ω. We also define α = 0 0 π, w = 0, β = αw = π 0.

4 4 MICHAEL M. SCHEIN Recall that the distinct irreducible F p -representations of GL 2 F p are σ r,w = det w Sym r F 2 p, where 0 w p 2 and 0 r p. A model for σ r,w is given by the r + -dimensional space V σr,w of homogeneous polynomials P F p [x, y] of degree r, where GL 2 F p acts as follows. a b If γ = GL c d 2 F p, then γp x, y = ad bc w P ax + cy, bx + dy. Given λ F p, let [λ] O be its canonical lift. For n define the sets I n = { [λ 0 ] + π[λ ] + + π n [λ n ] : λ 0,..., λ n F p n} O. We also set I 0 = {0}. Then for all n 0 and λ I n we set π gn,λ 0 = n λ 0, g n,λ = πλ π n+. In particular, g0,0 0 is the identity matrix and g0, = α. Also gn,λ = βg0 n,λw for all n 0 and λ I n. It follows from the Cartan decomposition that these gn,λ 0 and g n,λ comprise a set of coset representatives for KZ in G: G = gn,λkz. i i {0,} n 0,λ In For n 0, we define S 0 n = IZα n KZ = λ I n g 0 n,λ KZ and S n = IZβα n KZ = λ I n g n,λ KZ as in [Bre]. We also set S n = S 0 n S n and B n = B 0 n B n, where B 0 n = m n S0 m and B n = m n S m. Given an irreducible F p -representation σ of GL 2 F p, we can view it as a KZ-module where π 0 K acts via ω and the matrix acts trivially. Then a model for ind G 0 π KZσ is the space of functions f : G V σ that are compactly supported modulo KZ and satisfy fkg = σkfg for all k KZ and g G. The group G acts by hfg = fgh for h G. Such a function is clearly determined by its values on the gn,λ 0 and gn,λ. Note that ind G KZσ F p [G] V Fp[KZ] σ. If g G and v V σ, then the element g v corresponds to the function defined by { σhgv : h KZg g vh = 0 : h KZg. This is the element denoted [g, v] in [Bre]. Observe that any function f ind G KZσ may be written uniquely in the form f = n=0 λ I n g 0 n,λ v 0 n,λ + g n,λ v n,λ for suitable vn,λ 0, v n,λ V σ. We say that the support of f is the set of gn,λ i such that vi n,λ 0. We write f S n if the support of f is contained in S n, and similarly for B n, Sn, 0 etc. Observe that any element z O has a unique expansion z = z iπ i, where z i I. Let [z] n denote the truncation n z iπ i I n. We will sometimes write gn,z 0 to mean gn,[z] 0 n. Throughout this section and the following we assume that σ = σ r,0, with 0 r p. Then the formulae of section 2.5 and Lemme 3.. of [Bre] imply the following explicit expressions for the action of the canonical endomorphism T Endind G KZσ r. Lemma.4. Let v = r c ix r i y i V σr. If n and µ I n, then the action of T is given by: T gn,µ 0 v = r gn+,µ+π 0 n λ c i λ i x r + gn,[µ] 0 n c r µ n x + y r, T g n,µ v = r gn+,µ+π n λ c r i λ i y r + gn,[µ] n c 0 x + µ n y r.

5 AN IRREDUCIBILITY CRITERION 5 In the remaining cases the action of T is given by: T Id v = r g,λ 0 c i λ i x r + α c r y r, T α v = r g,λ c r i λ i y r + Id c 0 x r. Corollary.5. The endomorphism T Endind G KZσ r is injective. In particular, Proof. Immediate from Lemma.4. ind G KZσ r /T ind G KZσ r T e ind G KZσ r /T e ind G KZσ r. Lemma.6. Suppose that v = r c ix r i y i V σ and n 0. Let µ = [µ 0 ] + π[µ ] + + π n [µ n ] I n. If k, then r T k gn,µ 0 v = g 0 n+k,µ+π n ν + π n+k ν k c i ν i x r + B n+k, T k g n,µ v = ν,...,ν k I k ν,...,ν k I k r g n+k,µ+π n ν + π n+k ν k c r i ν i y r + B n+k. In particular, if k n and r > 0, then r T k gn,µ 0 v = g 0 n+k,µ+π n ν + π n+k ν k c i ν i x r + k m= ν,...,ν k I k ν,...,ν k m I k m g 0 n+k 2m,[µ] n m+ P k m j= πn m+j ν j r c r gn k,[µ] 0 n k c r µ n k x + y r, r T k gn,µ v = g n+k,µ+π n ν + π n+k ν k c r i ν i y r + k m= ν,...,ν k I k ν,...,ν k m I k m g n k,[µ] n k c 0 x + µ n k y r. g n+k 2m,[µ] n m+ P k m j= πn m+j ν j r c 0 Proof. This is a straightforward calculation using the formulae of Lemma Structured submodules and I-invariants r µ r i i n m ν i x r + r µ r i i n m ν i y r + Lemma 2.. Let n. Then for any set-theoretic map f : I n F p there exists a unique polynomial P F p [X 0,..., X n ] in which each variable appears with degree at most p and such that fµ = P µ 0,..., µ n for all µ I n. Proof. When n = this is Lemme 3..6 of [Bre]. Suppose the claim is known for n. By the claim for n =, for each µ I n there exist unique c µ 0,..., cµ p F p such that fµ + π n [λ] = p j=0 cµ j λj. But by induction the map µ c µ j is itself expressible as a unique polynomial in µ 0,..., µ n 2 for each 0 j p. Lemma 2.2. Let λ 0, λ,..., λ e F p. Then [λ 0 ] + π[λ ] + + π e [λ e ] + [λ 0 + ] + π[λ ] + + π e [λ e ] + π e [λ e + λp e 0 + λ 0 + pe π e ] mod π e+.

6 6 MICHAEL M. SCHEIN Proof. Using the isomorphism O lim O/π n, we see that [λ] can be viewed as the following sequence on the right hand side: λ + π, λ p + π 2, λ p2 + π 3,.... The claim then follows from a simple computation. Remark 2.3. An immediate consequence of the lemma is that if n e, then n n n [λ i ]π i + [µ i ]π i [λ i + µ i ]π i mod π n. The computations in the sequel rely on this observation. Remark 2.4. Observe that the binomial coefficient p e j is divisible by p but not by p 2 precisely when j = mp e for m =, 2,..., p. Hence, λ pe 0 + λ 0 + pe π e = p p e π e mp e λ mpe 0 = p p e π e mp e λ m 0. m= m= In particular, the expression above is a polynomial of degree p in λ 0. Lemma 2.5. Fix elements a, b, c, d O and write a = [a i]π i, and similarly for b, c, d. Suppose that n e and ε I n. Let µ ε = + aπ cεπ b + ε + dεπ. Then, n µ ε [ε u + P u ε 0,..., ε u ]π u mod π n. u=0 Here if l and x N, we define Jl, x to be the set of ordered l-tuples j,..., j l N l such that j + + j l = x. Then the polynomial P u ε 0,..., ε u is given by u P u ε 0,..., ε u = b u + ε j d u j + u m= b u m + ε u m + b 0 + ε 0 u l= Jl,u l l j=0 u m j=0 u ε j d u m j l a jk k= j k j=0 ε j c jk j. l= Jl,m l l l a jk k= j k j=0 ε j c jk j + Proof. Since n e we see from Lemma 2.2 that π-adic decompositions behave well under addition and multiplication modulo π n. For instance, ε + b n [ε i + b i ]π i mod π n, εa [ n i ] j=0 ε ja i j π i mod π n. The claim is then obtained by a straightforward calculation. For later reference we record here the first few polynomials P u : P 0 = b 0 P ε 0 = c 0 ε d 0 a 0 b 0 c 0 ε 0 + b + a 0 b 0 P 2 ε 0, ε = c 2 0ε b 0 c 2 0 2a 0 c 0 + c 0 d 0 + c ε c 0 ε 0 ε + d 0 a 0 b 0 c 0 ε + d b c 0 b 0 c a 0 d 0 + 2a 0 b 0 c 0 + a 2 0 a ε 0 + b 2 + a 0 b + a b 0 a 2 0b 0. A similar but easier computation produces the following result: Lemma 2.6. Suppose that n e and ν I n. Let λ I be such that λν 0 and set ν = [ν λν ] e. Denote u = λν 0. Then e ν = u ν 0 + u 2 ν i + R i ν 0,..., ν i π i, i=

7 AN IRREDUCIBILITY CRITERION 7 where i R i ν 0,..., ν i = l= ν i l l u j λ j j= j Jj,l j k= ν jk + Proof. This is a straightforward calculation. At its end the answer is simplified using the identity + λν 0 u = u. Definition 2.7. Let M e be a positive integer and let Q = q 0,..., q M be a sequence of integers such that 0 q i < p for each 0 i M. A G-invariant submodule W ind G KZσ is called Q-structured if every element f W such that f B 0 can be written in the form f = f 0 n + f n + f, where f B n, f 0 n S 0 n, f n S n, and f 0 n and f n satisfy the following condition: Let N = min{n, M}. For each 0 i N and each µ I n i there exist polynomials P 0 µ,i X, P µ,i X F p[x] of degree at most q i such that f 0 n = f n = µ I n N i= µ I n 2 i µ I n N gn,µ+π 0 n λ P µ,0λx 0 r + i gµ+π 0 n 2 i λ+π n i ν P µ,iλ 0 ν I i+ j= gn,µ+π n λ P µ,0λy r + i gµ+π n 2 i λ+π n i ν P µ,iλ ν I i+ j= i= µ I n 2 i. ν j ν j ν q0+ i x r, ν q0+ i y r. Moreover, we require that for every collection of polynomials Pµ,i 0 X, P µ,i X, for 0 i N and every µ, there exists an element f W of the above form. 2 A G-invariant submodule U ind G KZσ is called extended Q-structured if every element f W such that f B e can be written in the form f = f 0 n + f n + f, where f B n, and f 0 n and f n satisfy the following condition: Let N = min{m, n + e }. For each 0 i N and each µ I n i e there exist polynomials P 0 µ,i X, P µ,i X F p[x] of degree at most q i such that f 0 n = f n = µ I n e N i= gn,µ+π 0 n e λ+π n e+ ζ P µ,0λx 0 r + ζ I e i µ+π n e i λ+π n e i ν+π n e+ ζ P µ,iλ 0 µ I n e i µ I n N i= µ I n 2 i 0 g ν I i+ ζ I e gn,µ+π n e λ+π n e+ ζ P µ,0λy r + ζ I e i µ+π n e i λ+π n e i ν+π n e+ ζ P µ,iλ g ν I i+ ζ I e j= j= ν j ν j ν q0+ i x r, ν q0+ i y r. Again we require that for every collection of polynomials Pµ,i 0 X, P µ,i X, for all 0 i N and µ, there exists an element f U of the above form. Remark 2.8. From the formulae of Lemma.4 one sees that T ind G KZσ is a Q-structured submodule for M = and q 0 = r. Similarly, Lemma.6 shows that if W ind G KZσ is a Q-structured submodule, then T e W is extended Q-structured.

8 8 MICHAEL M. SCHEIN Suppose that σ = det w Sym r F 2 p and Q = q 0,..., q M. We will now define some special elements of ind G KZσ. Put X 0 0 = Id x r and X 0 = α y r, and for n e we define X 0 n = X n = gn+,µ 0 µ µ 2 µ n µ r+ n x r, µ I n+ gn+,µ µ µ 2 µ n µ r+ n y r. µ I n+ Observe that X n = β X 0 n. If l M, then for arbitrary n we set We also define X 0,+ = µ I n g 0 n,µ µ q l+ n l 2 X,+ = µ I n g n,µ µ q l+ n l 2 l µ n l +i µ q0+ n xr, i= l X 0, n,0 = µ I n g 0 n,µ µ q0+2 n xr, X, n,0 = µ I n g n,µ µ q0+2 n yr, X 0, µ n l +i i= = l gn,µ 0 µ 2 n l µ I n X, j= = l gn,µ µ 2 n l µ I n X 0, n,m = µ I n g 0 n,µ µ n M X, n,m = µ I n g n,µ µ n M j= µ n l +j µ n l +j M 2 µ n M+j j= M 2 µ n M+j j= µ q0+ n yr. µ q0+ n xr, µ q0+ n yr, µ q0+ n xr, µ q0+ n yr, where in the middle two lines we have l M 2. Define X n 0 = X n = 0 if n e. Note also that X,s n,j = βx0,s j,s n,j for all 0 j M and all s {+, }. If n e, then we define Y to be the part of T e X j,s n e+,l supported on S n, for j {0, }, all s {+, }, and all 0 l M for which this makes sense. Similarly we obtain Ỹ n j from X n. j Explicit expressions for these elements may be obtained from Lemma.6. For instance, if l M, then Y 0,+ = µ I n g 0 n,µ µ q l+ n e l l µ n e l+i i= µ q0+ n e x r. We state the following simple observation for later use; it implies that all the elements just defined are eigenvectors under the action of the set D K of diagonal matrices. Lemma 2.9. Let a, d F p and let ã, d O be any lifts of a, d. Let δ = diagã, d D. If P µ is a homogeneous polynomial of degree s in the variables µ 0,..., µ n and X 0 = µ I n g 0 n,µ P µx r ind G KZσ respectively, X = µ I n g n,µ P µy r, then δx 0 = a d s a r X 0, δx = ad s d r X.

9 AN IRREDUCIBILITY CRITERION 9 Proposition 2.0. Suppose that U ind G KZσ is an extended Q-structured G-submodule and that q 0 p 3. Let n e and let M = min{m, n e}. Let Y be the F p -vector subspace of ind G KZσ spanned by {Ỹ 0 n, Ỹ n } {Y 0,+ n,i, Y,+ n,i : i M } {Y 0, n,i, Y, n,i : 0 i M }. Suppose that f S n is such that γf n f n U +B n for all γ I. Then f n Y +U +B n. Proof. We largely follow the method of [Bre], Prop. 3.2., which considers the case of e = and U = T ind G KZσ. We write fn 0 = λ I n gn,λ 0 v λ. For λ = [λ 0 ] + π[λ ] + + π n [λ n ], define λ = [λ 0 ] + + π n e [λ n e ] + π n e [λ n e + ] + π n e+ [λ n e+ ] + + π n [λ n ]. A straightforward computation gives that π n π n e gn,λ 0 v λ gn,λ 0 v λ = gn,λ 0 g 0n,λ v λ g 0n, λ v λ = g 0n, λ v λ v λ, λ pe n e + λn e+pe π e v λ v λ The hypothesis on f n then implies the following equalities for all λ I n : v λ v λ F p x r, 2 λ pe n e + λn e+pe π e v λ v λ F p x r. 3 The equality 2 is easily seen to imply v λ F p x r + F p x r y, so we write v λ = c λ x r + d λ x r y. Then 3 implies that λ p e n e + λ n e + pe c λ c λ + d λ x r + d λ d λx r y F p x r. 4 π e Given λ I n, define λ = λ [λ n e ]π n e I n. In other words, λ is the same as λ, but with λ n e replaced by 0. Then the above formula implies that d λ is independent of λ n e, so we write d λ = d λ. Similarly, by Lemma 2. we can view c λ = c λ λ n e as a polynomial in λ n e of degree at most p. From the definition of an extended Q-structured module, we see that e c λ λ n e c λ λ n e + + d λ λp n e + λ n e + pe π e must be a polynomial of degree at most q 0 + in λ n e. Since c λ λ n e has degree at most p in λ n e, the difference c λ λ n e c λ λ n e + has degree at most p 2. But q 0 + p 2, so e n e + λn e+pe the remaining term d λ λp π must also have degree at most p 2, and this forces e d λ = 0 by the observation of Remark 2.4. Therefore c λ λ n e c λ λ n e + has degree at most q 0 + in the variable λ n e, and consequently c λ λ n e has degree at most q Using the deduction above we may rewrite fn 0 = gn,µ+π 0 n e λ+π n e+ ν c µ,νλx r, 5 ν I e µ I n e where c µ,ν X F p [X] is a polynomial of degree at most q From the definition of an extended Q-structured module it is easy to see cf. [Bre], Lemme 3..5 that we may modify f by an element of U + Bn 0 if necessary and assume without loss of generality that for all µ I n e we have c µ,0 X = a µ,0 X q0+ + b µ,0 X q0+2, where a µ,0, b µ,0 F p are constants. Now fix µ I n e and ν I e. Suppose that µ I n e, λ I, and ν I e are such that µ + π n e+ ν π n µ + π n e λ + π n e+ ν π n µ + π n e λ + π n e+ ν KZ.

10 0 MICHAEL M. SCHEIN for some λ I and ν I e. It is easy to see that, equivalently, µ + µ + π n e λ + π n e+ ν + ν µ+π n e λ+π n e+ ν mod π n. Considering this congruence modulo π, we find that µ 0 = 0, and it follows inductively that µ = 0. Similarly, λ = λ, and ν = ν ν by Remark 2.3. We conclude that the terms of h µ = µ + π n e+ ν f 0 n f 0 n with support in λ,ν KZg0 n,µ+π n e λ+π n e+ ν are precisely: ν I e g 0 n,µ+π n e λ+π n e+ ν c µ,ν νλ c µ,ν λx r. 6 By assumption, h µ U + B n and hence c µ,ν is independent of ν. Thus we may write c µ,ν X = c µ X = a µ X q0+ + b µ X q0+2. From 6 we see that a 0 a µ λ q0+ + b 0 b µ λ q0+2 is a polynomial of degree at most q 0 + in λ, and hence b µ = b 0 for all µ I n e. Thus we may write fn 0 = 0 n,µ+π n e λ+π n e+ ν aµ 0,..., µ n e λ q0+ + bλ q0+2 x r. 7 g µ I n e ν I e Here a is a polynomial in the indicated variables and b is a constant. For all 0 j n e denote π n e j γj = and note that the action of γj preserves S 0 m for each m. Using Lemma 2.2 and 7, we observe that if j e, then: γjfn 0 fn 0 = 8 0 n,µ,λ,ν aµ 0,..., µ n e aµ 0,..., µ n e j,..., µ n e λ q0+ x r. g µ I n e ν I e Here we have written gn,µ,λ,ν 0 for g0 n,µ+π n e λ+π n e+ ν. If j e, then we get a similar formula for γjfn 0 fn, 0 but with aµ 0,..., µ n e j,..., µ n e replaced by an expression of the form aµ 0,..., µ n e j, µ n e j, µ n e j+ + R n e j+,..., µ n e + R n e, where each R i is a polynomial in the variables µ n e j,..., µ i e. By assumption, γjfn 0 fn 0 U + B n. If M > 2 then it is evident from the case j = of the formula above that a has degree at most 2 in the variable µ n e. Therefore we may write a = a 0 +a µ n e +a 2 µ 2 n e, where each a i is a polynomial in the variables µ 0,..., µ n e 2. We claim that the polynomial a 2 is constant. Indeed, suppose it is not and consider the minimal j 2 such that µ n e j appears in a 2. Then we see from 8 and the remark following it that γjfn 0 fn 0 has a term of the form µ,λ,ν g0 n,µ,λ,ν Rµ 0,..., µ n e 2 µ 2 n e λ q0+ x r, contradicting γjfn 0 fn 0 U + B n. It is immediate from the case j = of 8 that a 0 and a have degrees at most q + and 2, respectively, in the variable µ n e 2. Modifying a 0 by an element of U + B n, we may assume that it has the form a 0 = â 0 µ 0,..., µ n e 3 µ q+ n e 2. But then we can show that â0 is a scalar by the same argument that was used for a 2. Therefore, after modifying fn 0 by an element of F p Y 0, n,0 + F p Y 0, n, + F p Y 0,+ n, + U + B n, we may assume that f 0 n = 0 g µ I n e ν I e n,µ,λ,ν a µ 0,..., µ n e 2 µ n e λ q0+ x r, where a has degree at most 2 in the variable µ n e. We may now go back to the expression 7 and repeat the entire argument with a in place of a.

11 AN IRREDUCIBILITY CRITERION Iterating the argument, we obtain inductively that, after adding to fn 0 an element of F p Y 0, n,0 + M 2 i= F p Y 0, n,i + F p Y 0,+ n,i + U + B n, we get fn 0 = M 2 0 n,µ,λ,ν aµ 0,..., µ n e M+ µ n e j λ q0+ x r, g µ I n e ν I e and this time a has degree at most in the variable µ n e M+. Thus we may write a = a 0 + a µ n e M+, where the a i are polynomials in the variables µ 0,..., µ n e M. We can show as before that, modulo U + B n, we may take a 0 = cµ q M + n e M for some scalar c F p. On the other hand, a must have degree at most in µ n e M. Using the same method as before, we show that a = d µ n e M + d 0 for scalars d 0, d F p. Moreover, since q M 0 we may modify fn 0 yet again by an element of U + B n and take d 0 = 0. This proves that f 0 F p Ỹ 0 n + i F py 0,+ n,i + i F py 0, n,i + U + B n. Observe that β f also satisfies the hypotheses of the lemma, and hence there exist scalars c, c,+ i, c, i F p such that β f = β f 0 cỹ n 0 + c, 0 Y 0, n,0 + M i= c,+ i Y 0,+ n,i + c, i Y 0, n,i modulo U + B n. But this means that f cỹ n + c, 0 Y, n,0 + M modulo U + B n. j= i= c,+ i Y,+ n,i + c, i Y, n,i Corollary 2.. Suppose that W ind G KZσ is a Q-structured G-submodule and that q 0 p 3. Let n and let M = min{m, n }. Let X be the F p -vector subspace of ind G KZσ spanned by { X 0 n, X n} {X 0,+ n,i, X,+ n,i : i M } {X 0, n,i, X, n,i : 0 i M }. Suppose that f S n is such that γf n f n W +B n for all γ I. Then f n X +U +B n. Proof. In view of Remark 2.8 and the injectivity of T Corollary.5, the claim is immediate from Proposition Construction of a quotient Let F nr be the maximal unramified extension of F, and let L = F nr π /p2. Choose a field embedding F p 2 F p. It induces a character ω 2 : I F GalL/F nr F p F 2 p of the inertia subgroup I F GalF /F. Then ω ep+ 2 is the restriction to I F of the mod p cyclotomic character. Suppose that 0 < r p 2e. Consider a continuous irreducible tamely ramified Galois representation ρ : GalF /F GL 2 F p whose restriction to I F has the form ω r+e 2 0 ρ IF 0 ω pr+e. 2 Recall that a Serre weight in this context is an irreducible F p -representation of the finite group GL 2 F p. In our case the author [Sch], [Sch2] has conjectured that the set of modular weights of ρ is the set D defined in. See [Sch], Def..2 for the definition of the modular weights of a Galois representation. In almost all of the cases under consideration here F/Q p totally ramified, with restrictions on r the conjecture has been proved by Gee and Savitt [GS]. We will inductively construct a sequence of quotients of V 0 = ind G KZσ 0 /T ind G KZσ 0 as follows. Let i e and suppose that V i has been constructed. We claim that the image of X0 i in V i is invariant under the action of I and, furthermore, that the KZ-submodule of V i generated by X i 0 is isomorphic to σ i. This gives us a map ψ Hom KZσ i, V i KZ determined by ψv = X i 0 I, where v Vσ is a highest weight vector. By Frobenius reciprocity we obtain a i map Ψ i Hom G ind G KZσ i, V i, and finally we define V i = V i /Ψ i T ind G KZσ i. Note that v is determined up to scalar, and hence V i is independent of the choice of v. In this section we show that the idea outlined here actually works. If V i has been constructed, let N i denote the kernel of the natural projection ind G KZσ 0 V i.

12 2 MICHAEL M. SCHEIN Lemma 3.. Suppose that n e and the quotient V n has been constructed. Then X 0 n V n I and X n V n I. Proof. Since X n = β X n 0 and β normalizes I, it suffices to show that X n 0 is an I-invariant. Let γ I, and write + aπ b γ =, cπ + dπ where a, b, c, d O. Expand a = [a i]π i, where [a i ] I, and do similarly for b, c, d. For any µ I n+, define ε µ = + dπ + µcπ b + µ + µaπ and expand ε µ = [ε i]π i. Observe that γgn+,µ 0 π n+ ε = µ + πa cεµ 0 + πd + µc On the other hand, π n+ ε µ = π n+ [ε µ ] n+ z 0 π n+2 c +πd+µc where z O. We conclude that γgn+,µ 0 = gn+,ε 0 µ u for some u I. Recall the definition of µ ε from Lemma 2.5 and observe that µ εµ = µ. Therefore, by the same lemma, γ X 0 n X 0 n = gn+,ε 0 µ µ µ n µ r+ n x r gn+,µ 0 µ µ n µ r+ n x r = µ I n+ g 0 n+,ε ε + P ε n + P n ε n + P n r+ ε ε n ε r+ n x r. ε I n+ Observe that we may write γ X n 0 X n 0 = n i= C n, where C n = gn+,ε 0 ε + P ε n + P n ε n + P n r+ ε r+ n x r ε I n+ and if i n then C i = i gn+,ε 0 ε j + P j P i ε 0,..., ε i ε I n+ j= n, j=i+ ε j ε r+ n x r. For each i n we claim that C i N n i N n. This would imply that the image in V n of γ X n 0 X n 0 vanishes for all γ I, hence that X n 0 is indeed an I-invariant in V n. Let c j ε 0,..., ε n, for 0 j r, be the polynomials such that j=0 c j ε n j = ε + P ε n + P n ε n + P n r+ ε r+ n. In particular, c r ε 0,..., ε n = r r + ε + P ε n + P n P n. For each ε = n ε iπ i I n, define v ε = r j=0 c jε 0,..., ε n x r j y j V σ0. Then it follows from Lemma.4 that C n = T gn, ε 0 v ε c r ε 0,..., ε n ε n x + y r. 9 ε I n ε I n ε n I It is easy to see from Lemma 2.5 that P ε 0 is a quadratic polynomial in ε 0, whereas if n > then ε n appears with degree at most one in any term of P n. Therefore, ε n appears with degree at most r + 3 in any term of c r ε n x + y r. Since r + 3 < p by assumption, we see that the second term on the right-hand side of 9 vanishes and thus C n T ind G KZσ 0 = N 0. One proves in a similar way that C i Ψ n i T ind G KZσ n i if i n. For more detail the reader is directed to the analogous argument for A i in the proof of Lemma 3.2..

13 AN IRREDUCIBILITY CRITERION 3 Lemma 3.2. For any λ I, the following identities hold in V n : h λ 0 n = X λ n 0 = gn+,ν 0 λν 0 p r 2n ν ν n νn r+ x r + ν I n+ gn,τ r+n λ p r 2n τ 0 τ n 2 τn r+ yr, τ I n h n = X 0 n 0 = gn+,ν 0 n ν p r 2n 0 ν ν n νn r+ x r + ν I n+ τ I n g n,τ τ 0 τ n 2 τ r+ n yr. Proof. For λ = 0 the claim is obvious, so assume λ 0. First we observe that 0 g 0 λ n+,µ = gn+,µλµ+ z λµ λµ + 0 π n+ z 2 if λµ+ O, where z, z 2 O. On the other hand, if λµ+ O in other words, if µ 0 = λ, then 0 g 0 λ n+,µ = gn,π 0 µ 0 0 µ +λµ z 0 µ 0 π n+ µ. Given our restrictions on r, it follows immediately that h λ n = gn+,µλµ+ 0 µ µ n µ r+ n λµ 0 + p r x r + µ I n+ gn,π µ λµ+ r µ µ n µ r+ n µ p r 0 y r. 0 µ I n+ λµ 0 +=0 For ν I n+, set ν = ν λν if this is defined. For τ I n we set τ = λ πτ, which exists for all τ since we have assumed λ 0. Then the expression above may be rewritten as h λ n = gn+,ν 0 λν 0 r ν ν n ν n r+ x r + gn,τ λ r τ τ n τ n r+ y r. ν I n+ τ I n Let ĥλ n denote the claimed expression for h λ n in the statement of the lemma. We need to show that h λ n ĥλ n lies in N n. By Lemma 2.6 we see that the first summand of h λ n consisting of the terms supported on S 0 n+ can be expressed as ν I n+ g 0 n+,ν λν 0 p 2n r ν + R ν n + R n ν n + R n r+ x r. As in the proof of the preceding lemma, the difference between this expression and the corresponding summand of ĥλ n can be written as a sum n i= A i, where A n = gn+,ν 0 λν 0 p r 2n ν + R ν n + R n ν n + R n r+ νn r+ ν I n+ and for i n we have A i = i gn+,ν 0 λν 0 p r 2n ν j + R j R i ν 0,..., ν i ν I n+ j= n j=i+ ν j x r ν r+ n x r. We claim that A i N n i N n for all i n. Indeed, consider first the case i = n. Define polynomials c j ν 0,..., ν n such that r j=0 c j ν n j = λν 0 p r 2n ν + R ν n +

14 4 MICHAEL M. SCHEIN R n ν n + R n r+ νn r+. For each ν I n define v ν = r j=0 c jx r j y j V σ0. Then we see from the formulae of Lemma.4 that A n = T gn,ν 0 v ν gn,[ν] 0 n Bν 0,..., ν n ν n x + y r = ν I n ν I n T gn,ν 0 v ν gn,ν 0 Bν 0,..., ν n ν n x + y r, ν I n ν I n ν n I where Bν 0,..., ν n = r r + λν 0 p r 2n ν + R ν n + R n R n. Observe that if n >, then ν n appears with degree at most two in any term of R n ν 0,..., ν n. Thus ν n appears with degree at most r + 3 in any term of Bν 0,..., ν n ν n x + y r. But r+3 < p by assumption, so ν n I Bν 0,..., ν n ν n x+y r = 0 and A n T ind G KZσ 0 = N 0. The case i = n = is handled separately but analogously. Now suppose that i < n. Observe that ν i does not appear in A i, and that the projection of A i to V n i is a scalar multiple of the image under Ψ n i of the element i gi,ν 0 λ 0 p r 2n ν j + R j R i ν 0,..., ν i ˆx p r 2n i ind G KZσ n i. ν I i j= Here we denote the usual basis of V σ n i by {ˆx p r 2n i j ŷ j : 0 j p r 2n i}. Using the assumption that p r 2n i > 3, we find that this element actually lies in T ind G KZσ n i and hence that A i N n i as claimed. The remaining terms of h λ n and ĥλ n, those supported on Sn, are shown to be equal modulo N n in a similar way, proving the claim about h λ n. The case of h n is also treated by easy but somewhat tedious computations. Proposition 3.3. The KZ-submodule U V n generated by X n 0 is irreducible and isomorphic to σ n. Proof. Since KZ = 0 I I and since λ 0 X n 0 V n is an I- invariant by Lemma 3., we see that U is spanned by the p + elements h λ n and h n. Inspecting the expressions of Lemma 3.2, we see easily by Corollary 2. that U I = F p X n. 0 Since I is a pro-p-group, any irreducible KZ-submodule U U must contain an I-invariant vector, hence X n 0 U, so U = U and U is irreducible. Finally, observe that for any a, b F p we have [a] 0 X 0 0 [b] n = µ I n+ g 0 n+,[ab ]µ µ µ n µ r+ n a r x r = ab r+n a p r 2n. The claim follows from the classification of irreducible KZ-representations. Proposition 4 of [BL]. See, for instance, Remark 3.4. It can be shown that the KZ-submodule of V n generated by X n is not irreducible, but rather a principal series of dimension p +. The preceding proposition shows that the inductive program described at the beginning of this section may indeed be carried out to produce a sequence of quotients ind G KZσ 0 V 0 V V e. Let κ i : ind G KZσ 0 V i be the natural surjections. For each 0 i e, let p i : V i V e be the natural surjection map. For i e and j {0, }, let X j i = κ e X j i. For any weight σ = σ s,w, let X 0 σ denote the element X σ = µ I2 g 0 2,µ µ s+ ˆx s ind G KZσ. We denote σ e = det e Sym r+2e F 2 p and σ e = det r+e Sym p r 2e F 2 p, compatibly with.

15 AN IRREDUCIBILITY CRITERION 5 Proposition 3.5. Let σ be a weight such that there exists a non-trivial G-equivariant map ind G KZσ/T ind G KZσ V e. Then σ D {σ e, σ e}. Proof. Observe that D acts on the I-invariants of σ i via the character χ i : diagã, d ad i a r and on the I-invariants of σ i via the character χ i : diagã, d a d i d r. So long as 0 < r < p 2e, each of these characters arises from a unique Serre weight. Observe also that N i is a Q-structured G-submodule of ind G KZσ 0 in the sense of Definition 2.7 for the i + -tuple Q = q 0,..., q i given by q 0 = r and q j = p r 2j for j > 0. Indeed, for i = 0 we already noted this above in Remark 2.8. If i > 0 this follows from the explicit expression of the map Ψ i see, for instance, equation 7 on page 266 of [BL] and the observation that, for any n 0 and any ν I n, Ψ i gn,ν 0 ˆx p r 2i = gn,ν 0 X i 0 = gn+i+,ν+π 0 n µ µ µ i µ r+ i x r, µ I i+ Ψ i g n,ν ŷ p r 2i = g n,ν w X 0 i = µ I i+ g n+i+,ν+π n µ µ µ i µ r+ i y r. Let 0 v Vσ I be a highest weight vector, and suppose that D acts on v via the character χ. By assumption there exists a non-zero map Φ Hom G ind G KZσ/T ind G KZσ, V e. By Frobenius reciprocity Φ corresponds to a non-zero map ϕ Hom KZ σ, V e KZ. Then ϕv V I e. For the e-tuple Q = q 0,..., q e just defined, we see by Lemma 2.9 that the subgroup D K of diagonal matrices acts as follows: 0,+ diagã, dx = ad l a r X 0,+,+ diagã, dx = a d l d r X 0,+, l e 0, diagã, dx n,0 = a d 2 d r X 0,, n,0 diagã, dx n,0 = ad 2 a r X, n,0 0, diagã, dx = a d l+2 d r X 0,, diagã, dx = ad l+2 a r X,, l e 2 0, diagã, dx n,e = a d e d r X 0,, n,e diagã, dx n,e = ad e a r X, n,e diagã, d X n 0 = a d n d r X0 n diagã, d X n = ad n a r X n n e By Corollary 2., ϕv must be a linear combination of the elements X j,+ for j {0, } and l e and X j, for j {0, } and 0 l e, as well as X n j if n is in the suitable range. Hence χ is one of the characters appearing in the list above, all of which are equal to χ i or χ i for some 0 i e. Therefore σ D {σ e, σ e}. Lemma 3.6. The element α y r ind G KZσ 0 /T ind G KZσ 0 generates a KZ-submodule isomorphic to σ 0. Proof. One shows by explicit computation, starting from Corollary 2., that if the image of f B ind G KZσ 0 is an I-invariant in ind G KZσ 0 /T ind G KZσ 0, then f F p Id x r F p α y r. The claim now follows by the argument of [Bre], Prop Proposition 3.7. Suppose we are given a map of G-modules τ : V 0 = ind G KZσ 0 /T ind G KZσ 0 W, where W satisfies soc K W = σ D σ and has no non-supersingular subrepresentations. Then τ factors through V e. Proof. We argue by induction. Let n e and suppose it is known that τ factors through V n. We claim it factors through V n. Let Xn 0 be the image in V n of X0 n. Note that the image of the map Ψ n : ind G KZσ n V n is contained in and in fact equal to the submodule of V n generated by the element Xn. 0 Hence if τ X n 0 = 0, then obviously τ factors through V n /Ψ n ind G KZσ n and hence through V n. Now suppose τ X n 0 0. By our assumption on the socle of W, the space Hom G ind G KZσ n, W Hom KZ σ n, W KZ is one-dimensional. Thus every non-zero element of this space, and in particular τ Ψ n, is an eigenvector for the action of the commutative algebra End G ind G KZσ n. It follows that τ Ψ n factors through ind G KZσ n/t ξind G KZσ n for some scalar ξ F p. But ξ = 0, since otherwise the classification of [BL] would imply that the image of τ Ψ n in W contains a non-supersingular representation.

16 6 MICHAEL M. SCHEIN Recall the elements f i defined in the introduction for 0 i e. Proposition 3.8. Suppose that τ : ind G KZσ 0 /T ind G KZσ 0 W is a quotient, that W has no non-supersingular subrepresentations, and that soc K W = σ D σ. If τf e 0, then for each 0 i e the K-submodule of W generated by τf i resp. τβf i is irreducible and isomorphic to σ i resp. σ i. Moreover, if i e and τ X i 0 0, then there exists a non-zero scalar c i F p such that τf i = c i τ X i and τβf i = c i τ X i 0. Proof. Recalling that β acts as an involution on ind G KZσ 0, observe from the definitions that f 0 = X 0 0 and f = X. Hence βf 0 = X and βf = X 0. Since these elements already generate irreducible K-submodules of V e isomorphic to the specified Serre weights by Proposition 3.3 and Lemma 3.6, and since τ factors through V e by Proposition 3.7, the claim is established for i {0, }. Now suppose that the claim is known for i. By Frobenius reciprocity the non-zero element τf i W defines a map Ψ i : ind G KZσ i W, which factors through ind G KZσ i /T ind G KZσ i as in the proof of the previous proposition. By the second part of Lemma., which follows from results that were proved earlier in this section, the element h = µ I 2 g2,µ 0 µ r+2i ˆx r+2i 2 ind G KZσ i /T ind G KZσ i generates an irreducible K-submodule isomorphic to σ i. Hence τβf i = Ψ i h W, which is non-zero by assumption, generates an irreducible K-submodule isomorphic to σ i. Moreover, τβf i is an I-invariant since h is. Similarly, τβf i W determines a map Ψ i : ind G KZσ i /T indg KZσ i W with Ψ if σ i = τβf i. Then Ψ i βf σ i = τf i generates an irreducible K-submodule isomorphic to σ i by Lemma 3.6. Since X i 0 is an I-invariant generating an irreducible K-submodule of V e isomorphic to σ i if i e, and since the Serre weights in soc K W appear with multiplicity one, τ X i 0 is necessarily a scalar multiple of τβf i. Remark 3.9. For each i e, define the elements Z i = λ p r 2i g 0 X 2,λ i 0 = gi+3,λ+π 0 2 µ λp r 2i µ µ i µ r+ i x r ind G KZσ 0. λ I 2 λ I 2 µ I i+ Let τ : ind G KZσ 0 /T ind G KZσ 0 W be such that W has no non-supersingular subrepresentations and that soc K W = σ D σ. Assume that τz i 0 for all i e. Then τ X 0 i 0 for all 0 i e, and one can show that W is irreducible by the same proof as that of Theorem.3, but with X i and Z i playing the roles of f i and z i respectively. To conclude, we observe that the construction of the quotient V e is very natural. Indeed, we began with a weight σ and took V 0 = ind G KZσ/T ind G KZσ. In each intermediate quotient V i, we used Corollary 2. to compute enough information about the I-invariants in V i to find a unique up to scalar multiplication pair of minimal I-invariants X i 0, X i on which D acts via characters that have not appeared in V I i 2. Here we denote V = ind G KZσ, and by minimal we mean that among all I-invariants with this property, X0 i and X i are supported closest to the origin of the Bruhat-Tits tree of G. One of the elements of this pair generates an irreducible KZ-module τ, which by Frobenius reciprocity gives a map Ψ : ind G KZτ V i. We defined V i = V i /ΨT ind G KZτ and repeated the process. Note that the quotient V e is very large; it is non-admissible if e >. However, the main idea behind this paper is that Breuil s [Bre] original computational proof of irreducibility still applies for a totally ramified extension of Q p if one works over V e rather than over V 0 = ind G KZσ 0 /T ind G KZσ 0 itself. References [BL] L. Barthel and R. Livné. Irreducible modular representations of GL 2 of a local field. Duke Math. J , [Bre] Christophe Breuil. Sur quelques représentations modulaires et p-adiques de GL 2 Q p. I. Compositio Math , [BP] Christophe Breuil and Vytautas Paskunas. Towards a modulo p Langlands correspondence for GL 2. To appear in Memoirs Amer. Math. Soc.

17 AN IRREDUCIBILITY CRITERION 7 [BDJ] Kevin Buzzard, Fred Diamond, and Frazer Jarvis. On Serre s conjecture for mod l Galois representations over totally real fields. Duke Math. J , [Eme] Matthew Emerton. On a class of coherent rings, with applications to the smooth representation theory of GL 2 Q p in characteristic p. Preprint, available at emerton/ pdffiles/frob.pdf. [GS] Toby Gee and David Savitt. Serre weights for mod p Hilbert modular forms: the totally ramified case. To appear in J. Reine Angew. Math. [Hu] Yongquan Hu. Diagrammes canoniques et représentations modulo p de GL 2 F. To appear in J. Inst. Math. Jussieu. [Oll] Rachel Ollivier. Le foncteur des invariants sous l action du pro-p-iwahori de GL 2 F. J. Reine Angew. Math , [Sch] Michael M. Schein. Weights in Serre s conjecture for Hilbert modular forms: the ramified case. Israel J. Math , [Sch2] Michael M. Schein. Reduction modulo p of cuspidal representations and weights in Serre s conjecture. Bull. London Math. Soc , Department of Mathematics, Bar Ilan University, Ramat Gan 52900, Israel address: mschein@math.biu.ac.il