aa r X i v : . [ m a t h . N T ] J u l GL ( Q p )-ordinary families and automorphy lifting Yiwen Ding a a B.I.C.M.R., Peking University, Beijing
Abstract
We prove automorphy lifting results for certain essentially conjugate self-dual p -adic Galois repre-sentations ρ over CM imaginary fields F , which satisfy in particular that p splits in F , and that therestriction of ρ on any decomposition group above p is reducible with all the Jordan-H¨older factorsof dimension at most 2. We also show some results on Breuil’s locally analytic socle conjecture incertain non-trianguline case. The main results are obtained by establishing an R = T -type resultover the GL ( Q p )-ordinary families considered in [7]. Contents1 Introduction 2 P -ordinary Galois deformations 53 P -ordinary automorphic representations 8 P -ordinary part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.5 P -ordinary Hecke algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 GL ( Q p ) -ordinary families 14 ( Q p )-ordinary families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4 Local-global compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Auxiliary primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.3 Patching I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.4 Patching II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.5 Automorphy lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.6 Locally analytic socle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 URL: [email protected] (Yiwen Ding) . Introduction Let p >
F/F + be a CM imaginary field such that p splits in F and that F/F + is unramified. For each place v | p of F + , we fix a place e v of F with e v | v . Let E be a sufficientlylarge finite extension of Q p . In this note, we prove automorphy lifting results for certain essentiallyconjugate self-dual p -adic Galois representations ρ of Gal F over E . For simplicity, we summarizeour main result for the case where dim E ρ = 3 in the the following theorem (see Theorem 5.7 forthe statement for general n ) . Theorem 1.1.
Let ρ : Gal F → GL ( E ) be a continuous representation satisfying the followingconditions:1. ρ c ∼ = ρ ∨ ε − n , where ρ c ( g ) = ρ ( cgc − ) , = c ∈ Gal(
F/F + ) , and ε denotes the cyclotomiccharacter.2. ρ is unramified for all but finitely many primes, and unramified for all the places that do notsplit over F + . We denote by S the union of the complement set and the places dividing p .3. ρ is absolutely irreducible, ρ (Gal F ( ζ p ) ) ⊆ GL ( k E ) is adequate and F Ker ad ρ does not contain F ( ζ p ) .4. For all v | p , ρ e v is reducible, i.e. is of the form ρ e v ∼ = (cid:18) ρ e v, ∗ ρ e v, (cid:19) . (1.1)
5. For all v | p , ρ e v is de Rham of distinct Hodge-Tate weights. Suppose moreover one of thefollowing two conditions holds(a) for all v | p , and i = 1 , , ρ e v,i is absolutely irreducible and the Hodge-Tate weights of ρ e v, are strictly bigger than those of ρ e v, ;(b) for all v | p , ρ e v is crystalline and generic in the sense of [8] .6. Let ρ e v,i be the mod p reduction of ρ e v,i (induced by ρ ), ω the modulo p cyclotomic character.Suppose for all v | p (a) Hom
Gal Q p ( ρ e v,i , ρ e v,j ) = 0 for i = j ;(b) Hom
Gal Q p ( ρ e v, , ρ e v, ⊗ k E ω ) = 0 ;(c) Hom
Gal Q p ( ρ e v , ρ e v, ⊗ k E ω ) = 0 .7. There exist a definite unitary group G/F + attached to F/F + such that G is quasi-split atall finite places of F + , and an automorphic representation π of G with the associated Galoisrepresentation ρ π : Gal F → GL ( E ) satisfying(a) ρ π ∼ = ρ ;(b) π v is unramified for all v / ∈ S ; Where we use the convention that the Hodge-Tate weight of the cyclotomic character is 1. I.e. the eigenvalues ( φ , φ , φ ) of the crystalline Frobeinus satisfy φ i φ − j / ∈ { , p } for i = j . We need some more technical assumptions when p = 3, that we ignore in the introduction. c) π is B -ordinary (cf. Definition 4.17, see also Lemma 4.19). Then ρ is automorphic, i.e. there exists an automorphic representation π ′ of G such that ρ ∼ = ρ π ′ . We make a few remarks on the assumptions. The assumptions in 1, 2, 3, the first part of 5, and7(a), 7(b) are standard for automorphy lifting theorems (e.g. see [15], [35], [36], [25], [1],...). Theassumption 4 is crucial for this paper, which gives a necessary condition such that ρ appears inthe GL ( Q p ) -ordinary family that we work with (see the discussions below). The assumption that p splits in F is also crucial because we use some results in p -adic Langlands program, those thatare only known for GL ( Q p ). The assumption 5(a) is a non-critical assumption, which is used fora classicality criterion; when ρ e v is crystalline and generic for all v | p (as in the assumption 5(b)),we apply the classicality result of Breuil-Hellmann-Schraen [9][8] to remove such non-critical as-sumption. The assumption 6 is rather technical, and we make this assumption so that the Galoisdeformations are easier to study. Finally the assumption 7(c) is to ensure that certain automorphylifting of ρ can appear in our GL ( Q p )-ordinary family. One can find analogues of these assump-tions (except for 5(b) and the generic assumption 6) in [25, Thm. 5.11] in ordinary case. Notethat, since we crucially use p -adic Langlands correspondence for GL ( Q p ), any base-change of F that we can use in this paper has to be split at p .We sketch the proof of the theorem. The main object that we work with is (a generalization/variationof) the GL ( Q p )-ordinary families considered in [7]. We fix a compact open subgroup U p of G ( A ∞ ,pF + ),and let b S ( U p , O E ) := { f : G ( F + ) \ G ( A ∞ F + ) /U p → O E | f is continuous } . This O E -module isequipped with a natural action of e T ( U p ) × G ( F + ⊗ Q Q p ) where e T ( U p ) is a (semi-local) completecommutative O E -algebra generated by certain Hecke operators outside p acting on b S ( U p , O E ). To ρ , we can associate a maximal ideal m ρ ⊂ e T ( U p ), and we denote by b S ( U p , O E ) ρ (resp. e T ( U p ) ρ )the localisation of b S ( U p , O E ) (resp. e T ( U p )) at m ρ . We have a natural surjection R ρ, S ։ e T ( U p ) ρ ,where R ρ, S denotes the universal deformation ring of a certain deformation problem S of ρ .Applying Emerton’s P -ordinary part functor (see footnote 4 for P ), we obtain an admissible Ba-nach representation Ord P ( b S ( U p , O E ) ρ ) of L P . We can further decompose Ord P ( b S ( U p , O E ) ρ ) usingthe theory of blocks of [30], in particular, we can associate to the block B (as in the theorem) a e T ( U p ) ρ × L P -equivariant direct summand Ord P ( b S ( U p , O E ) ρ ) B of Ord P ( b S ( U p , O E ) ρ ). The e T ( U p ) ρ -action on Ord P ( b S ( U p , O E ) ρ ) B factors through a certain quotient e T ( U p ) P − ord ρ, B .On the Galois side, we let R B := b ⊗ v | p R B , e v := b ⊗ v | p (cid:0) R pstr ρ e v, b ⊗ R pstr ρ e v, (cid:1) , where R pstr ρ e v,i denotes theuniversal deformation ring of the pseudo-character tr ρ e v,i . Let R P e v − ord , (cid:3) ρ e v , F e v be the framed universal P e v -ordinary deformation ring of ρ e v with respect to the P e v -filtration F e v on ρ e v induced by (1.1) (see § R B , e v → R P e v − ord , (cid:3) ρ e v , F e v . Adding the local conditions { R P e v − ord , (cid:3) ρ e v , F e v } v | p to the deformation problem S , we obtain a quotient R P − ord ρ, S , B of R ρ, S . There isa natural morphism R B → R P − ord ρ, S , B . One can prove that the composition R ρ, S ։ e T ( U p ) ρ ։ Where B is the block associated to { ρ e v,i } in the category of locally finite length smooth L P ( Q p )-representations(we refer to (4.16) and § L P is the Levi subgroup (containing the subgroup of diagonalmatrices) of P := Q v | p P e v ( Q p ) with P e v ⊆ GL the parabolic subgroup corresponding to the filtration (1.1). Inparticular, L P is equal, up to the order of the factors, to Q v | p (GL ( Q p ) × Q × p ). T ( U p ) P − ord ρ, B factors through R P − ord ρ, S , B −→ e T ( U p ) P − ord ρ, B . This is the “ R → T ” map of the paper.We then prove a local-global compatibility result on Ord P ( b S ( U p , O E ) ρ ) B . We use a similar formula-tion as in [28]. The first key point is that the L P -action on Ord P ( b S ( U p , E ) ρ ) B can be parameterizedby R B using p -adic local Langlands correspondence. More precisely, by the theory of Paˇsk¯unas,we can associate to the block B an L P -representation e P B (which is projective in a certain cate-gory, see § p -adic Langlands correspondence) R B ֒ → End L P ( e P B ). Put m ( U p , B ) := Hom L P (cid:0) e P B , Ord P ( b S ( U p , O E ) ρ ) d B (cid:1) where Ord P ( b S ( U p , O E ) ρ ) d B denotes the Schikhof dual of Ord P ( b S ( U p , O E ) ρ ) B (which lies in thesame category as e P B ). The natural action of R B on e P B induces an R B -action on m ( U p , B ). Onecan moreover show that m ( U p , B ) is a finitely generated R B -module. We remark that this R B -action is obtained in a purely local way, and characterizes the L P -action on Ord P ( b S ( U p , O E ) ρ ) B .On the other hand, m ( U p , B ) inherits from Ord P ( b S ( U p , O E ) ρ ) d B an action of e T ( U p ) P − ord ρ, B , hence isequipped with another R B -action via R B −→ R P − ord ρ, S , B −→ e T ( U p ) P − ord ρ, B . Note that this R B -action is obtained in a global way (since it comes from the global Galois defor-mation ring). Then we show that these two R B -actions on m ( U p , B ) coincide (up to a certain twist,that we ignore in the introduction). In summary, we find ourselves in a similar situation as Hida’sordinary families (see § § R B → e T ( U p ) P − ord ρ, B and afinitely generated e T ( U p ) P − ord ρ, B -module m ( U p , B ) with nice properties as R B -module.We then apply the Taylor-Wiles patching argument ([13] [34] [28]) to our GL ( Q p )-ordinary familiesand obtain the following data: S ∞ → R ∞ y m ∞ ( B ) , where S ∞ is a formal power series over O E , R ∞ is a patched global deformation ring, and m ∞ ( B )is a finitely generated R ∞ -module, which is flat over S ∞ . We remark thatdim R ∞ = dim S ∞ + dim( B p ∩ L P )where B p := Q v | p B ( Q p ). We also have a closed ideal a ⊂ S ∞ such that R ∞ / a ։ R P − ord S ,ρ, B , andwe have an R ∞ -equivariant isomorphism m ∞ ( B ) / a ∼ = m ( U p , B ) (where R ∞ acts on m ( U p , B ) viathe precedent projection). Using Taylor’s Ihara avoidance and arguments on supports of modules,one can show that ρ appears in the GL ( Q p )-ordinary family, in other words, ρ can be attached to P -ordinary p -adic automorphic representations. Finally we use the assumption 5 to show that ρ canbe attached to classical automorphic representations. Assuming 5(a), the result follows from theexistence of locally algebraic vectors in GL ( Q p )-representations in de Rham case and an adjunc-tion property of the functor Ord P ( − ). For 5(b), we first use p -adic local Langlands correspondencefor GL ( Q p ) (the result “trianguline implying finite slope”) to show that ρ appears in the eigen-variety, and then deduce the theorem from the classicality result of Breuil-Hellmann-Schraen [9] [8].4inally, under similar assumptions as in Theorem 1.1 except the assumption 5, and assuming ρ isautomorphic `a priori (which implies ρ e v is de Rham of distinct Hodge-Tate weights for v | p ), whenthe Hodge-Tate weights of ρ e v, are not bigger that those of ρ e v, (which is contrary to the assumption5(a), and is often referred to as the critical case), then the automorphy lifting method allows us tofind a non-classical point in the GL ( Q p )-ordinary family Spf e T ( U p ) P − ord ρ, B associated to ρ . We thendeduce from the existence of the non-classical point some results towards Breuil’s locally analyticsocle conjecture (cf. Theorem 5.10, Remark 5.11). Let us mention that when ρ e v is non-trianguline(i.e. ρ e v, or ρ e v, is not trianguline), these results provide probably a first known example (to theauthor’s knowledge) on the conjecture in the non-trianguline case. Acknowledgement
This work grows out from discussions with Lue Pan, and I want to thank him for the helpfuldiscussions and for answering my questions. I also want to thank Yongquan Hu for answering myquestions during the preparation of this note. The work was supported by Grant No. 7101502007and No. 8102600246 from Peking University.
Throughout the paper, E will be a finite extension of Q p , with O E its ring of integers, ̟ E auniformizer of O E , and k E := O E /̟ E . Let ε : Gal Q p → E × denote the cyclotomic character, ω : Gal Q p → k × E the modulo p cyclotomic character. We use the convention that the Hodge-Tateweight of ε is 1. We normalize local class field theory by sending a uniformizer to a (lift of the)geometric Frobenius. In this way, we view characters of Gal Q p as characters of Q × p without furthermention.For a torsion O E -module N , let N ∨ := Hom O E ( N, E/ O E ) be the Pontryagain dual of N . For M a p -adically complete torsion free O E -module (so M ∼ = lim ←− n M/̟ nE ), we let M d := Hom O E ( M, O E )equipped with the point-wise convergence topology, be the Schikhof dual of M . We haveHom O E ( M, O E ) ∼ = lim ←− n Hom O E /̟ nE ( M/̟ nE , O E /̟ nE ) ∼ = lim ←− n ( M/̟ nE ) ∨ , (1.2)where the map ( M/̟ nE ) ∨ ։ ( M/̟ n − E ) ∨ is induced by the injection M/̟ n − E ̟ E −−→ M/̟ nE . Wealso have (e.g. see the proof of [33, Thm. 1.2]) M ∼ = Hom cts O E ( M d , O E ) (1.3)where the right hand side is equipped with the compact-open topology. P -ordinary Galois deformations In this section, for P a parabolic subgroup of GL n , we study P -ordinary Galois deformations, andshow some (standard) properties of P -ordinary Galois deformation rings.5et L be a finite extension over Q p . We enlarge E such that E contains all the embeddings of L in Q p . Let B be the Borel subgroup of GL n of upper triangular matrices, and let P be a parabolicsubgroup containing B with a Levi subgroup L P given by (where P ki =1 n i = n ): GL n · · ·
00 GL n · · · · · · GL n k . (2.1)Denote by s i := P i − j =0 n j where we set n = 0 (hence s = 0).Let ( ρ, V k E ) be an n -dimensional P -ordinary representation of Gal L over k E in the sense of [7, Def.5.1], i.e. there exists an increasing Gal L -equivariant filtration F : 0 = Fil V k E ( Fil V k E ( · · · ( Fil k V k E = V k E such that dim k E gr i F := Fil i V k E / Fil i − V k E = n i . Let ( ρ i , gr i F ) be the Gal L -representation givenby the graded piece. We choose a basis { e , · · · , e n } of V k E such that { e , · · · , e s i } is a basis ofFil i − V k E for all i . Under this basis, ρ corresponds to a continuous morphism ρ : Gal L → P ( k E ).Let Art( O E ) be the category of local artinian O E -algebras with residue field k E and Def (cid:3) ρ the functorof framed deformations of ρ , i.e. the functor from Art( O E ) to sets which sends A ∈ Art( O E ) tothe set { ρ A : Gal L → GL n ( A ) | ρ A ≡ ρ (mod m A ) } . Let Def P − ord , (cid:3) ρ, F be the subfunctor of Def (cid:3) ρ which sends A ∈ Art( O E ) to the set of ρ A ∈ Def (cid:3) ρ ( A ) satisfying that the underlying A -module V A of ρ A admits an increasing filtration F A = Fil • V A by Gal L invariant free A submodules which aredirect summands as A -modules such that Fil i V A ∼ = Fil i V k E (mod m A ). We assume the followinghypothesis. Hypothesis 2.1.
Suppose
Hom
Gal L ( ρ i , ρ j ) = 0 for all i = j . By the same argument as in the proof of [7, Lem. 5.3], we have:
Lemma 2.2.
Assume Hypothesis 2.1 and let ρ A ∈ Def (cid:3) ρ .(1) ρ A ∈ Def P − ord , (cid:3) ρ, F if and only if there exists M ∈ GL n ( A ) such that M ρ A M − has image in P ( A ) , and M ρ A M − ≡ ρ (mod m A ) .(2) Suppose there exist M , M ∈ GL n ( A ) such that M i ρ A M − i has image in P ( A ) , and M i ρ A M − i ≡ ρ (mod m A ) , then M M − ∈ P ( A ) . In particular, if ρ A ∈ Def P − ord , (cid:3) ρ, F ( A ), then the associated increasing filtration Fil • V A is unique.Recall that Def (cid:3) ρ is pro-representable by a complete local noetherian O E -algebra R (cid:3) ρ of residuefield k E . Proposition 2.3.
Assume Hypothesis 2.1, the functor
Def P − ord , (cid:3) ρ, F is pro-representable by a com-plete local noetherian O E -algebra R P − ord , (cid:3) ρ, F , which is a quotient of R (cid:3) ρ .Proof. By Schlessinger’s criterion, the fact that Def P − ord , (cid:3) ρ, F is a subfunctor of Def (cid:3) ρ which is pro-representable, it suffices to show that given morphisms f : A → C , f : B → C in Art( O E ) with f surjective and small, the induced mapsDef P − ord , (cid:3) ρ, F ( A × C B ) −→ Def P − ord , (cid:3) ρ, F ( A ) × Def P − ord , (cid:3) ρ, F ( C ) Def P − ord , (cid:3) ρ, F ( B )6s surjective. Let ( ρ A , ρ B ) ∈ Def P − ord , (cid:3) ρ, F ( A ) × Def P − ord , (cid:3) ρ, F ( C ) Def P − ord , (cid:3) ρ, F ( B ) and let ˜ ρ be a liftingof ( ρ A , ρ B ) in Def (cid:3) ρ ( A × C B ). By Lemma 2.2 (1), there exists M A ∈ GL n ( A ) (resp. M B ) suchthat M A ρ A M − A (resp. M B ρ B M − B ) has image in P ( A ) (resp. in P ( B )) and that M A ρ A M − ≡ ρ (mod m A ) (resp. M B ρ B M − B ≡ ρ (mod m B )). Denote by M A and M B the image of M A , M B in GL n ( C ) respectively. By Lemma 2.2 (2), there exists N C ∈ P ( C ) such that M B = N C M A .Let N B ∈ P ( B ) be a lifting of N C . Then we see N B M B ρ B M − B N − B has image in P ( B ). Let˜ M be a lifting of ( M A , N B M B ) ∈ GL n ( A × C B ). It is easy to check that ˜ M ˜ ρ ˜ M − has image in P ( A × C B ) ∼ = P ( A ) × P ( C ) P ( B ). The proposition follows.Let Hom F ( V k E , V k E ) be the k E -vector subspace of Hom k E ( V k E , V k E ) consisting of morphisms offiltered k E -vector spaces, i.e. f : V k E → V k E lies in Hom F ( V k E , V k E ) if and only if f | Fil i V kE ⊆ Fil i V k E for all i . We have the following easy lemma. Lemma 2.4. dim k E Hom F ( V k E , V k E ) = P ki =1 ( n i ( n − s i )) .Proof. Using the basis { e , · · · , e n } of V k E , we identify the k E -vector space Hom k E ( V k E , V k E ) (resp.Hom F ( V k E , V k E )) with M n ( k E ) (resp. p ( k E )) (where p denotes the Lie algebra of P ). The lemmafollows.The k E -vector space Hom k E ( V k E , V k E ) is equipped with a natural Gal L -action given by( gf )( v ) = gf ( g − v ) , (2.2)and we denote by Ad ρ the corresponding representation. Since the filtration F on V k E is Gal L -equivariant, one easily check Hom F ( V k E , V k E ) is Gal L -invariant. We denote by Ad F ρ the corre-sponding Gal L -representation (which is a subrepresentation of Ad ρ ). Proposition 2.5.
Assume Hypothesis 2.1, we have a natural isomorphism of k E vector-spaces Def P − ord , (cid:3) ρ, F ( k E [ ǫ ] /ǫ ) ∼ = B (Gal L , Ad F ρ ) + Z (Gal L , Ad ρ ) , where we use the standard notation with B for the -coboundary and Z for the -cocycle.Proof. Let ˜ ρ ∈ Def (cid:3) ρ ( k E [ ǫ ] /ǫ ), and c ∈ B (Gal L , Ad ρ ) be the associated 1-th coboundary, i.e.˜ ρ ( g ) = ρ ( g )(1 + c ( g ) ǫ ) for g ∈ Gal L . It is sufficient to show ˜ ρ ∈ Def P − ord , (cid:3) ρ, F ( k E [ ǫ ] /ǫ ) if and only if c ∈ B (Gal L , Ad F ρ ) + Z (Gal , Ad ρ ).Suppose ˜ ρ ∈ Def P − ord , (cid:3) ρ, F ( k E [ ǫ ] /ǫ ). By Lemma 2.2, there exists M ∈ GL n ( k E [ ǫ ] /ǫ ) such that M ˜ ρM − has image in P ( k E [ ǫ ] /ǫ ), M ˜ ρM − ≡ ρ (mod ǫ ) and M modulo ǫ lies in P ( k E ). Thereexist then U ∈ P ( k E ) and A ∈ M n ( k E ) such that M = U (1 + Aǫ ). For any g ∈ Gal L , using(1 + Aǫ )˜ ρ ( g )(1 − Aǫ ) ∈ P ( k E [ ǫ ] /ǫ ), we deduce c ( g ) + ρ ( g ) − Aρ ( g ) − A ∈ p ( k E ) , (2.3)hence c ( g ) ∈ B (Gal L , Ad F ρ ) + Z (Gal , Ad ρ ).Conversely, if c ( g ) ∈ B (Gal L , Ad F ρ ) + Z (Gal , Ad ρ ), there exists A ∈ M n ( k E ) such that (2.3)holds, and it is easy to check (1 + Aǫ )˜ ρ (1 − Aǫ ) has image in P ( k E [ ǫ ] /ǫ ) and is equal to ρ modulo ǫ . This concludes the proof. Lemma 2.6.
Assume Hypothesis 2.1, if H (Gal L , Ad F ρ ) = 0 , then R P − ord , (cid:3) ρ, F is formally smoothover O E . roof. Let A ։ A/I be a small extension (i.e. I = ( ǫ ) with ǫ m A = 0). It is sufficient to show thatthe natural map Def P − ord , (cid:3) ρ, F ( A ) → Def P − ord , (cid:3) ρ, F ( A/I ) is surjective. Let ρ A/I ∈ Def P − ord , (cid:3) ρ, F ( A/I ),replacing ρ A/I be a certain conjugate of ρ A/I , we assume ρ A/I has image in P ( A/I ), and it issufficient to show there exists ρ A : Gal L → P ( A ) such that ρ A ≡ ρ A/I (mod ǫ ). Let ρ A : Gal L → P ( A ) be a set theoretic lift of ρ A/I . By standard arguments in Galois deformation theory, theobstruction for ρ A being a group homomorphism corresponds to an element c ∈ H (Gal L , Ad F ρ )given by ρ A ( g , g ) ρ A ( g ) − ρ A ( g ) − = 1 + c ( g , g ) ǫ for g , g ∈ Gal L . Since H (Gal L , Ad F ρ ) = 0,the existence of a homomorphism ρ A follows, from which we deduce the lemma.For 1 ≤ i < j ≤ k , we denote by ρ ji := Fil j ρ/ Fil i − ρ . Lemma 2.7.
Suppose that for any i , Hom
Gal L ( ρ i , ρ i ⊗ k E ω ) = 0 , then H (Gal L , Ad F ρ ) = 0 .Proof. We have a natural Gal L -equivariant exact sequence0 −→ Hom k E ( ρ k , ρ ) −→ Hom F ( ρ, ρ ) −→ Hom F (Fil k − ρ, Fil k − ρ ) −→ , where Fil k − ρ is equipped with the induced filtration. By assumption, Ext L ( ρ k , ρ ) = 0. Thelemma follows then by an easy d´evissage/induction argument. Corollary 2.8.
Assume Hypothesis 2.1, and keep the assumption in Lemma 2.7. Then R P − ord , (cid:3) ρ, F is formally smooth of relative dimension n + [ L : Q p ] P ki =1 ( n i ( n − s i )) over O E .Proof. It is not difficult to see that Hom k E ( ρ, ρ ) / Hom F ( ρ, ρ ) is isomorphic (as a Gal L -representation)to a successive extension of Hom k E ( ρ i , ρ j ) with i = j . By Hypothesis 2.1 and d´evissage, we deducethat H (Gal L , Ad F ρ ) ∼ −→ H (Gal L , Ad ρ ) and H (Gal L , Ad F ρ ) ֒ −→ H (Gal L , Ad ρ ) . Consequently, we deduce Z (Gal L , Ad ρ ) ∩ B (Gal L , Ad F ρ ) = Z (Gal L , Ad F ρ ). Hencedim k E ( B (Gal L , Ad F ρ ) + Z (Gal L , Ad ρ ))= dim k E H (Gal L , Ad F ρ ) + dim k E Z (Gal L , Ad ρ )= dim k E H (Gal L , Ad F ρ ) + n − dim k E H (Gal L , Ad ρ )= n + [ L : Q p ] k X i =1 ( n i ( n − s i )) , where the last equation follows from the Euler characteristic formula, Lemma 2.4 and Lemma 2.7.Together with Lemma 2.6, the corollary follows. P -ordinary automorphic representations In this section, we recall (and generalize) some results of [7, §
6] on P -ordinary automorphicrepresentations. 8 .1. Global setup We fix field embeddings ι ∞ : Q ֒ → C , ι p : Q ֒ → Q p . We also fix F + a totally real number field, F a quadratic totally imaginary extension of F + such that any place of F + above p is split in F ,and G/F + a unitary group attached to the quadratic extension F/F + as in [3, § G × F + F ∼ = GL n ( n ≥
2) and G ( F + ⊗ Q R ) is compact. For a finite place v of F + which is totallysplit in F , we fix a place ˜ v of F dividing v , and we have an isomorphism i G, e v : G ( F + v ) ∼ −→ GL n ( F ˜ v ).We let S p denote the set of places of F + dividing p . For an open compact subgroup U p = Q v ∤ p U v of G ( A p, ∞ F + ), we put b S ( U p , E ) := n f : G ( F + ) \ G ( A ∞ F + ) /U p −→ E, f is continuous o . Since G ( F + ⊗ Q R ) is compact, G ( F + ) \ G ( A ∞ F + ) /U p is a profinite set, and we see that b S ( U p , E ) isa Banach space over E with the norm defined by the (complete) O E -lattice: b S ( U p , O E ) := n f : G ( F + ) \ G ( A ∞ F + ) /U p −→ O E , f is continuous o . Moreover, b S ( U p , E ) is equipped with a continuous action of G ( F + ⊗ Q Q p ) given by ( g ′ f )( g ) = f ( gg ′ )for f ∈ b S ( U p , E ), g ′ ∈ G ( F + ⊗ Q Q p ), g ∈ G ( A ∞ F + ). The lattice b S ( U p , O E ) is obviously stable bythis action, so the Banach representation b S ( U p , E ) of G ( F + ⊗ Q Q p ) is unitary.Let S be a finite set of finite places of F + consisting of those v such that v | p , or v ramifies in F , or v is unramified and U v is not maximal hyperspecial. Let T ( U p ) := O E [ T ( j )˜ v ] be the commutativepolynomial O E -algebra generated by the formal variables T ( j )˜ v where j ∈ { , · · · , n } and v / ∈ S splits in F . The O E -algebra T ( U p ) acts on b S ( U p , E ) and b S ( U p , O E ) by making T ( j )˜ v act by thedouble coset operator: T ( j )˜ v := h U v g v i − G, ˜ v (cid:18) n − j ̟ ˜ v j (cid:19) g − v U v i (3.1)where ̟ ˜ v is a uniformizer of F ˜ v , and where g v ∈ G ( F + v ) is such that i G, ˜ v ( g − v U v g v ) = GL n ( O F ˜ v ).This action commutes with that of G ( F + ⊗ Q Q p ).Recall that the automorphic representations of G ( A F + ) are the irreducible constituents of the C -vector space of functions f : G ( F + ) \ G ( A F + ) −→ C which are: • C ∞ when restricted to G ( F + ⊗ Q R ) • locally constant when restricted to G ( A ∞ F + ) • G ( F + ⊗ Q R )-finite,where G ( A F + ) acts on this space via right translation. An automorphic representation π is isomor-phic to π ∞ ⊗ C π ∞ where π ∞ = W ∞ is an irreducible algebraic representation of (Res F + / Q G )( R ) = G ( F + ⊗ Q R ) over C and π ∞ ∼ = Hom G ( F + ⊗ Q R ) ( W ∞ , π ) ∼ = ⊗ ′ v π v is an irreducible smooth represen-tation of G ( A ∞ F + ). The algebraic representation W ∞ | (Res F + / Q G )( Q ) is defined over Q via ι ∞ and wedenote by W p its base change to Q p via ι p , which is thus an irreducible algebraic representation of(Res F + / Q G )( Q p ) = G ( F + ⊗ Q Q p ) over Q p . Via the decomposition G ( F + ⊗ Q Q p ) ∼ −→ Q v ∈ S p G ( F + v ),one has W p ∼ = ⊗ v ∈ S p W v where W v is an irreducible algebraic representation of G ( F + v ) over Q p .9ne can also prove π ∞ is defined over a number field via ι ∞ (e.g. see [3, § π ∞ ,p := ⊗ ′ v ∤ p π v , so that we have π ∞ ∼ = π ∞ ,p ⊗ Q π p (seen over Q via ι ∞ ), and by m ( π ) ∈ Z ≥ themultiplicity of π in the above space of functions f : G ( F + ) \ G ( A F + ) → C . Denote by b S ( U p , E ) lalg the subspace of b S ( U p , E ) of locally algebraic vectors for the (Res F + / Q G )( Q p ) = G ( F + ⊗ Q Q p )-action, which is stable by T ( U p ). We have an isomorphism which is equivariant under the actionof G ( F + ⊗ Q Q p ) × T ( U p ) (see e.g. [5, Prop. 5.1] and the references in [5, § b S ( U p , E ) lalg ⊗ E Q p ∼ = M π (cid:16) ( π ∞ ,p ) U p ⊗ Q ( π p ⊗ Q W p ) (cid:17) ⊕ m ( π ) (3.2)where π ∼ = π ∞ ⊗ Q π ∞ runs through the automorphic representations of G ( A F + ) and W p is associatedto π ∞ = W ∞ as above, and where T ( j )˜ v ∈ T ( U p ) acts on ( π ∞ ,p ) U p by the double coset operator (3.1).Following [15, § U p is sufficiently small if there is a place v ∤ p such that 1 is theonly element of finite order in U v . We have (e.g. see [7, Lem. 6.1]) Lemma 3.1.
Assume U p sufficiently small, then for any compact open subgroup U p of G ( F + ⊗ Q Q p ) there is an integer r ≥ such that b S ( U p , O E ) | U p is isomorphic to C ( U p , O E ) ⊕ r . In the following, we assume U p sufficiently small. Let G n be the group scheme over Z which is the semi-direct product of { , } acting on GL × GL n via ( µ, g ) − = ( µ, ( g t ) − µ ) . Denote by ν : G n → GL the morphism given by ( µ, g )
7→ − µ . Let ρ : Gal F + → G n ( k E ) be acontinuous representation such that • ρ (Gal F ) ⊆ GL ( k E ) × GL n ( k E ), • ρ is unramified outside S , • ρ ( c ) ∈ G n ( k E ) \ GL n ( k E ) (where c denote the complex conjugation), • the composition Gal F + ρ −→ G n ( k E ) ν −→ k × E is equal to ω − n δ nF/F + , where δ F/F + is the uniquenon-trivial character of Gal( F/F + ).Denote by ρ F the composition Gal F ρ −→ GL × GL n ։ GL n ( k E ) (where the second the map denotesthe natural projection), then we have ρ cF ∼ = ρ ∨ F ⊗ k E χ − n cyc . For a place v of F + such that v = e v e v c in F , denote by ρ e v := ρ F | F e v .Denote by S the following deformation problem (cf. [15, § § F/F + , S, e S, O E , ρ, ε − n δ F/F + , { R ¯ (cid:3) ρ e v } v ∈ S ) , where R ¯ (cid:3) ρ e v denotes the reduced quotient of the universal framed deformation ring of ρ e v , and e S = { e v | v ∈ S } . Suppose ρ is absolutely irreducible. By [15, Prop. 2.2.9], the deformation problem S is pro-represented by a complete local noetherian O E -algebra R ρ, S . Denote by R (cid:3) ρ, S the S -framed S -deformations (cf. [36, Def. 3.1]). By definition, R (cid:3) ρ, S is formally smooth over R ρ, S of relative dimension n | S | . Let R loc := b ⊗ v ∈ S R ¯ (cid:3) v , where the tensor product is taken over O E .We have by definition a natural morphism R loc −→ R (cid:3) ρ, S . We recall the definition of some useful pro- p -Hecke algebras and of their localisations.For s ∈ Z > and a compact open subgroup U p of G ( F + ⊗ Q Q p ) ∼ = Q v ∈ S p GL n ( F e v ), we let T ( U p U p , O E /̟ sE ) (resp. T ( U p U p , O E )) be the O E /̟ sE -subalgebra (resp. O E -subalgebra) of theendomorphism ring of S ( U p U p , O E /̟ sE ) (resp. S ( U p U p , O E )) generated by the operators in T ( U p ).Then T ( U p U p , O E /̟ sE ) is a finite O E /̟ sE algebra (cid:0) resp. T ( U p U p , O E ) is an O E -algebra which isfinite free as O E -module (cid:1) . We have T ( U p U p , O E ) ∼ −−→ lim ←− s T ( U p U p , O E /̟ sE ) , (3.3) e T ( U p ) := lim ←− s lim ←− U p T ( U p U p , O E /̟ sE ) ∼ = lim ←− U p lim ←− s T ( U p U p , O E /̟ sE ) ∼ = lim ←− U p T ( U p U p , O E ) . (3.4) We have as in [7, Lem. 6.3]:
Lemma 3.2.
The O E -algebra e T ( U p ) is reduced and acts faithfully on b S ( U p , E ) . To ρ (as in § m ρ of residue field k E of T ( U p ) such that for v / ∈ S splitting in F , the characteristic polynomial of ρ e v (Frob ˜ v ), where Frob ˜ v is a geometric Frobenius at˜ v , is given by: X n + · · · + ( − j (Nm ˜ v ) j ( j − θ ρ ( T ( j )˜ v ) X n − j + · · · + ( − n (Nm ˜ v ) n ( n − θ ρ ( T ( n )˜ v ) (3.5)where Nm ˜ v is the cardinality of the residue field at ˜ v and θ ρ : T ( U p ) / m ρ ∼ −→ k E . For a T ( U p )-module M , denote by M ρ the localisation of M at m ( ρ ). Recall a maximal ideal m ( ρ ) of T ( U p ) iscalled U p -automorphic if there exist s , U p as above such that the localisation S ( U p U p , O E /̟ sE ) ρ isnonzero. Suppose m ( ρ ) is U p -automorphic, then m ( ρ ) corresponds to a maximal ideal, still denotedby m ( ρ ), of e T ( U p ). The localisation e T ( U p ) ρ is a direct factor of e T ( U p ) (e.g. by [7, Lem. 6.5]), andthere is a natural isomorphism e T ( U p ) ρ ∼ = lim ←− s lim ←− U p T ( U p U p , O E /̟ sE ) ρ ∼ = lim ←− U p T ( U p U p , O E ) ρ . (3.6)Put b S ( U p , O E ) ρ := lim ←− s lim −→ U p S ( U p U p , O E /̟ sE ) ρ . We have as in [7, § Lemma 3.3.
Suppose ρ is U p -automorphic.(1) b S ( U p , O E ) ρ is a e T ( U p ) × G ( F ⊗ Q Q p ) -equivariant direct summand of b S ( U p , O E ) .(2) The action of e T ( U p ) on b S ( U p , O E ) ρ factors through e T ( U p ) ρ .(3) The O E -algebra e T ( U p ) ρ is reduced and acts faithfully on b S ( U p , O E ) ρ . We assume the following hypothesis:
Hypothesis 3.4.
We have p > , F/F + is unramified and G is quasi-split at all finite places of F + . Under the hypothesis, by [36, Prop. 6.7], there exists a natural surjection of complete O E -algebras R ρ, S − ։ e T ( U p ) ρ . In particular, e T ( U p ) ρ is a noetherian (local complete) O E -algebra.11 .4. P -ordinary part We fix a parabolic subgroup P ∼ = Y v ∈ S p Res F e v Q p P e v of Q v ∈ S p Res F e v Q p GL n , such that P e v is a parabolic subgroup of GL n containing the Borel subgroup B of upper triangular matrices. Let L e v ∼ = GL n e v, × · · ·× GL n e v,k e v be the Levi subgroup of P e v containingthe diagonal subgroup T , P e v be the parabolic subgroup of GL n opposite to P e v , N P e v (resp. N P e v )be the unipotent radical of P e v (resp. P e v ), and Z L e v be the center of L e v . For i = 1 , · · · , k e v , let s e v,i := P i − j =0 n e v,j , where n e v, := 0. Put L P := Y v ∈ S p Res F e v Q p L e v , P := Y σ ∈ S p Res F e v Q p P e v ,N P := Y v ∈ S p Res F e v Q p N P e v , N P := Y v ∈ S p Res F e v Q p N P e v , Z L P := Y v ∈ S p Res F e v Q p Z L e v . Thus L P is the Levi subgroup of P containing Q v ∈ S p Res F e v Q p T , P is the parabolic subgroup oppositeto P , Z L P is the center of L P , and N P (resp. N P ) is the unipotent radical of P (resp. of P ).For v ∈ S p , i ∈ Z ≥ , let K i, e v := { g ∈ GL n ( O F e v ) | g ≡ ̟ i e v ) } ,N i, e v := N P e v ( F e v ) ∩ K i, e v , L i, e v := L e v ( F e v ) ∩ K i, e v , N i, e v := N P e v ( F e v ) ∩ K i, e v . For i ≥ j ≥
0, put K i,j, e v := N i, e v L j, e v N , e v . Let Z + L e v := { ( a , · · · , a k e v ) ∈ Z L e v ( F e v ) | val p ( a ) ≥ · · · ≥ val p ( a k e v ) } . Finally, we put K i := Q v ∈ S p K i, e v , N i := Q v ∈ S p N i, e v , L i := Q v ∈ S p L i, e v , N i := Q v ∈ S p N i, e v , K i,j := Q v ∈ S p K i,j, e v ∼ = N i L j N , and Z + L P := Q v ∈ S p Z + L e v . The proof of the following lemma isstraightforward (and we omit). Lemma 3.5. (1) For i ∈ Z ≥ , K i is a normal subgroup of K , and N i × L i × N i ∼ −→ K i .(2) For i ≥ j ≥ , we have N i × L j × N ∼ −→ K i,j .(3) For z ∈ Z + L P , i ∈ Z ≥ , we have N i ⊆ zN i z − . Applying Emerton’s ordinary part functor ([21, § L P ( Q p ):Ord P ( b S ( U p , E ) ρ ) ∼ = Ord P ( b S ( U p , O E ) ρ ) ⊗ O E E. Let M be a finitely generated O E -module, equipped with an O E -linear action of Z + L P . Consider the O E -subalgebra B of End O E ( M ) generated by the image of ι : Z + L P → End O E ( M ). It is clear that B is a finite O E -algebra. Recall that a maximal ideal n of B is called ordinary if n ∩ ι ( Z + L P ) = ∅ .12enote by M ord := Q n ordinary M n , which is a direct summand of M . The induced action of Z + L P on M ord is invertible and hence extends naturally to an action of Z L P . We have thenOrd P ( b S ( U p , O E ) ρ ) ∼ = lim ←− k Ord P ( S ( U p , O E /̟ kE ) ρ ) ∼ = lim ←− k lim −→ i Ord P ( S ( U p , O E /̟ kE ) ρ ) L i ∼ = lim ←− k lim −→ i ( S ( U p , O E /̟ kE ) K i,i ) ord ∼ = lim ←− k lim −→ i ( S ( U p K i,i , O E /̟ kE ) ρ ) ord , (3.7)where the first isomorphism is by definition, the second from the fact that Ord P ( S ( U p , O E /̟ kE ) ρ )is a smooth representation of L P ( L ) over O E /̟ kE , the third isomorphism from step (c) in the proofof [7, Thm. 4.4], and the last isomorphism from S ( U p , O E /̟ kE ) K i,i ρ ∼ −→ S ( U p K i,i , O E /̟ kE ) ρ .We define as in [7, (4.15)]: Ord P ( S ( U p , O E ) ρ ) ∼ = lim −→ i (cid:0) S ( U p , O E ) K i,i ρ (cid:1) ord , which is equipped with a natural smooth action of L P ( Q p ). As in the proof of [7, Lem. 6.8 (1)],we have for k ≥
1: Ord P ( S ( U p , O E ) ρ ) /̟ kE ∼ −−→ Ord P ( S ( U p , O E /̟ kE ) ρ ) . (3.8)In particular, Ord P ( S ( U p , O E ) ρ is dense in Ord P ( b S ( U p , O E ) ρ ). Finally, by [7, Cor. 4.6], we have Lemma 3.6.
The representation
Ord P ( b S ( U p , O E ) ρ ) | L P ( Z p ) is isomorphic to a direct summand of C ( L P ( Z p ) , O E ) ⊕ r for some r ≥ .3.5. P -ordinary Hecke algebra Assume Ord P ( b S ( U p , O E ) ρ ) = 0. Note that by (3.8), this implies Ord P ( S ( U p , O E ) ρ ) = 0. By thelocal-global compatibility for classical local Langlands correspondence (see [36, Thm. 6.5 (v)], [12])and using [7, Prop. 5.10], we can deduce that ρ e v is P e v -ordinary, for v ∈ S p .For any i ≥ ∗ ∈ {O E , O E /̟ sE } , ( S ( U p , ∗ ) K i,i ρ ) ord = ( S ( U p K i,i , ∗ ) ρ ) ord is stable by T ( U p ) (sincethe action of T ( U p ) on S ( U p , ∗ ) K i,i ρ commutes with that of L + P ), and we denote by T ( U p K i,i , ∗ ) P − ord ρ the O E -subalgebra of the endomorphism ring of ( S ( U p , ∗ ) K i,i ρ ) ord generated by the operators in T ( U p ). From the natural T ( U p )-equivariant injection (cid:0) S ( U p , ∗ ) K i,i ρ (cid:1) ord ֒ −→ S ( U p , ∗ ) K i,i ρ ∼ = S ( U p K i,i , ∗ ) ρ , we have a natural surjection of local O E -algebras (finite over O E ): T ( U p K i,i , ∗ ) ρ ։ T ( U p K i,i , ∗ ) P − ord ρ . We have T ( U p K i,i , O E ) P − ord ρ ∼ = lim ←− s T ( U p K i,i , O E /̟ sE ) P − ord ρ . We set: e T ( U p ) P − ord ρ := lim ←− i T ( U p K i,i , O E ) P − ord ρ which is thus easily checked to be a quotient of e T ( U p ) ρ and is also a complete local O E -algebra ofresidue field k E . We have as in [7, Lem. 6.7, 6.8 (1)]: Lemma 3.7.
The O E -algebra e T ( U p ) P − ord ρ is reduced and the natural action of e T ( U p ) P − ord ρ on Ord P ( S ( U p , E ) ρ ) is faithful, and extends to a faithful action on Ord P ( b S ( U p , O E ) ρ ) . m be a maximal ideal of e T ( U p ) P − ord ρ [1 /p ] such that Ord P ( b S ( U p , E ) ρ )[ m ] = 0. Using the naturalcomposition R ρ, S − ։ e T ( U p ) ρ − ։ e T ( U p ) P − ord ρ , we associate to m a continuous representation ρ m : Gal F + → G n ( E ). Let ρ m ,F : Gal F ρ m −→ GL n ( E ) × GL ( E ) → GL n ( E ), and ρ m , e v := ρ m ,F | Gal F e v . Conjecture 3.8. ρ m , e v is P e v -ordinary for all v | p . Proposition 3.9.
Suppose that for all v | p , any P e v -filtration on ρ e v satisfies Hypothesis 2.1, thenConjecture 3.8 holds.Proof. For v | p , and a P e v -filtration F e v on ρ e v , by Hypothesis 2.1, (the proof of) Proposition 2.3, thereduced quotient R P − ord , ¯ (cid:3) ρ e v , F e v of R P e v − ord , (cid:3) ρ e v , F e v is a local deformation problem (cf. [36, Def. 3.2]). Denoteby I F e v the kernel of R (cid:3) ρ ։ R P e v − ord , ¯ (cid:3) ρ e v , F e v , and put I e v := ∩ F e v I F e v where F e v runs through all the (finitelymany) P e v -filtrations on ρ e v . Let R P e v − ord , ¯ (cid:3) ρ e v := R (cid:3) ρ /I e v , which, by [2, Lem. 3.2], is a local deformationproblem at the place e v . Let R P − ord ρ, S be the universal deformation ring of the deformation problem( F/F + , S, e S, O E , ρ, ε − n δ F/F + , { R ¯ (cid:3) ρ e v } v ∈ S \ S p , { R P e v − ord , ¯ (cid:3) ρ e v } v ∈ S p ) . Denote by I the kernel of the natural surjection R ρ, S ։ R P − ord ρ, S . By the same argument asin the proof of [7, Thm. 6.12 (1)] (which relies on [7, Prop. 5.10], noting also that since ρ is absolutely irreducible, for any continuous Gal F -representation ρ with modulo ̟ E reductionisomorphic to ρ , any P e v -filtration on ρ e v := ρ | Gal F e v naturally induces a P e v -filtration on ρ e v ), we have I (cid:0) Ord P ( b S ( U p , E ) ρ ) (cid:1) = 0. The proposition then follows by the same argument as for [7, Thm. 6.12(2)].
4. GL ( Q p )-ordinary families Keep the notation and assumptions in §
3, in particular, we assume Ord P ( b S ( U p , O E ) ρ ) = 0 andhence ρ e v is P e v -ordinary for all v | p . Assume moreover for all v ∈ S p , F + v ∼ = Q p and n e v,i ≤ i = 1 , · · · , k e v . In this section, using p -adic Langlands correspondence for GL ( Q p ), we constructGL ( Q p )-ordinary families from Ord P ( b S ( U p , E ) ρ ). We also show a local-global compatibility resultof these families, formulated in a similar way as in [28] (in particular, by using the theory ofPaˇsk¯unas [30]). Under more restrictive assumptions, similar results were essentially obtained in [7],but were stated in a different formulation, due to Emerton [22] (using deformations). As in [7, § P ( b S ( U p , E ) ρ ) L P ( Z p ) − alg+ := M σ Hom L P ( Z p ) ( σ, Ord P ( b S ( U p , E ) ρ )) ⊗ E σ ∼ = M σ (Ord P ( b S ( U p , E ) ρ ) ⊗ E σ ∨ ) L P ( Z p ) ⊗ E σ (4.1)where σ runs through algebraic representations of L P ( Q p ) ∼ = Q v ∈ S p L P e v ( Q p ) of highest weight( λ e v, , · · · , λ e v,n ) v ∈ S p satisfying λ e v,i ≥ λ e v,i +1 for all i ∈ { , · · · , n − } , v ∈ S p . We have as in [7,Prop. 7.2]: 14 roposition 4.1. Ord P ( b S ( U p , E ) ρ ) L P ( Z p ) − alg+ is dense in Ord P ( b S ( U p , E ) ρ ) . Definition 4.2. (1) A closed point x ∈ Spec e T ( U p ) P − ord ρ [1 /p ] is benign if: Ord P (cid:0) b S ( U p , E (cid:1) ρ [ m x ] (cid:1) L P ( Z p ) − alg+ = 0 . (2) A closed point x ∈ Spec e T ( U p ) ρ [1 /p ] is classical if b S ( U p , E ) ρ [ m x ] lalg = 0 . By the same argument as in the proof of [7, Prop. 7.5], we have
Proposition 4.3. (1) A benign point is classical.(2) The benign points are Zariski-sense in
Spec e T ( U p ) P − ord ρ [1 /p ] . Let x be a benign point, we can attach to x a dominant weight (i.e. λ e v, ≥ · · · ≥ λ e v,n ) λ = Y v ∈ S p λ e v = Y v ∈ S p ( λ e v, , · · · , λ e v,n )such that ρ x, e v is de Rham of Hodge-Tate weights ( λ e v, , λ e v, − , · · · , λ e v,n − n + 1) for v ∈ S p (e.g.by Proposition 4.3 and [36, Thm. 6.5(v)]). We have Proposition 4.4.
Let x be a benign point.(1) ρ x, e v is semi-stable for all v ∈ S p .(2) ρ x, e v is P e v -ordinary with a P e v -filtration F x, e v satisfying that ρ x, e v,i := gr i F x, e v (of dimension n e v,i )is crystalline of Hodge-Tate weights ( λ e v,s e v,i +1 − s e v,i , λ e v,s e v,i + n e v,i − ( s e v,i + n e v,i − . Moreover, let α e v,s e v,i +1 , α e v,s e v,i + n e v,i be the eigenvalues of ϕ on D cris ( ρ x, e v,i ) , then α e v,s e v,i +1 α − e v,s e v,i + n e v,i = p ± .(3) There exists an L P ( Q p ) -equivariant injection ⊗ v ∈ S p (cid:0) ⊗ i =1 , ··· ,k e v b π ( ρ x, e v,i ) lalg ⊗ k ( x ) ε s e v,i +1 − ◦ det (cid:1) ֒ −→ Ord P (cid:0) b S ( U p , E (cid:1) ρ [ m x ] (cid:1) , (4.2) where b π ( ρ x, e v,i ) denotes the continuous finite length representation of GL n e v,i ( Q p ) over k ( x ) (theresidue field at x ) associated to ρ x, e v,i via the p -adic local Langlands correspondence for GL ( Q p ) ([16]) normalized as in loc. cit. and [30] (so that the central character of b π ( ρ x, e v,i ) is equalto ( ∧ ρ x, e v,i ) ε − ) when n e v,i = 2 , via local class filed theory normalized by sending p to a (lift of )geometric Frobenius when n e v,i = 1 .Proof. The proposition follows by verbatim of the proof of [7, Prop. 7.6, Cor. 7.10]. Note thatthe strict P -ordinary assumption of loc. cit. is only used to compare ρ x, e v,i with a representationobtained by another way (which we don’t use here). Proposition 4.5. (1) If x is benign, then there exists r ( x ) ≥ such that (cid:16) ⊗ v ∈ S p (cid:0) ⊗ i =1 , ··· ,k e v b π ( ρ x, e v,i ) lalg ⊗ k ( x ) ε s e v,i +1 − ◦ det (cid:1)(cid:17) ⊕ r ( x ) ∼ −−→ Ord P (cid:0) b S ( U p , E (cid:1) ρ [ m x ] lalg (cid:1) , where we refer to [7, § Ord P (cid:0) b S ( U p , E (cid:1) ρ [ m x ] lalg (cid:1) .(2) The action of e T ( U p ) ρ [1 /p ] on Ord P ( b S ( U p , E ) ρ ) L P ( Z p ) − alg+ is semi-simple.Proof. (1) follows from the same argument as in [7, Lem. 7.8, Prop. 7.9, Cor. 7.10]. (2) followsfrom the proof of [7, Prop. 7.5] (which proves that all the vectors in Ord P ( b S ( U p , E ) ρ ) L P ( Z p ) − alg+ come from locally algebraic vectors in b S ( U p , E ) ρ via the adjunction property [7, Prop. 4.21], onwhich the action of e T ( U p ) ρ [1 /p ] is semi-simple by (3.2)). note that the normalization is slightly different from that in [7]. .2. Paˇsk¯unas’ theory We recall Paˇsk¯unas’ theory of blocks ([30]), which we will use to construct our GL ( Q p )-ordinaryfamilies.Let H be a p -adic analytic group. Denote by Mod sm H ( O E ) the category of smooth representationsof H over O E in the sense of [21, Def. 2.2.1], and Mod lfin H ( O E ) the full subcategoy of Mod sm H ( O E )consisting of those objects which are locally of finite length. For an irreducible representation π ∈ Mod lfin H ( O E ), denote by J π the injective enveloppe of π in Mod lfin H ( O E ). A block B of Mod lfin H ( O E )is a set of irreducible representations, such that if τ ∈ B , then τ ′ ∈ B if and only if there exists asequence of irreducible representaitons τ = τ , τ , · · · , τ m = τ ′ such that τ i = τ i +1 , Ext H ( τ i , τ i +1 ) =0 or Ext H ( τ i +1 , τ i ) = 0. We have a decompositionMod lfin H ( O E ) ∼ = Y B Mod lfin H ( O E ) B , where B runs through the blocks of Mod lfin H ( O E ), and Mod lfin H ( O E ) B denotes the full subcategoryof Mod lfin H ( O E ) consisting of those objects such that all the irreducible subquotients lie in B . Inparticular, for any τ ∈ Mod lfin H ( O E ), τ ∼ = ⊕ B τ B with τ B ∈ Mod lfin H ( O E ). If τ is moreover admissible,then there exists a finite set I of blocks such that τ ∼ = ⊕ B ∈I τ B . For a block B , denote by π B := ⊕ π ∈ B π . Denote by J B ∼ = ⊕ π ∈ B J π the injective envelope of π B inMod lfin H ( O E ), and e E B := End H ( J B ).By [21, (2.2.8)], taking Pontryagain dual induces an anti-equivalence of categories between thecategory Mod sm H ( O E ) and the category Mod pro aug H ( O E ) of profinite augmented H -representationsover O E (cf. [21, Def. 2.1.6]). Denote by C H ( O E ) the full subcategory of Mod pro aug H ( O E ) consistingof those objects that are the Pontryagain duals of the representations in Mod lfin H ( O E ). For a block B of Mod lfin H ( O E ), we see e P B := J ∨ B ∼ = ⊕ π ∈ B J ∨ π ∼ = ⊕ π ∈ B e P π ∨ is a projective envelope of π ∨ B in C H ( O E ), where e P π ∨ denotes the projective enveloppe of π ∨ in C H ( O E ). And we have End C H ( O E ) ( e P B ) ∼ = e E B . Denote by C H ( O E ) B the full subcategory of C H ( O E )consisting of those objects whose Pontryagain dual lies in Mod lfin H ( O E ) B . The functor sending M ∈ C H ( O E ) B to Hom C H ( O E ) ( e P B , M ) induces an anti-equivalence of categories between C H ( O E ) B and the category of pseudo-compact e E B -modules, with the inverse given by M M b ⊗ e E B e P B (cid:0) cf.[30, Lem. 2.9, 2.10], note that a similar argument as in the proof of [30, Lem. 2.10] also shows thatHom C H ( O E ) ( e P B , M ) b ⊗ e E B e P B ∼ −→ M for M ∈ C H ( O E ) B (cid:1) .By [30, § lfin Q × p ( O E ) that contain an absolutely irreducible representation aregiven by B = { χ : Q × p → k × E } . For such B , we have e E B ∼ = O E [[ x, y ]], and that e P B is a free e E B -module of rank 1. Actually, let Def χ : Art( O E ) → { Sets } denote the standard deformation functorof χ , then Def χ is pro-represented by e E B and e P B is isomorphic to the universal deformation of χ over e E B . We denote by 1 univ the universal deformation of the trivial character over Λ := O E [[ x, y ]].By the local class field theory, we have an isomorphism between Def and the deformation functorof the trivial character of Gal Q p , which we also denote by Def . The Q × p action on 1 univ naturallyextends to a Gal Q p -action, and the resulting Gal Q p -representation over Λ is actually the universal16eformation of 1.By [31, Cor. 1.2], the blocks of Mod lfinGL ( Q p ) ( O E ) that contain an absolutely irreducible representa-tion are given by (when p > B = { π } , supersingular,(2) B = { Ind GL ( Q p ) B ( Q p ) ( χ ω − ⊗ χ ) , Ind GL ( Q p ) B ( Q p ) ( χ ω − ⊗ χ ) } , χ χ − = 1, ω ± ,(3) B = { Ind GL ( Q p ) B ( Q p ) ( χω − ⊗ χ ) } ,(4) B = { η ◦ det , Sp ⊗ k E η ◦ det , (Ind GL ( Q p ) B ( Q p ) ω − ⊗ ω ) ⊗ k E η ◦ det } if p ≥ B = { η ◦ det , Sp ⊗ k E η ◦ det , ( ηω ) ◦ det , Sp ⊗ k E ( ηω ) ◦ det } if p = 3.For each B as above, we can attach a 2-dimensional semi-simple representation ρ B of Gal Q p over k E such that • if π ∈ B supersingular, then ρ B = V ( π ), where V is the Colmez’s functor normalized as in[30, § • if Ind GL ( Q p ) B ( Q p ) ( χ ω − ⊗ χ ) ∈ B with χ χ − = ω ± , then ρ B = χ ⊕ χ , • if η ◦ det ∈ B , then ρ B = η ⊕ ηω .Note that under this normalization, if π ∈ B has central character ζ : Q × p → k × E , then ∧ ρ B = ζω .Suppose p ≥
3. Denote by R ps B the universal deformation ring which parametrizes all 2-dimensionalpseudo-representations of Gal Q p lifting tr ρ B (cf. [26, Lem. 1.4.2]). Theorem 4.6 (Paˇsk¯unas) . Suppose the block B of Mod lfinGL ( Q p ) ( O E ) lies in case (1) (2) (3) (4).(1) There exists a natural isomorphism between the centre of e E B and R ps B .(2) e E B is a finitely generated module over R ps B .Proof. Let ζ : Q × p → O × E be a continuous character, let Mod lfinGL ( Q p ) ,ζ ( O E ) be the full subcat-egory of Mod lfinGL ( Q p ) ( O E ) consisting of those representations that have central character ζ , and C GL ( Q p ) ,ζ ( O E ) be the full subcategory of C GL ( Q p ) ( O E ) consisting of the objects whose dual liesin Mod lfinGL ( Q p ) ,ζ ( O E ). We denote by e P B ,ζ the projective enveloppe of π ∨ B in C GL ( Q p ) ,ζ ( O E ) (recall π B = ⊕ π ∈ B π ), and put e E B ,ζ := End C GL2( Q p ) ,ζ ( O E ) ( e P B ,ζ ). For π ∈ B , denote by e P π ∨ ,ζ the projectiveenveloppe of π ∨ in C GL ( Q p ) ,ζ ( O E ), thus e P B ,ζ ∼ = ⊕ π ∈ B e P π ∨ ,ζ .Suppose B is in case (1) (2) (4):Let π be an arbitrary representation in B if B is in the case (1) or (2), and let π := (Ind GL ( Q p ) B ( Q p ) ω − ⊗ ω ) ⊗ k E η ◦ det ∈ B if B is in the case (4). By [14, Prop. 6.18], we have e P π ∨ ∼ = e P π ∨ ,ζ b ⊗ O E univ ◦ det.Denote by ρ π the (unique) two dimensional representation of Gal Q p over k E satisfying that ρ ss π ∼ = ρ B and that if B is moreover in the case (2) or (4), then V ( π ) − ωζ ∼ = cosoc Gal Q p ρ π . By [14, Cor.6.23], we have a natural isomorphism (see also Remark 4.7)End C GL2( Q p ) ( O E ) ( e P π ∨ ) ∼ = R ρ π , (4.3)where R ρ π denotes the universal deformation ring of ρ π (note that by the assumption on π ,End Gal Q p ( ρ π ) ∼ = k E hence R ρ π exists). 17 If B is in case (1), then e P B ∼ = e P π ∨ , and e E B ∼ = R ρ π ∼ = R ps ρ B . • If B is in case (2), we write B = { π , π } . The statement follows by the same argument asin [30, Cor. 8.11] replacing the isomorphism in [30, Cor. 8.7] by (4.3) applied to π and π . • If B is in case (4). The statement follows by the same arguments as for [30, Thm. 10.87]replacing the isomorphism in [30, Thm. 10.71] by (4.3).Suppose B is in case (3). The statement in this case follows by a similar argument. We include aproof (with several steps) for the convenience of the reader.(a) Let π ∈ B . We first show e P π ∨ ∼ = e P π ∨ ,ζ b ⊗ O E univ ◦ det. Put e P ′ := e P π ∨ ,ζ ⊗ O E univ , equipped withthe diagonal action of GL ( Q p ), where GL ( Q p ) acts on the second factor via det : GL ( Q p ) → Q × p .It is not difficult to see e P ′ ∈ C GL ( Q p ) ( O E ). Indeed, we can write e P π ∨ ,ζ ∼ = lim ←− n e P π ∨ ,ζ,n (resp. Λ ∼ =lim −→ n Λ n ) such that the Pontryagain dual of each e P π ∨ ,ζ,n (resp. Λ n ) is a finite length representationof GL ( Q p ) (resp. of Q × p ), and hence e P ′ ∼ = lim ←− n ( e P π ∨ ,ζ,n ⊗ O E Λ n ) ∈ C GL ( Q p ) ( O E ). By [30, Thm.3.26], e P π ∨ ,ζ is a deformation of π ∨ over e E B ,ζ . We see by definition that e P ′ is a deformation of π ∨ over e E ′ := e E B ,ζ b ⊗ O E Λ.We show cosoc GL ( Q p ) e P ′ ∼ = π ∨ . By the proof of [24, Lem. B.8], we knowHom C GL2( Q p ) ,ζ ( O E ) × C Q × p ( O E ) ( e P π ∨ ,ζ b ⊗ O E univ , π ∨ ⊗ k E k E ) ∼ = k E . We deduce then Hom C GL2( Q p ) ( O E ) ( e P ′ , π ∨ ) ֒ → k E . Since any irreducible constituent of e P ′ is isomor-phic to π ∨ , we deduce then cosoc GL ( Q p ) e P ′ ∼ = π ∨ .We have thus a projection e P π ∨ ։ e P ′ . Applying Hom C GL2( Q p ) ( O E ) ( e P π ∨ , − ), we obtain a surjection e E B − ։ Hom C GL2( Q p ) ( O E ) ( e P π ∨ , e P ′ ) . (4.4)By the same argument as in [30, Lem. 3.25], we have an isomorphism of e E ′ -module:Hom C GL2( Q p ) ( O E ) ( e P π ∨ , e P ′ ) ∼ = e E B ,ζ b ⊗ O E Λ . (4.5)such that the composition of (4.4) with (4.5) gives a surjective homomorphism of O E -algebra δ : e E B ։ e E ′ , and e P ′ ∼ = e P π ∨ b ⊗ e E B f E ′ . By the same argument as in [30, § k E Ext ( Q p ) ( π, π ) = 4(which for example follows from the fact dim k E Ext ( Q p ) ,Z ( π, π ) = 2 ([30, Prop. 9.1]), and thesame argument as in the proof of [7, Lem. A.3]), we have e E B ։ e E ab B ∼ = O E [[ x , x , x , x ]] (where e E ab B is the maximal commutative quotient of e E ). Moreover, one can check that [30, Lem. 9.2, Lem.9.3] hold with e E of loc. cit. replaced by e E B , and O [[ x, y ]] replaced by O E [[ x , x , x , x ]]. By [30,Lem. 9.2], there exists t ∈ e E B ,ζ such that0 → e E B ,ζ t −−→ e E B ,ζ → ( e E B ,ζ ) ab → , which then induces (noting Λ is flat over O E , and e E B ,ζ is O E -torsion free)0 → e E ′ t −−→ e E ′ → ( e E B ,ζ ) ab b ⊗ O E Λ → . (4.6)18sing the same argument as in the proof of [30, Lem. 9.3], we deduce then δ is an isomorphism,and e P π ∨ ∼ = e P ′ ∼ = e P π ∨ ,ζ b ⊗ O E Λ.(b) Let χ : Q × p → k × E be such that π ∼ = Ind GL ( Q p ) B ( Q p ) ( χω − ⊗ χ ) (hence ζ = χ ω − ). Thus R ps B = R ps2 χ , the universal deformation ring of the pseudo-character 2 χ . Denote by R ps ,ζε χ the universaldeformation ring parameterizing 2-dimensional pseudo-characters of Gal Q p with determinant ζε lifting 2 χ . We show there is a natural isomorphism R ps2 χ ∼ −→ R ps ,ζε χ b ⊗ O E Λ. Let T univ ,ζε : Gal Q p → R ps ,ζε χ be the universal deformation with determinant ζε of 2 χ , and put T ′ : Gal Q p → R ps ,ζε χ b ⊗ O E Λbe the pseudo-character sending g to T univ ,ζε ( g ) ⊗ univ ( g ). By the universal property of R ps2 χ , weobtain a morphism of complete O E -algebras: R ps2 χ −→ R ps ,ζε χ b ⊗ O E Λ . (4.7)By [30, Cor. 9.13], R ps ,ζε χ ∼ = O E [[ x , x , x ]]. Using the fact that taking determinant induces a sur-jective map R ps2 χ ( k [ ǫ ] /ǫ ) ։ Λ( k [ ǫ ] /ǫ ) (since it is easy to construct a section of this map), it is notdifficult to see the tangent map of (4.7) is bijective, from which we deduce (4.7) is an isomorphism.(c) By [30, Cor. 9.27], we have a natural isomorphism (which is unique up to conjugation by e E × B ,ζ ) (cid:0) R ps ,ζε χ [[ G ]] /J ζε (cid:1) op ∼ −−→ e E B ,ζ , (4.8)where G denotes the maximal pro- p quotient of Gal Q p , which is a free pro- p group generated by 2elements γ , δ , and where J ζε denotes the closed two-sided ideal generated by g − T univ ,ζε ( g ) g + ζε ( g ),for all g ∈ G . By (a) and (b), we have e E B ∼ = e E B ,ζ b ⊗ O E Λ ∼ = (cid:0) R ps ,ζε χ [[ G ]] /J ζε (cid:1) op b ⊗ O E Λ ∼ = (cid:0) R ps2 χ [[ G ]] /J ζε (cid:1) op . (4.9)In particular, we see by [30, Cor. 9.25] that e E B is a free R ps2 χ -module of rank 4. Composingwith an automorphism of Λ if needed, we assume 1 univ ( γ ) = 1 + x , and 1 univ ( δ ) = 1 + y (recall1 univ : G ab → Λ, and where we use γ , δ to denote their images in G ab ). The induced isomorphism1 univ : O E [[ G ab ]] ∼ −→ O E [[ x, y ]] lifts to an isomorphism O E [[ G ]] ∼ −→ O E [[ x, y ]] nc sending γ to 1 + x and δ to 1 + y (“nc” means non-commutative). Let e J be the closed two-sided ideal of R ps2 χ [[ G ]]generated by g − T univ ( g ) g + det( T univ )( g ). Consider the following isomorphism of R ps2 χ -algebras R ps2 χ [[ G ]] ∼ −−→ (cid:0) R ps ,ζε χ b ⊗ O E O E [[ x, y ]] (cid:1) [[ G ]]which sends γ to γ (1 + x ) and δ to δ (1 + y ). One can check (using T univ = T univ ,ζε ⊗ univ ) thatthis isomorphism induces an isomorphism R ps2 χ [[ G ]] / e J ∼ −−→ R ps ,ζε χ [[ G ]] /J ζε b ⊗ O E Λ . Using the same argument as in [30, Cor. 9.24], we have that the center of R ps2 χ [[ G ]] / e J (hence of e E B )is equal to R ps2 χ .(d) We show the injection R ps2 χ ֒ → e E B is independent of the choice of ζ , by unwinding a littlethe isomorphism in (4.8). Let η : Q × p → O × E be such that η ≡ ̟ E ). We have a naturalGL ( Q p )-equivariant isomorphism e P π ∨ ,ζη ∼ = e P π ∨ ,ζ ⊗ O E ( η − ◦ det), which induces an isomorphism19w η : e E B ,ζη ∼ −→ e E B ,ζ . Twisting η also induces an isomorphism tw η : R ps ,ζη χ ∼ −→ R ps ,ζ χ . Denote by ˇ V ζ (resp. ˇ V ζη ) the functor V of [30, § C GL ( Q p ) ,ζ ( O E ) (resp. on C GL ( Q p ) ,ζη ( O E )) associatedto ζ (resp. to ζη ). By definition (cf. loc. cit. ), we have a Gal Q p -equivariant isomorphismˇ V ζη ( e P π ∨ ,ζη ) ∼ = ˇ V ζ ( e P π ∨ ,ζ ) ⊗ O E η. (4.10)As in the discussion below [30, Lem. 9.3], we can deduce from (4.10) a commutative diagram O E [[ G ]] op id −−−−→ O E [[ G ]] op ϕ ˇ V ζη y η ⊗ ϕ ˇ V ζ ye E B ,ζη tw η −−−−→ e E B ,ζ , (4.11)where “ η ” in the right vertical map denotes the composition O E [[ G ]] op η −→ O E ֒ → e E B ,ζ , and where ϕ ˇ V ζη (resp. ϕ ˇ V ζ ) is the map ϕ ˇ V of [30, § e E × B ,ζη (resp. e E × B ,ζ ), but we can choose the maps so that (4.11) commutes). Hence ϕ ˇ V ζη is equal to thecomposition O E [[ G ]] op η ⊗ id −−−→ O E [[ G ]] op ϕ ˇ V ζ −−−→ e E B ,ζ tw η − −−−−→ ∼ e E B ,ζη . We deduce that the following diagram commutes O E [[ G ]] η ⊗ id −−−−→ O E [[ G ]] y y R ps ,ζη χ [[ G ]] /J ζη ε −−−−→ R ps ,ζ χ [[ G ]] /J ζε ∼ y ∼ y(cid:0) e E B ,ζη (cid:1) op tw η −−−−→ (cid:0) e E B ,ζ (cid:1) op where the middle horizontal map sends g to η ( g ) g for g ∈ G , and sends a to tw η ( a ) for a ∈ R ps ,ζη χ ,where the vertical maps in the top square are the surjections given as in [30, (150)], and where thevertical maps in the bottom square are given as in (4.8), induced by ϕ ˇ V ζη , ϕ ˇ V ζ respectively (see[30, § R ps ,ζη χ tw η −−−−→ R ps ,ζ χ y y(cid:0) e E B ,ζη (cid:1) op tw η −−−−→ (cid:0) e E B ,ζ (cid:1) op . Together with similar commutative diagrams as in [14, (6.4)] replacing “ R p ” by e E B and R ps2 χ , wededuce that the composition R ps2 χ ∼ = R ps ,ζη χ b ⊗ O E Λ ֒ −→ (cid:0) e E B ,ζη (cid:1) op b ⊗ O E Λ ∼ = e E op B coincides with the one induced by (4.9). This concludes the proof.20 emark 4.7. Let π be the GL ( Q p ) -representation in the proof of Theorem 4.6 for the case(1)(2)(4). Let ζ : Q × p → O × E be such that ζ is equal to the central character of π . Let ˇ V ζ bethe functor ˇ V of [30, § C GL ( Q p ) ,ζ ( O E ) (which depends on the choice of ζ ). As in [30, Prop.6.3, Cor. 8.7, Thm. 10.71], the functor ˇ V ζ induces an isomorphism R ζερ π ∼ −−→ End C GL2( Q p ) ( O E ) ( e P π ∨ ) where R ζερ π denotes the universal deformation over ρ of deformations with determinant equal to ζε .The isomorphism in (4.3) is given by the composition R ρ π ∼ = R ζερ π b ⊗ O E Λ ∼ −−→ End C GL2( Q p ) ,ζ ( O E ) ( e P π ∨ ) b ⊗ O E Λ ∼ = End C GL2( Q p ) ,ζ ( O E ) ( e P π ∨ ) . We deduce by [14, (6.4)] and the diagram in the proof of [14, Lem. 6.9] that the isomorphism in(4.3) is (also) independent of the choice of ζ . For m a maximal ideal of R ps B [1 /p ] with p := m ∩ R ps B , we denote by b π B , m the multiplicity free directsum of the irreducible constituents of the finite length Banach representationHom cts O E (cid:0) e P B b ⊗ R ps B ( R ps B / p ) , E (cid:1) . Let H = Q i H i be a finite product with H i ∼ = Q × p or GL ( Q p ). By [28, Lem. 3.4.10, Cor. 3.4.11](and the proof), we have Proposition 4.8.
Any block B of Mod lfin H ( O E ) is of the form B = ⊗ i B i := {⊗ π i ∈ B i π i } where B i is a block of Mod lfin H i ( O E ) . And we have e P B ∼ = b ⊗ i e P B i , e E B ∼ = b ⊗ i e E B i .4.3. GL ( Q p ) -ordinary families We apply Paˇsk¯unas’ theory to construct GL ( Q p )-ordinary families.Let C := C L P ( Q p ) ( O E ). We decompose the space of P -ordinary automorphic representations usingthe theory of blocks. We haveOrd P ( b S ( U p , O E ) ρ ) d = Hom O E (Ord P ( b S ( U p , O E ) ρ ) , O E ) ∈ C . For k ∈ Z ≥ , the Pontryagain dual Ord P ( b S ( U p , O E /̟ kE ) ρ ) ∨ is also an onject in C . We have an e T ( U p ) P − ord ρ -equivariant isomorphism in C (cf. (1.2)):Ord P ( b S ( U p , O E ) ρ ) d ∼ = lim ←− k Ord P ( b S ( U p , O E /̟ kE ) ρ ) ∨ . (4.12)For M ∈ Mod lfin L P ( Q p ) ( O E ) (resp. in C ), and B a block of Mod lfin L P ( Q p ) ( O E ) (hence can also be viewedas a block of C ), we denote by M B the maximal direct summand of M such that all the irreduciblesubquotients π of M B satisfy π ∈ B (resp. π ∨ ∈ B ). We have thus decompositions (cf. [30, Prop.5.36]) Ord P ( b S ( U p , O E ) ρ ) d ∼ = ⊕ B Ord P ( b S ( U p , O E ) ρ ) d B , (4.13)Ord P ( b S ( U p , O E /̟ kE ) ρ ) ∼ = ⊕ B Ord P ( b S ( U p , O E /̟ kE ) ρ ) B , Ord P ( b S ( U p , O E /̟ kE ) ρ ) ∨ ∼ = ⊕ B Ord P ( b S ( U p , O E /̟ kE ) ρ ) ∨ B .
21t is also clear that the isomorphism in (4.12) respects the decompositions.Recall that Ord P ( b S ( U p , O E ) ρ ) is admissible, hence Ord P ( b S ( U p , O E ) ρ ) d is finitely generated over O E [[ L P ( Z p )]]. By [21, (2.2.12)], the Pontryagain dual of Ord P ( b S ( U p , O E ) ρ ) d is a smooth admissiblerepresentation of L P ( Q p ) over O E . Hence there are finitely many blocks B of Mod lfin L P ( Q p ) ( O E ) suchthat Ord P ( b S ( U p , O E ) ρ ) d B = 0 (cid:0) which is equivalent to Ord P ( b S ( U p , O E /̟ kE ) ρ ) ∨ B = 0, by (4.12) (cid:1) .Put Ord P ( b S ( U p , O E ) ρ ) B := Hom cts O E (Ord P ( b S ( U p , O E ) ρ ) d B , O E ) , Ord P ( b S ( U p , E ) ρ ) B := Ord P ( b S ( U p , O E ) ρ ) B ⊗ O E E, which are equipped with the supreme norm. For ∗ ∈ {O E , E, O E /̟ kE } , Ord P ( b S ( U p , ∗ ) ρ ) B is adirect summand of Ord P ( b S ( U p , ∗ ) ρ ) (e.g. using 4.13 and (1.3)), and we haveOrd P ( b S ( U p , O E ) ρ ) B ∼ = lim ←− n Ord P ( S ( U p , O E /̟ nE ) ρ ) B . (4.14)The following lemma follows easily from Lemma 3.6 and [7, Cor. 7.7]. Lemma 4.9.
Let B be a block of Mod lfin L P ( Q p ) ( O E ) such that Ord P ( b S ( U p , O E ) ρ ) B = 0 .(1) Ord P ( b S ( U p , O E ) ρ ) B | L P ( Z p ) is isomorphic to a direct summand of C ( L P ( Z p ) , O E ) ⊕ r for some r > .(2) (cid:0) Ord P ( b S ( U p , O E ) ρ ) B (cid:1) L P ( Z p ) − alg+ is dense in Ord P ( b S ( U p , O E ) ρ ) B . We haveOrd P ( b S ( U p , O E ) ρ ) d B ∼ = e P B b ⊗ e E B Hom C (cid:0) e P B , Ord P ( b S ( U p , O E ) ρ ) d B (cid:1) ∼ = e P B b ⊗ e E B Hom C (cid:0) e P B , Ord P ( b S ( U p , O E ) ρ ) d (cid:1) ֒ −→ Ord P ( b S ( U p , O E ) ρ ) d (4.15)where the first isomorphism follows from [30, Lem. 2.10] (applying Hom C ( e P B , − ) to [30, (6)]with M = Ord P ( b S ( U p , O E ) ρ ) d B , we easily deduce that the kernel is zero), the second isomor-phism follows from (4.13) and Hom C (cid:0) e P B , Ord P ( b S ( U p , O E ) ρ ) d B ′ ) = 0 for B ′ = B , and where thelast injection is the evaluation map (indeed, applying Hom C ( e P B ′ , − ) to the kernel of this map,we get zero for all B ′ , from which we deduce that the kernel has to be zero). We see thatOrd P ( b S ( U p , O E ) ρ ) d B inherits a natural e T ( U p ) P − ord ρ -action from Ord P ( b S ( U p , O E ) ρ ) d (via the firsttwo isomorphisms in (4.15)), so that the decomposition (4.13) is in fact e T ( U p ) P − ord ρ -equivariant.Similarly, for all k ≥
1, Ord P ( S ( U p , O E /̟ kE ) ρ ) B is also a e T ( U p ) P − ord ρ -equivariant direct sum-mand of Ord P ( S ( U p , O E /̟ kE ) ρ ). For i ≥ (cid:0) Ord P ( S ( U p , O E /̟ kE ) ρ ) B (cid:1) L i is hence a e T ( U p ) P − ord ρ -equivariant direct summand of (cf. (3.7))Ord P ( S ( U p , O E /̟ kE ) ρ ) L i ∼ = S ( U p K i,i , O E /̟ kE ) ρ, ord . Denote by T ( U p K i,i , O E /̟ kE ) P − ord ρ, B the image of T ( U p ) ρ −→ End O E (cid:0)(cid:0) Ord P ( S ( U p , O E /̟ kE ) ρ ) B (cid:1) L i (cid:1) , and put e T ( U p ) P − ord ρ, B := lim ←− k lim ←− i T ( U p K i,i , O E /̟ kE ) P − ord ρ, B . It is clear that e T ( U p ) P − ord ρ, B is a quotient of e T ( U p ) P − ord ρ hence is also a complete local noetherian O E -algebra of residue field k E . Similarly as in Lemma 3.7, we have22 emma 4.10. The O E -algebra e T ( U p ) P − ord ρ, B is reduced and the natural action of e T ( U p ) P − ord ρ, B on Ord P ( b S ( U p , O E ) ρ, B ) and Ord P ( b S ( U p , E ) ρ, B ) is faithful. Since Ord P ( b S ( U p , E ) ρ ) B is an L P ( Q p ) × e T ( U p ) P − ord ρ, B -equivariant direct summand of Ord P ( b S ( U p , E ) ρ ),by the same argument, we have as in Proposition 4.3, 4.4: Proposition 4.11. (1) The benign points are Zariski-dense in
Spec e T ( U p ) P − ord ρ, B [1 /p ] .(2) Let x be a benign point of Spec e T ( U p ) P − ord ρ, B [1 /p ] , the statements in Proposition 4.3 (2), Propo-sition 4.4 hold with Ord P (cid:0) b S ( U p , E (cid:1) ρ [ m x ] (cid:1) replaced by Ord P (cid:0) b S ( U p , E (cid:1) ρ [ m x ] (cid:1) B . By Proposition 4.8, there exist blocks B e v,i for v ∈ S p , i = 1 , · · · , k e v such that B = ⊗ v ∈ S p ( ⊗ i =1 , ··· ,k e v B e v,i ( s e v,i +1 − ⊗ v ∈ S p B e v , (4.16)where B ( r ) denotes the block { π ⊗ k E ω r ◦ det | π ∈ B } for a block B . If p = 3, we assume that B e v,i isnot in case (4’) for all v , i with n e v,i = 2. For v ∈ S p , i = 1 , · · · , k e v , twisting ε − s e v,i +1 : Gal Q p → O × E induces an isomorphism tw e v,i : R ps B e v,i ∼ −→ R ps B e v,i ( s e v,i +1 − . Put R p, B := b ⊗ v ∈ S p (cid:0) b ⊗ i =1 , ··· ,k e v R ps B e v,i (cid:1) . (4.17)Denote by m ( U p , B ) := Hom C (cid:0) e P B , Ord P ( b S ( U p , O E ) ρ ) d B (cid:1) , which is a compact e E B -module. By Theorem 4.6, Proposition 4.8 and the fact Ord P ( b S ( U p , O E ) ρ ) B is admissible, m ( U p , B ) is finitely generated over R p, B where the action of R p, B is induced from thenatural action of e E B via R p, B (tw e v,i ) −−−−→ ∼ b ⊗ v ∈ S p (cid:0) b ⊗ i =1 , ··· ,k e v R ps B e v,i ( s e v,i +1 − (cid:1) ֒ −→ e E B . The (faithful) e T ( U p ) P − ord ρ, B -action on Ord P ( b S ( U p , O E ) ρ ) d B (commuting with L P ( Q p )) induces afaithful e E B -linear action of e T ( U p ) P − ord ρ, B on m ( U p , B ). We denote by A the R p, B -subalgebra ofEnd R p, B (cid:0) m ( U p , B ) (cid:1) generated by the image of the composition e T ( U p ) P − ord ρ, B ֒ −→ End e E B (cid:0) m ( U p , B ) (cid:1) ֒ −→ End R p, B (cid:0) m ( U p , B ) (cid:1) . (4.18)By definition, A is commutative and finite over R p, B , and it is not difficult to see that A is O E -torsion free and ∩ j ≥ ̟ jE A = 0 (using similar properties for m ( U p , B ), and the fact that the A -actionon m ( U p , B ) is faithful). We also have a surjective map e T ( U p ) P − ord ρ, B b ⊗ O E R p, B − ։ A , and hence a natural embedding:(Spf A ) rig ֒ −→ (Spf e T ( U p ) P − ord ρ, B ) rig × (Spf R p, B ) rig such that the composition (which one can view as an analogue of the weight map of Hida families) κ : (Spf A ) rig ֒ −→ (Spf e T ( U p ) P − ord ρ, B ) rig × (Spf R p, B ) rig −→ (Spf R p, B ) rig is finite. For each point z e v,i of (Spf R ps B e v,i ) rig , we can attach a representation b π z e v,i of GL n e v,i ( Q p ) suchthat if n e v,i = 1, b π z e v,i is the corresponding continuous (unitary) character of Q × p , and if n e v,i = 2, b π z e v,i := b π B e v,i , m z e v,i (see the discussion below Remark 4.7).23 roposition 4.12. Let y = ( x, z ) = ( x, ( z e v,i )) ∈ (Spf e T ( U p ) P − ord ρ, B ) rig × (Spf R p, B ) rig . Then y ∈ (Spf A ) rig if and only if Hom L P ( Q p ) (cid:16) b ⊗ v ∈ S p i =1 , ··· ,k e v ( b π z e v,i ⊗ k ( y ) ε s e v,i +1 − ◦ det) , Ord P (cid:0) b S ( U p , E ) ρ (cid:1) B [ m x ] (cid:17) = 0 . (4.19) Proof.
Without loss of generality, we assume k ( y ) = E . Denote by p x ⊂ e T ( U p ) P − ord ρ, B (resp. p z ⊂ R p, B , resp. p z e v,i ⊂ R ps B e v,i ) the prime ideal associated to x (resp. z , resp. z e v,i ). By definition of b π z e v,i ,we see (4.19) is equivalent toHom C (cid:16) Ord P ( b S ( U p , O E ) ρ [ p x ]) d B , b ⊗ v ∈ S p i =1 , ··· ,k e v (cid:0) e P B e v,i b ⊗ e E B e v,i ( e E B e v,i / p z e v,i ) tf (cid:1)(cid:17) = 0 , (4.20)where “tf” denotes the O E -torsion free quotient. Suppose (4.20) holds. Let f be a non-zeromorphism in (4.20), and consider the compositionOrd P ( b S ( U p , O E ) ρ ) d B − ։ Ord P ( b S ( U p , O E ) ρ [ p x ]) d B f −−→ b ⊗ v ∈ S p i =1 , ··· ,k e v (cid:0) e P B e v,i b ⊗ e E B e v,i ( e E B e v,i / p z e v,i ) tf (cid:1) . Applying Hom C ( e P B , − ), we obtain a non-zero e T ( U p ) P − ord ρ, B × R p, B -equivariant map m ( U p , B ) / p x −→ Hom C (cid:0) e P B , b ⊗ v ∈ S p i =1 , ··· ,k e v (cid:0) e P B e v,i b ⊗ e E B e v,i ( e E B e v,i / p z e v,i ) tf (cid:1)(cid:1) ∼ = ⊗ v ∈ S p i =1 , ··· ,k e v ( e E B e v,i / p z e v,i ) tf , (4.21)where the tensor product on the right hand side is over O E and the second isomorphism followsfrom [30, Lem. 2.9] and the proof of [24, Lem. B.8]. Note also the R p, B -action on the righthand side of (4.21) factors through R p, B / p z . In particular, m ( U p , B ) admits an O E -torsion freequotient on which e T ( U p ) P − ord ρ, B (resp. R p, B ) acts via e T ( U p ) P − ord ρ, B / p x (resp. R p, B / p z ). We deducethen y = ( x, z ) ∈ (Spf A ) rig .Conversely, suppose y = ( x, z ) ∈ (Spf A ) rig , and let p y ⊂ A be the prime ideal associated to y . Wehave m ( U p , B ) − ։ ( m ( U p , B ) ⊗ A A / p y ) tf = 0 . From which we deduceOrd P ( b S ( U p , O E ) ρ ) d B ∼ = e P B b ⊗ e E B m ( U p , B ) − ։ e P B b ⊗ e E B ( m ( U p , B ) ⊗ A A / p y ) tf . Considering the e T ( U p ) P − ord ρ, B -action, we see the above map factors throughOrd P ( b S ( U p , O E ) ρ [ p x ]) d B − ։ e P B b ⊗ e E B ( m ( U p , B ) ⊗ A A / p y ) tf . (4.22)Applying Hom cts O E ( − , E ) to (4.22), we obtain an injectionHom cts O E ( e P B b ⊗ e E B ( m ( U p , B ) ⊗ A A / p y ) tf , E ) ֒ −→ Ord P ( b S ( U p , E ) ρ [ m x ]) B . (4.23)Since the R p, B -action on m ( U p , B ) ⊗ A A / p y factors through R p, B / p z , we see the right hand side of(4.22) is a quotient of e P B b ⊗ e E B ( e E B / p z ) ⊕ r tf for certain r . Together with the fact that b π z e v,i is semi-simple for all v , i , we see the right hand side of (4.23) contains a direct summand of a copy of b ⊗ v ∈ S p i =1 , ··· ,k e v ( b π z e v,i ⊗ E ε s e v,i +1 − ◦ det). This concludes the proof.24et x be a benign point of Spec e T ( U p ) P − ord ρ, B [1 /p ]. By Proposition 4.4, for v ∈ S p , and i =1 , · · · , k e v , ρ x, e v,i is crystalline. We denote by b π ( ρ x, e v,i ) the universal completion of b π ( ρ x, e v,i ) lalg (see [4] [29], noting that by Proposition 4.4(2), b π ( ρ x, e v,i ) lalg is isomorphic to the tensor productof an irreducible algebraic representation of GL ( Q p ) with a smooth irreducible principal series).By [10, Lem. 3.4(i)], b ⊗ v ∈ S p (cid:0) b ⊗ i =1 , ··· ,k e v b π ( ρ x, e v,i ) ⊗ E ε s e v,i +1 − ◦ det (cid:1) is the universal completion of ⊗ v ∈ S p i =1 , ··· ,k e v ( b π ( ρ z e v,i ) lalg ⊗ k ( y ) ε s e v,i +1 − ◦ det). The injection (4.2) (see Corollary 4.11 (2)) induces hencea non-zero morphism of L P ( Q p )-representations b ⊗ v ∈ S p (cid:0) b ⊗ i =1 , ··· ,k e v b π ( ρ x, e v,i ) ⊗ E ε s e v,i +1 − ◦ det (cid:1) −→ Ord P (cid:0) b S ( U p , E (cid:1) ρ [ m x ] (cid:1) B . (4.24)Let Λ be a Gal F -equivariant lattice of ρ x,F = ρ m x ,F (where m x denotes the associated maximalideal of e T ( U p ) P − ord ρ, B [1 /p ]). Since ρ F is absolutely irreducible, Λ is unique up to scalar, and we haveΛ /̟ k ( x ) ∼ = ρ F and hence Λ /̟ k ( x ) | Gal F e v ∼ = ρ e v . The P e v -filtration on ρ x, e v induces a P e v -filtration onΛ, and hence induces a P e v -filtration on ρ e v :¯ F x, e v : 0 = Fil x ρ e v ( Fil x ρ e v ( · · · ( Fil k e v x ρ e v = ρ e v (4.25)such that the graded piece gr i ¯ F x is a reduction of ρ x, e v,i . Proposition 4.13. (1) For v ∈ S p , i = 1 , · · · , k e v , we have ρ ss x, e v,i ∼ = ρ B e v,i , where ρ ss x, e v,i denotes thesemi-simplification of one (or any) modulo ̟ k ( x ) reduction of ρ x, e v,i .(2) We have ( x, { z e v,i } ) ∈ (Spf A ) rig where z e v,i is the point associated to tr ρ x, e v,i .Proof. We have b π ( ρ x, e v,i ) ֒ → b π ( ρ x, e v,i ). Let Θ x, e v,i be a GL n e v,i ( Q p )-invariant lattice of b π ( ρ x, e v,i ), thusΘ dx, e v,i ∈ C GL n e v,i ( Q p ) ( O E ) B x, e v,i where B x, e v,i is the block corresponding to ρ ss x, e v,i . Let Θ x, e v,i, := Θ x, e v,i ∩ b π ( ρ x, e v,i ) . Let Θ x be the preimage of Ord P (cid:0) b S ( U p , O E (cid:1) ρ [ p x ] (cid:1) B via (4.24) ( p x = m x ∩ e T ( U p ) P − ord ρ, B ).By (4.24), we deduce Hom C ( e P B , Θ dx ) = 0. Since Θ x and Θ ′ x := b ⊗ v ∈ S p i =1 , ··· ,k e v (Θ x, e v,i, ⊗ O E ε s e v,i +1 − ◦ det)are commensurable, we see Hom C ( e P B , (Θ ′ x ) d ) = 0. We have a natural projectionHom C (cid:16) e P B , (cid:16) b ⊗ v ∈ S p i =1 , ··· ,k e v (Θ x, e v,i ⊗ O E ε s e v,i +1 − ◦ det) (cid:17) d (cid:17) − ։ Hom C (cid:0) e P B , (Θ ′ x ) d (cid:1) ( = 0) . (4.26)By [28, Lem. 3.4.9] (see also [24, Lem. B.8]) and the fact (e.g. using (1.2), and noting that “ b ⊗ ”on the left hand side is the p -adic completion of the tensor product, while “ b ⊗ ” on the right handside is defined in the following way: M b ⊗ M = lim ←− ( M /U ⊗ O E M /U ) for compact O E -modules M i , where U i runs through open O E -submodules of M i ): (cid:16) b ⊗ v ∈ S p i =1 , ··· ,k e v (Θ x, e v,i ⊗ O E ε s e v,i +1 − ◦ det) (cid:17) d ∼ = b ⊗ v ∈ S p i =1 , ··· ,k e v (Θ x, e v,i ⊗ O E ε s e v,i +1 − ◦ det) d we deduce Hom C GL n e v,i ( Q p ) (cid:0) e P B e v,i , Θ dx, e v,i (cid:1) = 0, and hence B e v,i = B x, e v,i for all v ∈ S p , i = 1 , · · · , k e v .(1) follows. Part (2) of the proposition follows from part (1), (4.24) and Proposition 4.12. Corollary 4.14.
The set Y of points y = ( x, z ) = ( x, { z e v,i } ) as in Proposition 4.13 (2) is Zariski-dense in Spec A [1 /p ] . roof. For y ∈ Y , we denote by m y ⊂ A [1 /p ] (resp. p y ⊂ A ) the maximal ideal (resp. theprime ideal) associated to y . To show ∩ y ∈Y m y = 0, it is sufficient to show ∩ y ∈Y p y = 0. Let f ∈ ∩ y ∈Y p y , which by definition (see (4.18)) corresponds to an e E B -equivariant morphism f : m ( U p , B ) → m ( U p , B ) such that the composition m ( U p , B ) f −→ m ( U p , B ) ։ m ( U p , B ) / p y is equal tozero for all y ∈ Y . Using the first isomorphism in (4.15), the morphism f induces a continuous L P ( Q p )-equivariant morphism f : Ord P ( b S ( U p , O E ) ρ ) B −→ Ord P ( b S ( U p , O E ) ρ ) B . (4.27)For y = ( x, z ) ∈ Y , let i x be an injection as in (4.2), which induces a non-zero L P ( Q p )-equivariantmorphism as in (4.24), still denoted by i x . Let Θ x be the preimage of Ord P (cid:0) b S ( U p , O E (cid:1) ρ [ p x ] (cid:1) B via i x . We use the notation of the proof of Proposition 4.13. Since the R p, B -action on the lefthand side of (4.26) factors through R p, B / p z (where p z is the prime ideal of R p, B associated to z ),the same holds for the R p, B -action on Hom C ( e P B , (Θ ′ x ) d ). Since Θ x and Θ ′ x are commensurable, wededuce that the R p, B -action on m := Hom C ( e P B , Θ dx ) factors through R p, B / p z as well. It is also clearthat the e T ( U p ) P − ord ρ, B -action on m (as a quotient of m ( U p , p ) via the projection induced by i x ) factorsthrough e T ( U p ) P − ord ρ, B / p x . Hence the A -action on m factors through A / p y , and m is thus a quotientof m ( U p , p ) / p y . So the following composition is zero: e P B b ⊗ e E B m ( U p , B ) f −−→ e P B b ⊗ e E B m ( U p , B ) − ։ e P B b ⊗ e E B m . Applying Hom cts O E ( − , E ), we deduce that the composition of f with i x is zero. Consequently, wededuce that the image of any injection as in (4.2) is annihilated by (4.27) (for any benign point x ).By the same argument as in the proof of [7, Prop. 7.5 (2)], we deduce then the map (4.27) is zero,and hence f = 0. This concludes the proof.We have the following classicality criterion. Proposition 4.15 (Classicality) . Let y = ( x, { z e v,i } ) ∈ (Spf A ) rig . Suppose • for all v ∈ S p , i = 1 , · · · , k e v , the pseudo-character associated to z e v,i is absolutely irreducible,and let ρ z e v,i : Gal Q p → GL n e v,i ( k ( y )) be the associated absolutely irreducible representation(enlarging k ( y ) if necessary); • ρ z e v,i is de Rham of distinct Hodge-Tate weights, and any Hodge-Tate weight of ρ z e v,i is strictlybigger than that of ρ z e v,j for j > i .Then x is classical.Proof. The proposition follows by the same argument of the proof of [7, Cor. 7.34]. We give asketch for the convenience of the reader. By Proposition 4.12, we have a non-zero map b ⊗ v ∈ S p i =1 , ··· ,k e v ( b π z e v,i ⊗ k ( y ) ε s e v,i +1 − ◦ det) −→ Ord P (cid:0) b S ( U p , E ) ρ (cid:1) B [ m x ] . (4.28)By assumption, b π z e v,i ∼ = b π ( ρ z e v,i ) is absolutely irreducible. Consider the restriction of (4.28): ⊗ v ∈ S p i =1 , ··· ,k e v ( b π ( ρ z e v,i ) lalg ⊗ k ( y ) ε s e v,i +1 − ◦ det) −→ Ord P (cid:0) b S ( U p , E ) ρ (cid:1) B [ m x ] . (4.29)26f (4.29) is zero, there exists u = ⊗ v ∈ S p i =1 , ··· ,k e v u i ∈ b ⊗ v ∈ S p i =1 , ··· ,k e v ( b π z e v,i ⊗ k ( y ) ε s e v,i +1 − ◦ det) which is sentto zero via (4.28). Since b π ( ρ z e v,i ) is absolutely irreducible, it is easy to see that the left hand sideof (4.28) can be topologically generated by u under the action of L P ( Q p ), which contradicts tothat (4.28) is non-zero. So (4.29) is non-zero (and is actually injective). Using the assumption onthe Hodge-Tate weights and the adjunction property [7, Prop. 4.21], the proposition follows from(3.2).By Lemma 4.9 (2) and the same argument of the proof of [28, Thm. 3.6.1] (one can also use aninfinite fern argument as in [7, § Theorem 4.16.
Each irreducible component of A is of characteristic zero and of dimension atleast X v ∈ S p (cid:0) |{ i | n e v,i = 1 }| + 3 |{ i | n e v,i = 2 }| (cid:1) = 1 + X v ∈ S p (2 n − k e v ) . We end this section by some discussions on criterions of Ord P ( b S ( U p , O E ) ρ ) B = 0. . Definition 4.17.
Let π = π ∞ ⊗ π ∞ = ( ⊗ ′ v ∤ ∞ π v ) ⊗ π ∞ be an automorphic representation of G (where π is defined over Q p , and π v is defined over E for v ∤ ∞ ) with W p = ⊗ v ∈ S p W v the associatedalgebraic representation of G ( F + ⊗ Q Q p ) ∼ = Q v ∈ S p GL n ( Q p ) . Let U p ⊆ G ( A ∞ ,pF + ) be a sufficientlysmall compact open subgroup such that ( π ∞ ) U p = 0 . Then π is called B -ordinary if there is aninjection ι : ⊗ v ∈ S p ( π v ⊗ E W v ) ֒ −→ b S ( U p , E ) ρ [ m π ] such that Ord P (cid:0) ⊗ v ∈ S p ( π v ⊗ E W v ) (cid:1) ∨ B = 0 where m π is the maximal idea of e T ( U p ) ρ [1 /p ] attached to π , and Ord P (cid:0) ⊗ v ∈ S p ( π v ⊗ E W v ) (cid:1) ∈ Mod lfin L P ( Q p ) ( O E ) denotes the modulo ̟ E reduction of (where we also use ι to denote the inducedmorphism on Ord( − ) , cf. [7, Lem. 4.18]) ι (cid:0) ⊗ v ∈ S p Ord P e v ( π v ⊗ E W v ) (cid:1) ∩ Ord P ( b S ( U p , O E )) ρ . Remark 4.18.
Note that the definition is independent of the choice of U p (e.g. by using anisomorphism induced by (5.4) below via taking inverse limit on r ). However, it is not clear to theauthor if the definition depends only on the p -factor ⊗ v ∈ S p π v of π (unless under further assumptions,see Remark 4.20 below). Hence it is not clear to the author if a p -split base-change of a B -ordinaryautomorphic representation is still B -ordinary. Lemma 4.19.
For a block B of Mod lfin L P ( Q p ) ( O E ) and a sufficiently small subgroup U p ⊂ G ( A ∞ ,pF + ) ,the followings are equivalent:(1) there exists a B -ordinary automorphic representation π = π ∞ ⊗ π ∞ such that ( π ∞ ) U p = 0 ,(2) Ord P ( b S ( U p , O E ) ρ ) B = 0 ,(3) there exists a benign point of x of Spec e T ( U p ) P − ord ρ [1 /p ] such that ρ ss x, e v,i ∼ = ρ B e v,i for v ∈ S p , i = 1 , · · · , k e v . roof. (1) ⇒ (2) is clear. (2) ⇒ (3) follows from Proposition 4.13 (1). We show (3) ⇒ (1). Bythe decomposition (4.13), there exists a block B ′ such that the following composition is non-zero(hence injective since the left hand side is irreducible) ⊗ v ∈ S p (cid:0) ⊗ i =1 , ··· ,k e v b π ( ρ x, e v,i ) lalg ⊗ k ( x ) ε s e v,i +1 − ◦ det (cid:1) ( . ) −−−→ Ord P (cid:0) b S ( U p , E (cid:1) ρ [ m x ] (cid:1) − ։ Ord P (cid:0) b S ( U p , E (cid:1) ρ [ m x ] (cid:1) B ′ . (4.30)We deduce hence a similar non-zero map as in (4.24) with B replaced by B ′ . Using the sameargument as in the proof of Proposition 4.13 (1), we deduce ρ ss x, e v,i ∼ = ρ B ′ e v,i for all v ∈ S p and i , andhence B ′ = B (by the statement in (3)). By (4.30) (with B ′ = B ), it is easy to deduce that anyautomorphic representation π associated to the point x is B -ordinary and satisfies π U p = 0. Thisconcludes the proof. Remark 4.20.
The condition (3) is equivalent to that there exists an automorphic representation π = ( ⊗ ′ v ∤ ∞ π v ) ⊗ π ∞ such that (cid:0) where ( − ) L P ( Z p ) − alg+ , ( − ) L P e v ( Z p ) − alg+ are defined similarly as in (4.1),and the first equality easily holds by definition (cid:1) (a) Ord P ( ⊗ v ∈ S p ( π v ⊗ E W v )) L P ( Z p ) − alg+ = ⊗ v ∈ S p Ord P e v ( π v ⊗ E W v ) L P e v ( Z p ) − alg+ = 0 ,(b) the unique Hodge-Tate weights descending P e v -filtration on ρ π, e v (where the existence followsfrom Proposition 4.4 (2)) satisfies gr i F e v ss ∼ = ρ B e v,i , where ρ π denotes the Gal F -representationassociated to π ,(c) ( ⊗ ′ v ∤ p, ∞ π v ) U p = 0 .Note that the conditions (a) and (b) depend only on the p -factor of π and { ρ π, e v } v ∈ S p .4.4. Local-global compatibility We show a local-global compatibility result for our GL ( Q p )-ordinary families under certain genericassumptions. Definition 4.21.
We call ρ is B -generic if for all v ∈ S p ,(1) ρ e v admits a unique filtration F B e v such that (gr i F B e v ) ss ∼ = ρ B e v,i for i = 1 , · · · , k e v ;(2) the filtration F B e v satisfies Hypothesis 2.1. We assume ρ is B -generic. By Proposition 4.13 (1), we see ¯ F x, e v = F B e v (cf. (4.25)) for all benignpoints x . Consider the deformation problem (where R P e v − ord , ¯ (cid:3) ρ e v , F B e v denotes the reduced quotient of R P e v − ord , (cid:3) ρ e v , F B e v , cf. § (cid:0) F/F + , S, e S, O E , ρ, ε − n δ F/F + , { R P e v − ord , ¯ (cid:3) ρ e v , F B e v } v ∈ S p ∪ { R ¯ (cid:3) ρ e v } v ∈ S \ S p (cid:1) . (4.31)By [36, Prop. 3.4], the corresponding deformation functor is represented by a complete localnoetherian O E -algebra, denoted by R P − ord ρ, S , B , which is a quotient of R ρ, S . By Proposition 4.11 (2)and Proposition 4.13(1), for any benign point x ∈ Spec e T ( U p ) P − ord ρ, B [1 /p ], the action of R ρ, S onOrd P ( b S ( U p , E ) ρ ) B [ m x ] factors through R P − ord ρ, S , B . Since (Ord P ( b S ( U p , E ) ρ ) B ) L P ( Z p ) − alg+ is dense inOrd P ( b S ( U p , E ) ρ ) B (Proposition 4.1), we deduce by the same argument as in [7, Thm. 6.12]:28 roposition 4.22. (1) The action of R ρ, S on Ord P ( b S ( U p , E ) ρ ) B factors through R P − ord ρ, S , B .(2) Let x be a closed point of Spec e T ( U p ) ρ [1 /p ] such that Ord P ( b S ( U p , E ) ρ [ m x ]) B = 0 , then for v ∈ S p , ρ x, e v admits a P e v -filtration with the induced P e v -filtration on ρ e v equal to F B e v . Let A be an artinian local O E -algebra A , and ρ A ∈ R P − ord ρ, S , B ( A ). For v ∈ S p , by definition ρ A, e v :=pr ◦ ρ A | Gal F e v (with pr : GL × GL n ։ GL n ) admits a P e v -filtration F e v such that tr(gr i F e v ) is adeformation of tr(gr i F B e v ) over A for i = 1 , · · · , k e v . We obtain thus a natural morphism R p, B −→ R P − ord ρ, S , B . (4.32)The composition R p, B −→ R P − ord ρ, S , B −→ e T ( U p ) P − ord ρ, B . (4.33)equips with m ( U p , B ) another R p, B -action. The following theorem follows by the same argument asin the proof of [28, Thm. 3.5.5] [32, Prop. 5.5]. Indeed, one can easily obtain an analogue of [28,Lem. 3.5.6] with p of loc. cit. replaced by the benign points, using Lemma 4.9 (2), Corollary 4.11(2). Note also that an analogue of [28, Lem. 3.5.7] is already contained in Proposition 4.5 (1) (whoseproof builds upon the local-global compatibility result in classical local Langlands correspondence,see the proof of [7, Prop. 7.6 (2)]). Theorem 4.23 (local-global compatibility) . The two actions of R p, B on m ( U p , B ) coincide, i.e.the composition R p, B −→ e T ( U p ) P − ord ρ, B ֒ −→ End R p, B ( m ( U p , B )) coincides with the structure map. Remark 4.24.
One might prove (e.g. by putting more hypothesis on ρ p ) a stronger local-globalcompatibility result (e.g. by replacing the universal pseudo-deformation rings by the universal de-formation rings of certain Galois representations) as in [7, § By definition, Theorem 4.23 and Theorem 4.16, we have
Corollary 4.25.
We have e T ( U p ) P − ord ρ, B ∼ −→ A , and each irreducible component of e T ( U p ) P − ord ρ, B is ofcharacteristic zero and of dimension at least P v ∈ S p (2 n − k e v ) . We end this section by a B -generic criterion, which is easier to check in certain cases. Lemma 4.26.
For v ∈ S p , let F e v be a P e v -filtration on ρ e v satisfying Hypothesis 2.1 and that (gr i F e v ) ss ∼ = ρ B e v,i for all i . Suppose moreover B e v,i = B e v,j for all i = j and v ∈ S p . Then ρ is B -generic.Proof. Suppose we have another filtration F ′ e v satisfying the same property.(1) We show first gr F ′ e v ∼ = gr F e v . If ρ B e v, is irreducible, then it is clear. Suppose ρ B e v, ∼ = χ ⊕ χ ,and we prove the statement case by case.(a) If gr F ′ e v ∼ = χ ⊕ χ , and if gr F e v is not isomorphic to gr F ′ e v . Without loss of generality, weassume gr F e v is a non-split extension of χ by χ . We seeHom Gal F e v ( χ , ρ e v ) −→ Hom
Gal F e v ( χ , ρ e v / gr F e v ) ֒ −→ Hom
Gal F e v (gr F e v , , ρ e v / gr F e v ) , (4.34)29here the first map is non-zero by assumption (using gr F ′ e v ֒ → ρ e v ).We deduce thus the right handside of (4.34) is non-zero, which leads to a contradiction with Hypothesis 2.1.(b) Suppose gr F ′ e v is a non-split extension of χ by χ , and suppose gr F e v is an extension of χ by χ (e.g. if gr F e v is a direct sum of χ and χ ). If χ = χ = χ , the sum gr F ′ e v + gr F e v will be asubrepresentation of ρ e v of dimension bigger than 3. We deduce that Hom Gal Q p ( χ, gr F ′ e v / (gr F ′ e v ∩ gr F e v )) = 0, and hence Hom Gal Q p (gr F e v , ρ e v / gr F e v ) = 0, a contradiction with Hypothesis 2.1. If χ = χ , then we haveHom Gal F e v ( χ , gr F e v ) ֒ −→ Hom
Gal F e v ( χ , ρ e v ) −→ Hom
Gal F e v ( χ , ρ e v / gr F ′ e v )with the second map non-zero. There exists thus i > χ ֒ → gr i F ′ e v , thusHom Gal F e v (gr F ′ e v , gr i F ′ e v ) = 0 , which (again) contradicts with Hypothesis 2.1.(c) Consider the last case where both of gr F ′ e v and gr F e v are isomorphic to a non-split extensionof χ by χ , and gr F ′ e v ≇ gr F e v . In this case, χ χ − = 1 or ω .If χ = χ = χ . It is easy to see gr F e v + gr F ′ e v is isomorphic to a successive extension of χ ofdimension bigger three. Using the same argument in (b) (in the case χ = χ = χ ), we easilydeduce a contradiction.Suppose χ = χ ω . Denote by ι : χ ֒ → gr F e v ֒ → ρ e v , and ι ′ : χ ֒ → gr F ′ e v ֒ → ρ e v .If Im( ι ) = Im( ι ′ ), since gr F ′ e v ≇ gr F e v , we deduce the composition χ ∼ = gr F e v / Im( ι ) ֒ → ρ e v / Im( ι ′ ) −→ ρ e v / gr F ′ e v is non-zero. Hence there exists i > χ ֒ → gr i F ′ e v , thus Hom Gal F e v (gr F ′ e v , gr i F ′ e v ) = 0, acontradiction with Hypothesis 2.1.If Im( ι ′ ) = Im( ι ). Let i > χ ι ′ −→ ρ e v → ρ e v / Fil i − F e v ρ e v is non-zero. Thus χ is an irreducible sub of gr i F e v . Since F e v satisfies Hypothesis 2.1, we deduce dim k E gr i F e v = 2 andsoc Gal F e v gr i F e v ∼ = χ (otherwise, Hom Gal F e v (gr i F e v , gr F e v ) = 0). Consider χ ∼ = gr F ′ e v /χ ֒ −→ ρ e v / Im( ι ′ ) − ։ ρ e v / (Im( ι ) ⊕ Im( ι ′ )) . (4.35)If the image of (4.35) lies in gr F e v /χ , then it is not difficult to see the first injection in (4.35) hasimage in gr F e v / Im( ι ′ ) ∼ = gr F e v . But if so, gr F e v is split, a contradiction. So there exists j > χ ∼ = gr F ′ e v /χ −→ ρ e v / (cid:0) Im( ι ′ ) + Fil j − F e v ρ e v (cid:1) is non-zero. We let j be the maximal integer satisfying this property. Then χ is an irreducibleconstituent of gr j F e v . Using (again) Hypothesis 2.1, we easily deduce dim k E gr j F e v = 2 andcosoc Gal F e v gr j F e v ∼ = χ (otherwise, Hom Gal F e v (gr j F e v , gr F e v ) = 0). If j < i , let V j be the kernelof Fil j − F e v ։ gr j F e v ։ χ , then we see gr F ′ e v −→ ρ e v /V j is injective, and the injection χ ֒ → ρ e v /V j induces a section of gr F ′ e v ։ χ , a contradiction. If j > i , since χ ֒ → ρ e χ / Im( ι ′ ) ։ ρ e χ / Fil j − F e v , we deduce gr j F e v splits, a contradiction. Finally if j = i ,then B v,i = B v, a contradiction. 302) We show Fil F e v = Fil F ′ e v (as subrepresentation of ρ e v ), from which the lemma follows by an easyinduction argument. Suppose Fil F e v = Fil F ′ e v , consider V := (Fil F e v + Fil F ′ e v ) / Fil F e v , which is thus anon-zero subrepresentation of ρ e v / Fil F e v , and whose irreducible constituents appear in gr F e v . If V is irreducible, it has to be a sub of gr j F e v for certain j >
1. However, we havegr F e v ∼ = Fil F ′ e v ։ (Fil F e v + Fil F ′ e v ) / Fil F e v = V. Thus Hom
Gal F e v (gr F e v , gr j F e v ) = 0, a contradiction with Hypothesis 2.1. If V is isomorphic to adirect sum of characters, we similarly obtain a contradiction. Suppose there exist characters χ , χ such that V is a non-split extension of χ by χ . In this case we see V ∼ = gr F e v . Using the sameargument as in (c) for the case Im( ι ) = Im( ι ′ ), we can also obtain a contradiction. This concludesthe proof.
5. Patching and automorphy lifting
We apply the Taylor-Wiles patching argument to our GL ( Q p )-ordinary families, and prove ourmain result on automorphy lifting. p Let B be a block of Mod lfin L P ( Q p ) ( O E ) such that Ord P ( b S ( U p , O E ) ρ ) B = 0. Lemma 5.1.
Let H be a finite group, r ≥ , M be a finitely generated flat O E /̟ rE [ H ] -module.Then M ∨ is also a finitely generated flat O E /̟ rE [ H ] -modules.Proof. It is easy to check that if M is a finite free O E /̟ rE [ H ]-module, then M ∨ ∼ = M , and henceis also a finite free O E /̟ rE [ H ]-module. The lemma follows.Let Y p = Q v ∤ p Y v be a compact open normal subgroup of U p (which is thus also sufficiently small).For any compact open subgroup U p of Q v ∈ S p GL n ( Z p ), by [15, Lem. 3.3.1], S ( Y p U p , O E /̟ rE ) is afinite free O E /̟ rE [ U p /Y p ]-module. Lemma 5.2.
Let i ∈ Z ≥ , r ∈ Z ≥ .(1) Ord P ( S ( Y p , O E /̟ rE ) ρ ) L i B and (cid:0) Ord P ( S ( Y p , O E /̟ rE ) ρ ) L i B (cid:1) ∨ are finite flat O E /̟ rE [ U p /Y p ] -modules.(2) We have a natural isomorphism (cid:0) Ord P ( S ( Y p , O E /̟ rE ) ρ ) L i B (cid:1) ∨ U p /Y p ∼ −−→ (cid:0) Ord P ( S ( U p , O E /̟ rE ) ρ ) L i B (cid:1) ∨ . Proof.
Recall (cf. (3.7))Ord P ( S ( Y p , O E /̟ rE ) ρ ) L i ∼ = S ( Y p K i,i , O E /̟ rE ) ρ, ord ∼ = ⊕ m ordinary S ( Y p K i,i , O E /̟ rE ) ρ, m , where m runs through the ordinary maximal ideals of the O E /̟ rE -subalgebra B ( Y p ) ofEnd O E /̟ rE (cid:0) S ( Y p K i,i , O E /̟ rE ) ρ (cid:1) generated by the operators in Z L + P . The same statement holds with Y p replaced by U p . Note alsothat the natural inclusion S ( U p K i,i , O E /̟ rE ) ρ ∼ = S ( Y p K i,i , O E /̟ rE ) U p /Y p ρ ֒ −→ S ( Y p K i,i , O E /̟ rE ) ρ B ( Y p ) ։ B ( U p ). By definition, it is straightforward to see that for a max-imal ideal m of B ( U p ), m is ordinary if and only if pr − ( m ) is ordinary. Since the action of B ( Y p )and U p /Y p commute, we deduce that Ord P ( S ( Y p , O E /̟ rE ) ρ ) L i is an O E /̟ rE [ U p /Y p ]-equivariantdirect summand of S ( Y p K i,i , O E /̟ rE ) ρ , and hence is a finite flat O E /̟ rE [ U p /Y p ]-module.Using the isomorphism (which follows by the same argument as for (4.15))Ord P ( S ( Y p , O E /̟ rE ) ρ ) ∨ B ∼ = e P B b ⊗ e E B Hom C (cid:0) e P B , Ord P ( S ( Y p , O E /̟ rE ) ρ ) ∨ (cid:1) , we see that Ord P ( S ( Y p , O E /̟ rE ) ρ ) ∨ B inherits a natural U p /Y p -action from Ord P ( S ( Y p , O E /̟ rE ) ρ ) ∨ satisfying that the natural injection (given by the evaluation map)Ord P ( S ( Y p , O E /̟ rE ) ρ ) ∨ B ∼ = e P B b ⊗ e E B Hom C (cid:0) e P B , Ord P ( S ( Y p , O E /̟ rE ) ρ ) ∨ (cid:1) ֒ −→ Ord P ( S ( Y p , O E /̟ rE ) ρ ) ∨ is U p /Y p -equivariant. Hence, the decompositionOrd P ( S ( Y p , O E /̟ rE ) ρ ) ∼ = ⊕ B Ord P ( S ( Y p , O E /̟ rE ) ρ ) B is U p /Y p -equivariant. This implies that Ord P ( S ( Y p , O E /̟ rE ) ρ ) L i B is a U p /Y p -equivariant directsummand of Ord P ( S ( Y p , O E /̟ rE ) ρ ) L i , hence is also a finite flat O E /̟ rE [ U p /Y p ]-module. Togetherwith Lemma 5.1, (1) follows.For each i ≥
1, we have a natural isomorphism S ( U p K i,i , O E /̟ rE ) ρ ∼ −−→ S ( Y p K i,i , O E /̟ rE ) U p /Y p ρ . (5.1)By the discussion in the first paragraph, we deduce that (5.1) induces an isomorphismOrd P ( S ( U p , O E /̟ rE ) L i ρ ) ∼ −−→ Ord P ( S ( Y p , O E /̟ rE ) L i ρ ) U p /V p . (5.2)Indeed, for all maximal ideals of B ( U p ), we have S ( U p K i,i , O E /̟ rE ) ρ, m ֒ −→ (cid:0) S ( Y p K i,i , O E /̟ rE ) ρ, pr − ( m ) (cid:1) U p /Y p , (5.3)hence ⊕ m S ( U p K i,i , O E /̟ rE ) ρ, m ֒ → ⊕ m (cid:0) S ( Y p K i,i , O E /̟ rE ) ρ, pr − ( m ) (cid:1) U p /Y p ֒ → S ( Y p K i,i , O E /̟ rE ) U p /Y p ρ . By (5.1), the above composition is surjective, hence each map in (5.3) has to be bijective. Theisomorphism (5.2) follows. Taking direct limit on i , (5.2) induces an isomorphismOrd P ( S ( U p , O E /̟ rE ) ρ ) ∼ −−→ Ord P ( S ( Y p , O E /̟ rE ) ρ ) U p /V p . (5.4)Which together with the obvious injections (with the composition bijective) ⊕ B Ord P ( S ( U p , O E /̟ rE ) ρ ) B ֒ −→ ⊕ B (cid:0) Ord P ( S ( Y p , O E /̟ rE ) ρ ) B (cid:1) U p /V p ֒ −→ Ord P ( S ( Y p , O E /̟ rE ) ρ ) U p /V p imply Ord P ( S ( U p , O E /̟ rE ) ρ ) B ∼ = (cid:0) Ord P ( S ( Y p , O E /̟ rE ) ρ ) B (cid:1) U p /V p for all B . HenceOrd P ( S ( U p , O E /̟ rE ) ρ ) L i B ∼ −−→ (cid:0) Ord P ( S ( Y p , O E /̟ rE ) ρ ) L i B (cid:1) U p /Y p . (5.5)By taking Pontryagain dual, (2) follows. 32 .2. Auxiliary primes Choose q ≥ [ F + : Q ] n ( n − . By [36, Prop. 4.4] (see also the proof of [36, Thm. 6.8]), for all N ≥ Q N (resp. e Q N ) of primes of F + (resp. of F ) such that • | Q N | = q , Q N is disjoint from S , and any primes in Q N is split in F ; • e Q N = { e v | v | v ∈ Q N } ; • Nm( v ) ≡ p N ) for v ∈ Q N ; • for e v ∈ e Q N , ρ e v ∼ = s e v ⊕ ψ e v , where ψ e v is the (generalized) eigenspace of Frobenius of aneigenvalue α e v on which Frobenius acts semisimply.For e v ∈ e Q N , denote by D e v the local deformation problem such that for A ∈ Art( O E ), D e v ( A )consists of all lifts which are 1 + M n ( m A )-conjugate to one of the form s e v ⊕ ψ e v where s e v is anunramified lift of s e v and where ψ e v is a (possibly ramified) lift of ψ e v satisfying that the image ofinertial under ψ e v is contained in the set of scalar matrices. The deformation problem D e v is pro-represented by a quotient of R ¯ (cid:3) ρ e v , denoted dy R ψ e v ρ e v . Let ψ e v be as above, then the restriction of ψ e v to the inertial subgroup I e v of Gal F e v gives a character χ e v : I e v → m A (noting χ e v ≡ ψ e v | I e v = 1(mod m A )). We can prove (e.g. using χ | F e v | e v = χ e v since Frob − e v σ Frob e v = σ | F e v | for all σ ∈ I e v /P e v ,and using the fact that 1 + m A is a p -group) that χ e v factors through I e v /P e v − ։ F × e v − ։ F e v ( p ) ∼ = Z /p N Z where F e v ( p ) denotes the maximal p -power order quotient of F × e v , and F e v denotes the residue field of F at e v . We deduce thus a natural morphism χ univ e v : Z /p N Z −→ ( R ψ e v ρ e v ) × . (5.6)Denote by S Q N the following deformation problem (cid:0) F/F + , S ∪ Q N , e S ∪ e Q N , O E , ρ, ε − n δ F/F + , { R ¯ (cid:3) ρ e v } v ∈ S ∪ { D e v } v ∈ Q N (cid:1) . Let R ρ, S QN be the corresponding universal deformation ring, and R (cid:3) S ρ, S QN the S -framed universaldeformation ring. By [36, Prop. 4.4], we can and do choose Q N , e Q N satisfying moreover (recall R loc = b ⊗ v ∈ S R ¯ (cid:3) e v ) R (cid:3) S ρ, S QN is topologically generated over R loc by g := q − [ F + : Q ] n ( n − Q N := Q v ∈ Q N F e v ( p ) ∼ = ( Z /p N Z ) ⊕ q . We deduce from (5.6) a natural morphism ∆ Q N → ( R (cid:3) S ρ, S QN ) × . Using the fact χ univ e v does not depend on the choice of basis, it is not difficult to see thismorphism factors through ( R ρ, S QN ) × . We have thus morphisms of O E -algebras O E [∆ Q N ] −→ R ρ, S QN −→ R (cid:3) S ρ, S QN . Denote by a Q N the augmentation ideal of O E [∆ Q N ]. Then we have R ρ, S QN / a Q N ∼ = R ρ, S , R (cid:3) S ρ, S QN / a Q N ∼ = R (cid:3) ρ, S . v ∈ Q N , denote by p e vN := (cid:26) g ∈ GL n ( O F e v ) | g (mod ̟ F e v ) ∈ GL n − d e vN ∗ d e vN ! (cid:27) , where d e vN := dim k E ψ e v . Denote by p e vN, the kernel of the following composition p e vN − ։ GL d e vN ( F e v ) det −−→ F × e v − ։ F e v ( p ) , where the first map is given by the composition of the modulo ̟ F e v map and the natural projection.Put U ( Q N ) e v := i − e v ( p e vN ) , U ( Q N ) e v := i − e v ( p e vN, ) ,U i ( Q N ) p := (cid:0) Y v ∤ pv / ∈ Q N U v (cid:1)(cid:0) Y v ∈ Q N U i ( Q N ) e v (cid:1) ⊂ U p , i = 0 , . We have U ( Q N ) p /U ( Q N ) p ∼ = ∆ Q N .We have by definition (cf. § T ( U i ( Q N ) p ) ֒ → T ( U p ), and we use m ( ρ ) to denote m ( ρ ) ∩ T ( U i ( Q N ) p ) which is the maximal ideal of T ( U i ( Q N ) p ) associated to ρ via the relations (3.5). As be-fore, we also use the subscript ρ to denote the localizations at the maximal ideal m ( ρ ) ⊂ T ( U i ( Q N ) p ).Let i ∈ { , } . For a compact open subgroup U p of G ( F + ⊗ Q Q p ), and for a uniformiser ̟ e v of O F e v for v ∈ Q N , we have as in [36, Prop. 5.9] (see also [23, § § ̟ e v ∈ End O E (cid:0) S ( U i ( Q N ) p U p , O E /̟ rE ) ρ (cid:1) (defined using Hecke operators at e v ). We denote by pr N := Q v ∈ Q N pr ̟ e v . By [36, Prop. 5.9], thefollowing composition S ( U p U p , O E /̟ rE ) ρ ֒ −→ S ( U ( Q N ) p U p , O E /̟ rE ) ρ pr N −−→ pr N ( S ( U ( Q N ) p U p , O E /̟ rE ) ρ ) (5.8)is an isomorphism. Since pr N is defined using Hecke operators for e v ∈ e Q N , (5.8) is T ( U ( Q N ) p )-equivariant. We also havepr N (cid:0) S ( U i ( Q N ) p , O E /̟ rE ) ρ (cid:1) ∼ = lim −→ U p pr N (cid:0) S ( U i ( Q N ) p U p , O E /̟ rE ) ρ (cid:1) , as a e T ( U i ( Q N ) p ) ρ × G ( F + ⊗ Q Q p )-equivariant direct summand of S ( U i ( Q N ) p , O E /̟ rE ) ρ (see [13, § N (cid:0) Ord P (cid:0) S ( U i ( Q N ) p , O E /̟ rE ) ρ (cid:1)(cid:1) ∼ = lim −→ j pr N (cid:0) S ( U i ( Q N ) p K j,j , O E /̟ rE ) ρ, ord (cid:1) ∼ = Ord P (cid:0) pr N (cid:0) S ( U i ( Q N ) p , O E /̟ rE ) ρ (cid:1)(cid:1) , (5.9)and we see that the object in (5.9) is a e T ( U i ( Q N ) p ) P − ord ρ × L P ( Q p )-equivariant direct summand ofOrd P (cid:0) S ( U i ( Q N ) p , O E /̟ rE ) ρ (cid:1) . It is also clear (e.g. by a similar argument as in the proof of Lemma 5.2) that the decompositionOrd P (cid:0) S ( U i ( Q N ) p , O E /̟ rE ) ρ (cid:1) ∼ = ⊕ B Ord P (cid:0) S ( U i ( Q N ) p , O E /̟ rE ) ρ (cid:1) B N , and hence we have that V i ( N, B , r ) := pr N (cid:0) Ord P (cid:0) S ( U i ( Q N ) p , O E /̟ rE ) ρ (cid:1) B (cid:1) ∼ = lim −→ j pr N (cid:0) Ord P ( S ( U i ( Q N ) p , O E /̟ rE ) ρ ) L j B (cid:1) is a e T ( U i ( Q N ) p ) P − ord ρ, B × L P ( Q p )-equivariant direct summand of Ord P (cid:0) S ( U i ( Q N ) p , O E /̟ rE ) ρ (cid:1) B .The isomorphism (5.8) induces a T ( U ( Q N ) p ) × L P ( Q p )-equivariant isomorphismOrd P ( S ( U p , O E /̟ rE ) ρ ) B ∼ −−→ V ( N, B , r ) . (5.10)Note that V ( N, B , r ) is naturally equipped with a U ( Q N ) p /U ( Q N ) p ∼ = ∆ Q N -action, which com-mutes with e T ( U ( Q N ) p ) P − ord ρ, B × L P ( Q p ). Lemma 5.3.
Let j ∈ Z ≥ .(1) V ( N, B , r ) L j and ( V ( N, B , r ) L j ) ∨ are finite flat O E /̟ rE [∆ Q N ] -modules.(2) There is a natural isomorphism ( V ( N, B , r ) L j ) ∨ ∆ QN ∼ −−→ V ( N, B , r ) L j .Proof. (1) follows from Lemma 5.2 and the fact that V ( N, B , r ) L j ∼ = pr N (cid:0) Ord P ( S ( U ( Q N ) p , O E /̟ rE ) ρ ) L j B (cid:1) is a ∆ Q N -equivariant direct summand of Ord P ( S ( U ( Q N ) p , O E /̟ rE ) ρ ) L j B . We have (e.g. see [13, § N (cid:0) S ( U ( Q N ) p K j,j , O E /̟ rE ) ρ (cid:1) ∆ QN ∼ −−→ pr N (cid:0) S ( U ( Q N ) p K j,j , O E /̟ rE ) ρ (cid:1) . Using the same argument as for (5.2), we deduce an isomorphismpr N (cid:0) Ord P (cid:0) S ( U ( Q N ) p , O E /̟ rE ) ρ (cid:1) L j (cid:1) ∆ QN ∼ −−→ pr N (cid:0) Ord P (cid:0) S ( U ( Q N ) p , O E /̟ rE ) ρ (cid:1) L j (cid:1) . (2) follows then by the same argument as for (5.5) (and Pontryagain duality).Put V i ( N, B ) := lim ←− r V i ( N, B , r ), and we have a natural isomorphism V i ( N, B ) ∼ = pr N (cid:0) Ord P ( b S ( U i ( Q N ) p , O E ) ρ, B (cid:1) . Put M i ( N, B , r, L j ) := (cid:0) V i ( N, B , r ) L j (cid:1) ∨ , and M i ( N, B ) := lim ←− j,r M i ( N, B , r, L j ) ∼ = V i ( N, B ) d . ByLemma 5.3(1), M ( N, B , r, L j ) is a finite flat O E /̟ rE [∆ Q N ]-module. By [28, Lem. 4.4.4], we deduce(where the conditions of loc. cit. are easy to verify in our case) Proposition 5.4. M ( N, B ) is a flat O E [∆ Q N ] -module, and M ( N, B ) / a Q N M ( N, B ) ∼ −−→ M ( N, B ) . .3. Patching I By [23, Prop. 5.3.2] (see also the proof [36, Thm. 6.8]), we have a natural surjection R ρ, S ( Q N ) − ։ e T ( U ( Q N ) p ) ρ . By the local-global compatibility in classical local Langlands correspondance, for any compact opensubgroup U p of Q v | p GL n ( O F e v ), the induced action of O E [∆ Q N ] on S ( U ( Q N ) p U p , O E /̟ rE ) ρ via O E [∆ Q N ] −→ R ρ, S ( Q N ) −→ e T ( U ( Q N ) p ) ρ , (5.11)coincides with the O E [∆ Q N ]-action coming from the natural ∆ Q N ∼ = U ( Q N ) p /U ( Q N ) p -action.We assume henceforth ρ is B -generic (cf. Definition 4.21). By Proposition 4.22 (applied with U p = U ( Q N )), we can deduce that the morphism R ρ, S ( Q N ) − ։ e T ( U ( Q N ) p ) P − ord ρ, B factors through R P − ord ρ, S ( Q N ) , B := R ρ, S ( Q N ) ⊗ R ρ, S R P − ord ρ, S , B , which is also the universal deformation ringof the deformation problem (cid:0) F/F + , S, e S, O E , ρ, ε − n δ F/F + , { R P e v − ord , ¯ (cid:3) ρ e v , F B e v } v ∈ S p ∪ { R ¯ (cid:3) ρ e v } v ∈ S \ S p ∪ { D e v } v ∈ Q N (cid:1) . (5.12)Since V ( N, B , r ) L j is a e T ( U ( Q N ) p ) ρ × U ( Q N ) p /U ( Q N ) p -equivariant direct summand of S ( U ( Q N ) p K j,j , O E /̟ rE ) ρ , the two O E [∆ Q N ]-actions on V ( N, B , r ) L j , obtained by the following two ways (noting the firstcomposition is compatible with (5.11)) O E [∆ Q N ] −→ R P − ord ρ, S ( Q N ) , B −→ e T ( U ( Q N ) p ) P − ord ρ, B , O E [∆ Q N ] ∼ = O E [ U ( Q N ) p /U ( Q N ) p ] , coincide. By taking limit, we obtain a similar statement for V ( N, B ) and M ( N, B ).Denote by R (cid:3) S ,P − ord ρ, S , B (cid:0) resp. by R (cid:3) S ,P − ord ρ, S ( Q N ) , B (cid:1) the S -framed deformation ring of the deformationproblem (4.31) (resp. of (5.12)). For ∗ ∈ {S , S ( Q N ) } , the composition R loc → R (cid:3) S ρ, ∗ → R (cid:3) S ,P − ord ρ, ∗ , B thus factors through R loc ,P − ord B := (cid:0) b ⊗ v ∈ S \ S p R ¯ (cid:3) ρ e v (cid:1) b ⊗ O E (cid:0) b ⊗ v ∈ S p R ¯ (cid:3) ,P e v − ord ρ e v, B e v (cid:1) . For v ∈ S p , we have a natural morphism b ⊗ i =1 , ··· ,k e v R ps B e v,i → R ¯ (cid:3) ,P e v − ord ρ e v, B e v , which induces (see (4.17) for R p, B ) R p, B ֒ −→ R B −→ R loc ,P − ord B −→ R (cid:3) S ,P − ord ρ, S ( Q N ) , B , (5.13)where R B := R p, B b ⊗ O E (cid:0) b ⊗ v ∈ S \ S p R ¯ (cid:3) ρ e v (cid:1) . It is clear that (5.13) factors through R P − ord ρ, S ( Q N ) , B (e.g. seethe argument above (4.32)). Note that we have R P − ord ρ, S ( Q N ) , B / a Q N ∼ = R P − ord ρ, S , B , R (cid:3) S ,P − ord ρ, S ( Q N ) , B / a Q N ∼ = R (cid:3) S ,P − ord ρ, S , B . O ∞ := O E [[ z , · · · , z n | S | ]] with the maximal ideal b , S ∞ := O E [[ y , · · · , y q , z , · · · , z n | S | ]] withthe maximal ideal a , and R ∞ := R loc ,P − ord B [[ x , · · · , x g ]]. Denote by a = ( y , · · · , y q ) the kernel of S ∞ ։ O ∞ , and a := ( z , · · · , z n | S | , y , · · · , y q ). For an open ideal c of S ∞ , we denote by s ( c ) bethe integer such that S ∞ / c ∼ = O E /̟ s ( c ) E .For each N ≥
1, we fix a surjection O E [[ y , · · · , y q ]] ։ O E [∆ Q N ] with kernel c N = ((1 + y ) p N − , · · · , (1 + y q ) p N − O E [∆ Q N ] → R P − ord ρ, S ( Q N ) , B , induce O E [[ y , · · · , y q ]] −→ R P − ord ρ, S ( Q N ) , B . Together with the isomorphism R (cid:3) S ,P − ord ρ, S ( Q N ) , B ∼ = R P − ord ρ, S ( Q N ) , B b ⊗ O E O ∞ , we obtain a morphism of com-plete noetherian O E -algebras S ∞ −→ R (cid:3) S ,P − ord ρ, S ( Q N ) , B . (5.14)By (5.7), R (cid:3) S ,P − ord ρ, S ( Q N ) , B can be topologically generated by g elements over R loc ,P − ord B , hence thereexists a surjective map R ∞ = R loc ,P − ord B [[ x , · · · , x g ]] − ։ R (cid:3) S ,P − ord ρ, S ( Q N ) , B ∼ = R P − ord ρ, S ( Q N ) , B [[ z , · · · , z n | S | ]] . (5.15)We lift the morphism (5.14) to a morphism S ∞ → R ∞ . For i ∈ { , } , j ≥ k >
0, we put M (cid:3) i ( N, B , k, L j ) := M i ( N, B , k, L j ) ⊗ O E O ∞ . Since M i ( N, B , k, L j ) is equipped with a natural R P − ord ρ, S ( Q N ) , B -action via R P − ord ρ, S ( Q N ) , B −→ e T ( U ( Q N ) p ) P − ord ρ, B , we see M (cid:3) i ( N, B , k, L j ) is equipped with a natural R (cid:3) S ,P − ord ρ, S ( Q N ) , B ( ∼ = R P − ord ρ, S ( Q N ) , B b ⊗ O E O ∞ )-action, andhence with an S ∞ -action via (5.14). Moreover, for any open ideal of S ∞ containing c N and ̟ kE ,by Lemma 5.3, we deduce that M (cid:3) ( N, B , k, L j ) /I is a finite flat S ∞ /I -module of rank equal tork O E /̟ kE M ( N, B , k, L j ) (and hence has finite cardinality).We use the language of ultrafilters for the patching argument (cf. [34, § F be a non-principal ultrafilter of I := Z ≥ , R := Q I O E . Let S F ⊂ R be the multiplicative set consisting of allidempotents e I with I ∈ F where e I ( i ) = 1 if i ∈ I , and e I ( i ) = 0 otherwise. Denote by R F := S − F R .For k ∈ Z ≥ , put (noting that the cardinality of M (cid:3) ( N, B , k, L j ) / a k and M (cid:3) ( N, B , k, L j ) / b k isfinite and bounded by a certain integer independent of N ) M ∞ ( B , k ) := lim ←− j M ∞ ( B , k, L j ) := lim ←− j (cid:0) ( Y N ∈I M (cid:3) ( N, B , k, L j ) / a k ) ⊗ R R F (cid:1) ,M ∞ ( B , k ) := lim ←− j (cid:0) ( Y N ∈I M (cid:3) ( N, B , k, L j ) / b k ) ⊗ R R F (cid:1) . The diagonal S ∞ -action (resp. O ∞ -action) on Y N ∈I M (cid:3) ( N, B , k, L j ) / a k (cid:0) resp. Y N ∈I M (cid:3) ( N, B , k, L j ) / b k (cid:1) induces an S ∞ -module (resp. an O ∞ -module) structure on M ∞ ( B , k ) (resp. on M ∞ ( B , k )).Moreover, M ∞ ( B , k ) (resp. M ∞ ( B , k )) is equipped with a natural S ∞ -linear (resp. O ∞ -linear) L P ( Q p )-action. By similar (and easier) arguments as in [28, § roposition 5.5. (1) M ∞ ( B , k ) ∼ = M ( N, B ) ⊗ O E O ∞ / b k ∼ = Ord P ( b S ( U p , O E ) ρ ) d B ⊗ O E O ∞ / b k .(2) M ∞ ( B , k ) is a flat S ∞ / a k -module, M ∞ ( B , k ) / a ∼ = M ∞ ( B , k ) and M ∞ ( B , k ) / a ∼ = Ord P ( b S ( U p , O E /̟ s ( a k + a ) E ) ρ ) ∨ B . (5.16) (3) M ∞ i ( B , k ) is a finitely generated O E [[ L P ( Z p )]] -module, and in particular, M ∞ i ( B , k ) ∈ C . For j ≥ k ≥
1, there exists d ( j, k ) > N ) such that the R (cid:3) S ,P − ord ρ, S ( Q N ) , B -actionon M (cid:3) ( N, B , k, L j ) / a k factors through R (cid:3) S ,P − ord ρ, S ( Q N ) , B / m d ( j,k ) N where m N denotes the maximal ideal of R (cid:3) S ,P − ord ρ, S ( Q N ) , B . Actually, when k = 1, it follows easily from the fact (by (5.10) and Proposition 5.4): M (cid:3) ( N, B , , L j ) / a ∼ = (cid:0) Ord P ( S ( U p , O E /̟ E ) ρ ) L j B (cid:1) ∨ ;the general case then follows by considering the a -adic filtration on M (cid:3) ( N, B , k, L j ) / a k . Note that d ( j, k ) → + ∞ when k → + ∞ , and we choose d ( j, k ) such that d ( j ′ , k ′ ) ≥ d ( j, k ) if j ′ ≥ j and k ′ ≥ k . Denote by R ( N, k, d ( j, k )) := R (cid:3) S ,P − ord ρ, S ( Q N ) , B / m d ( j,k ) N ⊗ S ∞ S ∞ / a k . When N is sufficiently large(satisfying c N ⊂ a k ), then we have R ( N, k, d ( j, k )) / a ∼ = (cid:0) R (cid:3) S ,P − ord ρ, S ( Q N ) , B / m d ( j,k ) N ⊗ O ∞ [∆ QN ] O ∞ [∆ Q N ] / a k (cid:1) / a ∼ = R P − ord ρ, S , B / ( m d ( j,k ) , ̟ s ( a k + a ) E ) , (5.17)where m denotes the maximal ideal of R P − ord ρ, S , B . In particular, we see R ( N, k, d ( j, k )) is an S ∞ / a k -module of bounded rank (with N varying). Put R ( ∞ , k, d ( j, k )) := ( Y N ∈I R ( N, k, d ( j, k ))) ⊗ R R F , which naturally acts on M ∞ ( B , k, j ). We have a natural injection S ∞ / a k ֒ → R ( ∞ , k, d ( j, k )) (since S ∞ / a k ֒ → R ( N, k, d ( j, k )) for all N ). By [24, Lem. 2.2.4], we deduce from (5.17) an isomorphism R ( ∞ , k, d ( j, k )) / a ∼ = R P − ord ρ, S , B / ( m d ( j,k ) , ̟ s ( a k + a ) E ) . For N sufficiently large, we have M (cid:3) ( N, B , k, L j ) / a ∼ = (cid:0) Ord P ( S ( U p , O E /̟ s ( a + a k ) E ) ρ ) L j B (cid:1) ∨ , (5.18)and the isomorphism is R ( N, k, d ( j, k ))-equivariant, where R ( N, k, d ( j, k )) acts on the right handside via the isomorphism (5.17). We deduce then the isomorphism (which is obtained via the sameway as in Proposition 5.5 (2)) M ∞ ( B , k, L j ) / a ∼ = (cid:0) Ord P ( b S ( U p , O E /̟ s ( a k + a ) E ) ρ ) L j B (cid:1) ∨ is R ( ∞ , k, d ( j, k ))-equivariant. The map R ∞ ։ R ( N, k, d ( j, k )) induces a natural projection R ∞ ։ R ( ∞ , k, d ( j, k )), and equips M ∞ ( B , k, L j ) with a natural R ∞ -action. Taking inverse limit on j , wesee M ∞ ( B , k ) is equipped with a natural R ∞ -action via R ∞ − ։ lim ←− j R ( ∞ , k, d ( j, k )) , (5.19)38atisfying that the isomorphism in (5.16) is R ∞ -equivariant, where the R ∞ -action on the right handside is induced from the natural action of R P − ord ρ, S , B via R ∞ / a − ։ lim ←− j R ( ∞ , k, d ( j, k )) / a ∼ = lim ←− j R P − ord ρ, S , B / ( m d ( j,k ) , ̟ s ( a k + a ) E ) . (5.20)Note also we have S ∞ / a k ֒ → lim ←− j R ( ∞ , k, d ( j, k )).Let M ∞ i ( B ) := lim ←− k M ∞ i ( B , k ). By Proposition 5.5 (3), M ∞ i ( B ) ∈ C . Using M (cid:3) ( N, B , k, L j ) / a k ∼ = (cid:0) M (cid:3) ( N, B , k + 1 , L j ) / a k +1 (cid:1) / a k for all N , we see M ∞ ( B , k, L j ) ∼ = M ∞ ( B , k + 1 , L j ) / a k . By [28, Lem. 4.4.4 (1)], we have thus M ∞ ( B , k ) ∼ = M ∞ ( B , k + 1) / a k for k >
0. By [28, Lem. 4.4.4 (2)], we see M ∞ ( B ) is a flat S ∞ -module. By Proposition 5.5 (1) (2) and [28, Lem. 4.4.4 (1)], we also have M ∞ ( B ) / a ∼ = M ∞ ( B ),and M ∞ ( B ) / a ∼ = Ord P ( b S ( U p , O E ) ρ ) d B . (5.21)We have a natural injection S ∞ ∼ = lim ←− k S ∞ / a k ֒ −→ lim ←− k lim ←− j R ( ∞ , k, d ( j, k )) , (5.22)and M ∞ ( B ) is equipped with a natural S ∞ -linear action of lim ←− k lim ←− j R ( ∞ , k, d ( j, k )). The mor-phism (5.19) induces R ∞ − ։ lim ←− k lim ←− j R ( ∞ , k, d ( j, k )) . (5.23)We can hence lift (5.22) to an injection S ∞ ֒ → R ∞ . The R ∞ -action on M ∞ ( B ) (induced by (5.23))is then S ∞ -linear. By (5.20) (and taking inverse limit over k ), we have a projection R ∞ / a − ։ R P − ord ρ, S , B , (5.24)and the isomorphism (5.21) is equivariant under the R ∞ -action, where the R ∞ -action on the righthand side of (5.21) is induced from the natural R P − ord ρ, S , B -action via (5.24).We apply Paˇsk¯unas’ theory. Put m ∞ i ( B , k ) := Hom C (cid:0) e P B , M ∞ i ( B , k ) (cid:1) , m ∞ i ( B ) := Hom C (cid:0) e P B , M ∞ i ( B ) (cid:1) . We have m ∞ i ( B ) ∼ = lim ←− k m ∞ i ( B , k ). By the same argument as in [28, Lem. 4.7.4], m ∞ ( B ) is a flat S ∞ -module. By the same argument as in the proof of [28, Prop. 4.7.7 (1)], m ∞ ( B ) is a finitelygenerated R ∞ -module. Denote by b := ( z , · · · , z n | S | ) ⊂ O ∞ . We have by (5.21) m ∞ ( B ) / a ∼ = m ∞ ( B ) / b ∼ = m ( U p , B ) . (5.25)It is clear that y , · · · , y q , z , · · · , z n | S | form a regular sequence of m ∞ ( B ). Hence by [11, Prop.1.2.12], they can extend to a system of parameters of m ∞ ( B ). By [11, Prop. A.4], we havedim R ∞ m ∞ ( B ) = dim R ∞ m ( U p , B )+ q + n | S | . Note that the R ∞ -action on m ( U p , B ) factors through R P − ord ρ, S , B . By Corollary 4.25 (and the fact A is finite over R P − ord ρ, S , B ), we deduce:39 roposition 5.6. We have dim R ∞ m ∞ ( B ) ≥ q + n | S | + X v ∈ S p (2 n − k e v )= 1 + g + n | S | + X v ∈ S p (cid:0) |{ i | n e v,i = 1 }| + 3 |{ i | n e v,i = 2 }| + n ( n − (cid:1) = 1 + g + n | S | + X v ∈ S p (cid:0) k e v X i =1 n e v,i ( n − s e v,i ) (cid:1) . We construct certain patched modules to apply Taylor’s Ihara avoidance.Let Ω be a finite set of finite places v of F + satisfying that • v = e v e v c in F , • p | (Nm( e v ) − p m k (Nm( e v ) −
1) then n ≤ p m .Let U Ω := Q v ∈ Ω Iw( e v ), Y Ω := Q v ∈ Ω Iw ( e v ), where Iw( e v ) (resp. Iw ( e v )) is the standard Iwahorisubgroup (resp. pro- ℓ Iwahori subgroup with e v | ℓ ) of G ( F e v ), i.e. the preimage via i − G, e v of thematrices in GL n ( O F e v ) (recalling that we assume G quasi-split at all finite places of F + ) that areupper triangular (resp. upper triangular unipotent) modulo ̟ e v . We have thus∆ Ω := Y v ∈ Ω ( F × e v ) n ∼ −−→ U Ω /Y Ω . Enlarging E if necessary, we assume E contains p m -th roots of unity if p m k (Nm( e v ) −
1) for v ∈ Ω.Let U p = U Ω × Q v / ∈ S p ∪ Ω U v , and Y p = Y Ω × Q v / ∈ S p ∪ Ω U v , and suppose U Ω ,p := Q v / ∈ S p ∪ Ω U v issufficiently small. For any compact open subgroup U p of G ( F + ⊗ Q Q p ), S ( Y p U p , O E /̟ kE ) ρ isequipped with a natural action of ∆ Ω . For a continuous character χ : ∆ Ω → O × E , we denote by S χ ( U p U p , O E /̟ kE ) the sub O E /̟ kE -module of S ( Y p U p , O E /̟ kE ) on which ∆ Ω acts via χ . Usingthe fact U Ω ,p is sufficiently small, we have S χ ( U p U p , O E /̟ kE ) ∼ = S χ ( U p U p , O E /̟ k +1 E ) ⊗ O E /̟ k +1 E O E /̟ kE . (5.26)Consequently, we see b S χ ( U p , O E ) := lim ←− k lim −→ U p S χ ( U p U p , O E /̟ kE )is also the subspace of b S ( U p , O E ) on which ∆ S acts via χ . By the same argument as in [7, Lem.6.1], one can show that b S χ ( U p , O E ) is a finite projective O E [[ G ( Z p )]]-module. Let S be a finite Using that U Ω ,p is sufficiently small, we can reduce to the following fact: let H be a finite cyclic group, then O E /̟ kE [ H ] ∼ = O E /̟ kE [ x ] / ( x m −
1) (using a generator σ of H ); let χ be a character of H , then O E /̟ kE [ H ] χ ∼ = x m − x − χ ( σ ) O E /̟ kE [ H ], and hence O E /̟ kE [ H ] χ ∼ = O E /̟ k +1 E [ H ] χ ⊗ O E /̟ k +1 E O E /̟ kE . Actually, by loc. cit. , we can show b S ( U p , O E ) is a finite projective O E [∆ Ω ][[ G ( Z p )]]-module, from which wededuce the statement. S p and the places v such that U v is not hyperspecial. We define T χ ( U p U p , ∗ ) ρ inthe same way as T ( U p U p , ∗ ) ρ with S ( U p U p , ∗ ) ρ replaced by S χ ( U p U p , ∗ ) ρ for ∗ ∈ {O E , O E /̟ kE } ,and e T χ ( U p ) ρ := lim ←− U p T χ ( U p , O E ) ρ (where ρ is as in § e T χ ( U p ) P − ord ρ, B which acts faithfully on Ord P ( b S χ ( U p , O E ) ρ ) B . We have natural projections R ρ, S − ։ e T χ ( U p ) ρ − ։ e T χ ( U p ) P − ord ρ, B (5.27)satisfying that the composition factors through R P − ord ρ, S , B (assuming ρ is B -generic). We have thusa natural morphism R p, B → e T χ ( U p ) P − ord ρ, B . By the same arguments, we have a similar local-globalcompatibility result as in Theorem 4.23 for the e T χ ( U p ) P − ord ρ, B -action on m ( U p , B ) χ := Hom C ( e P B , Ord P ( b S χ ( U p , O E ) ρ ) d B ) , (5.28)and we have dim e T χ ( U p ) P − ord ρ, B [1 /p ] ≥ P v ∈ S p (2 n − k e v ) (noting that Ord P ( b S χ ( U p , O E ) ρ ) B is afinite projective O E [[ L P ( Z p )]]-module by [7, Cor. 4.6]).Suppose for v ∈ Ω, ρ e v is trivial. Let χ e v = ( χ e v,i ) : ∆ v := ( F × e v ) n → O × E . We view χ e v,i as a characterof I e v via I e v − ։ I e v /P e v − ։ F × e v χ e v,i −−→ O × E . Denote by D χ e v the deformation problem consisting of liftings ρ (over artinian O E -algebras) of ρ e v such that for all σ ∈ I e v the characteristic polynomial of ρ ( σ ) is given by Q ni =1 ( X − χ e v,i ( σ )).Denote by R ¯ (cid:3) ρ e v ,χ e v the reduced universal deformation ring of D χ e v , which is a quotient of R ¯ (cid:3) ρ e v . Let χ := Q v ∈ Ω χ e v . For the global Galois deformation rings considered in the previous sections, we add χ in the subscript to denote the corresponding universal deformation ring with the local deformationproblem R ¯ (cid:3) ρ e v replaced by D χ e v for v ∈ Ω( ⊂ S ). For example, we have deformation rings R ρ, S ,χ , R (cid:3) ρ, S ,χ , R P − ord ρ, S , B ,χ , R P − ord , (cid:3) S ρ, S , B ,χ etc. By [36, Prop. 8.5], the morphism R ρ, S ։ e T χ ( U p ) ρ (resp. thecomposition in (5.27)) factors through R ρ, S ,χ (resp. through R P − ord ρ, S , B ,χ ).We are particularly interested in the following setting. Let v is a finite place of F + split in F (with v = e v e v c ) such that p ∤ (Nm( e v ) − U p has the following form U p = U Ω × Iw( e v ) × Y v / ∈ Ω ∪ S p ∪{ v } U v (5.29)where U v is hyperspecial for all v / ∈ Ω ∪ S p ∪{ v } (hence S = Ω ∪ S p ∪{ v } ). By the assumption on v ,one can check that U p is sufficiently small. For i = 0 , k ≥ j ≥ N ≥
1, we define V i ( N, B , k ) L j χ by the same way as V i ( N, B , k ) L j with S ( U i ( Q n ) p , O E /̟ kE ) replaced by S χ ( U i ( Q N ) p , O E /̟ kE ). Put M i ( N, B , k, L j ) χ := ( V i ( N, B , k ) L j χ ) ∨ . Let R loc ,P − ord B ,χ := (cid:0) b ⊗ v ∈ Ω R ¯ (cid:3) ρ e v ,χ e v (cid:1) b ⊗ O E R ¯ (cid:3) ρ e v b ⊗ O E (cid:0) b ⊗ v ∈ S p R P e v − ord , ¯ (cid:3) ρ e v , B e v (cid:1) . The morphisms (5.14), (5.15), and S ∞ → R ∞ (lifting (5.14)) induce morphisms S ∞ −→ R (cid:3) S ,P − ord ρ, S ( Q N ) , B ,χ ,R ∞ ,χ := R loc ,P − ord B ,χ [[ x , · · · , x g ]] − ։ R (cid:3) S ,P − ord ρ, S ( Q N ) , B ,χ , S ∞ → R ∞ ,χ respectively. The module M (cid:3) i ( N, B , r, L j ) χ := M i ( N, B , r, L j ) χ ⊗ O E O ∞ is equipped with a natural S ∞ -linear action of R ∞ ,χ . We can run the patching argument as in § { S ∞ , R ∞ , { M (cid:3) i ( N, B , k, L j ) }} replaced by { S ∞ , R ∞ ,χ , { M (cid:3) i ( N, B , k, L j ) χ }} , to obtain R ∞ ,χ -modules m ∞ i ( B ) χ replacing the R ∞ -modules m ∞ i ( B ). By the same arguments, we have that m ∞ ( B ) χ is flat over S ∞ and (cf. (5.28)) m ∞ ( B ) χ / a ∼ = m ( U p , B ) χ . (5.30)As in Proposition 5.6, we have (if m ∞ ( B ) χ = 0)dim R ∞ ,χ m ∞ ( B ) χ ≥ g + n | S | + X v ∈ S p (cid:0) k e v X i n e v,i ( n − s e v,i ) (cid:1) . (5.31)Let χ ′ : ∆ R → O × E be another character such that χ ′ ≡ χ (mod ̟ E ). We have natural isomor-phisms R ρ, S ,χ /̟ E ∼ = R ρ, S ,χ ′ /̟ E , (5.32) R ∞ ,χ /̟ E ∼ = R ∞ ,χ ′ /̟ E . (5.33)We have natural isomorphisms (compatible with (5.32)) M i ( N, B , , L j ) χ ∼ = M i ( N, B , , L j ) χ ′ for all N ∈ Z ≥ , i ∈ { , } , j ∈ Z ≥ , from which we deduce (using (5.26)) natural isomorphisms M (cid:3) i ( N, B , k, L j ) χ /̟ E ∼ = M (cid:3) i ( N, B , k, L j ) χ ′ /̟ E which are compatible with (5.33). These isomor-phisms finally induce an isomorphism m ∞ ( B ) χ /̟ E ∼ −−→ m ∞ ( B ) χ ′ /̟ E (5.34)which is compatible with (5.33). In this section, we prove our main results on automorphy lifting. Recall that S ⊃ S p is a finite setof finite places of F + which split in F , and for all v ∈ S , we fix a place e v of F above v , and thatwe assume Hypothesis 3.4. Theorem 5.7.
Let ρ : Gal F → GL n ( E ) be a continuous representation satisfying the followingconditions:1. ρ c ∼ = ρ ∨ ε − n .2. ρ is unramified outside S .3. ρ absolutely irreducible, ρ (Gal F ( ζ p ) ) ⊆ GL n ( k E ) is adequate and F Ker ad ρ does not contain F ( ζ p ) . . For all v ∈ S p , ρ e v is P e v -ordinary, i.e. ρ e v ∼ = ρ e v, ∗ · · · ∗ ρ e v, · · · ∗ ... ... . . . ... · · · ρ e v,k e v (5.35) with dim E ρ e v,i = n e v,i ( ≤ .5. For all v ∈ S p , ρ e v is de Rham of distinct Hodge-Tate weights. Suppose moreover one of thefollowing two conditions holds(a) for all v ∈ S p , and i = 1 , · · · , k e v , ρ e v,i is absolutely irreducible and the Hodge-Tate weightsof ρ e v,i are strictly bigger than those of ρ e v,i − ;(b) for all v | p , ρ e v is crystalline with the eigenvalues ( φ , · · · , φ n ) of the crystalline Frobeniussatisfying φ i φ − j = 1 , p ± for i = j .6. Let ¯ F e v be P e v -filtration on ρ e v induced by (5.35), B e v,i be the block associated to tr(gr i ¯ F e v ) (cf. § B := ⊗ v ∈ S p i =1 , ··· ,k e v B e v,i ( s e v,i +1 − (which is a block of Mod lfin L P ( Q p ) ( O E ) ). Suppose:(a) ρ is B -generic (Definition 4.21, see also Lemma 4.26);(b) Hom
Gal Q p (Fil i ¯ F e v ρ e v , ρ e v,i ⊗ k E ω ) = 0 for all v | p , i = 1 , · · · , k e v .7. There is an automorphic representation π of G with the associated representation ρ π : Gal F → GL n ( E ) satisfying(a) ρ π ∼ = ρ ;(b) π v is unramified for all v / ∈ S ;(c) π is B -ordinary (cf. Definition 4.17, see also Lemma 4.19).Then ρ is automorphic, i.e. there exists an automorphic representation π ′ of G such that ρ ∼ = ρ π ′ .Proof. Step (1): Let U p = Q v / ∈ S p U v be a sufficiently small compact open subgroup of G ( A ∞ ,pF + )such that U v is hyperspecial for all v / ∈ S (enlarging S if necessary). By Lemma 4.19, the condition7 is equivalent to the following condition7’. there exists an automorphic representation π ′′ such that 7(a), 7(b) hold for π ′′ and that theconditions (a) (b) in Remark 4.20 hold for π ′′ .We replace π by π ′′ , and hence assume π satisfies the condition 7’. By solvable base change, wecan reduce to the case where • S = Ω ∪ S p ∪ { v } is as in § p ∤ ( | F e v | − • for v ∈ Ω, ρ e v is trivial, and ρ e v | ss I e v ∼ = ρ π, e v | ss I e v ∼ = 1 ⊕ n , • ad ρ (Frob e v ) = 1 (hence ρ e v and ρ π, e v are unramified, e.g. by [15, Lem. 2.4.9, Cor. 2.4.21]), • the conditions stay unchanged (with the condition 7 replaced by 7’).43ctually, for v such that ρ e v or ρ π, e v is ramified (we denote by S the set of such places, thus S ⊆ S ),there exists a finite extension M v of F + v such that ρ | Gal Mv is trivial, and ρ e v | ss I Mv ∼ = ρ π, e v | ss I Mv ∼ = 1 ⊕ n . If p ∤ ( | F e v | − M v such that ρ e v | Gal Mv and ρ π, e v | Gal Mv are unramified; otherwise,we enlarge M v such that if p m k ( | F e v | −
1) for m ≥
1, then n ≤ p m . Since F Ker ad ρ does not contain F ( ζ p ), we choose a finite place v ′ / ∈ S of F + , split in F (with v ′ = e v ′ ( e v ′ ) c ) such that e v ′ does notsplit completely in F ( ζ p ) (hence p ∤ ( | F e v | − ρ (Frob e v ′ ) = 1. By [35, Lem. 4.1.2],we let L + /F + be a solvable totally real extension linearly disjoint from F Ker( ρ ) ( ζ p ), and that for afinite place w of L + , and v the place of F + with w | v , we have • if v ∈ S , then L + w ∼ = M v , • if v ∈ S p , then L + w ∼ = F + v ( ∼ = Q p ), • if v = v ′ , then L + w ∼ = F + v ′ .Let v be a place of L + above v ′ . We replace P by Q w | p P e w where P e w = P e v for a p -adic place w of L + with v the place of F + such that w | v (and where e w is a place of L above w that we fix aswe have done for places in F , cf. § F/F + by L/L + . Using [25, Prop. 2.7] [15,Lem. 4.2.2], we reduce to the situation below the condition 7’ (noting that the base change of π to L + still satisfies the condition 7’).Step (2): Let U p be as in (5.29), and let χ = Q v ∈ Ω χ e v : ∆ Ω → O × E with χ e v = ( χ e v,i ) i =1 , ··· ,n satisfyingthat χ e v,i ≡ ̟ E ) and that the χ e v,i ’s are distinct for i = 1 , · · · , n . We have as in § R ∞ , -module m ∞ ( B ) , and an R ∞ ,χ -module m ∞ ( B ) χ . By the assumptions (i.e. π is unramifiedfor places not in S p ∪ Ω, and for v ∈ Ω, we have π Iw( e v ) e v = 0 (since ρ π, e v | ss I e v ∼ = 1 ⊕ n )), π U p = 0. By thecondition 7’ and Lemma 4.19, we have thusOrd P ( b S ( U p , O E ) ρ ) B = Ord P ( b S ( U p , O E ) ρ ) B = 0 . Hence m ( U p , B ) = m ( U p , B ) = 0, and m ∞ ( B ) = 0. Let m x be the maximal ideal of R P − ord ρ, S , B , [1 /p ]associated to ( ρ, { ρ e v,i } ), m ∞ x be the preimage of m x via the projection R ∞ , [1 /p ] ։ R P − ord ρ, S , B , [1 /p ](cf. (5.24)). Let x ∈ Spec R P − ord ρ, S , B , [1 /p ], x ∞ ∈ Spec R ∞ , [1 /p ] be the closed points associated to m x , m ∞ x respectively.By Corollary 2.8 and Condition 6(b) (for places in S p ), [35, Prop. 3.1] (for places in Ω) and [15,Lem. 2.4.9, Cor. 2.4.21] (for v ), R ∞ , , R ∞ ,χ are both equidimensional of relative dimension g + n | S | + X v ∈ S p (cid:0) k e v X i =1 n e v,i ( n − s e v,i ) (cid:1) over O E . By (5.31), we see m ∞ ( B ) is supported on a union of irreducible components of Spec R ∞ , .Giving an irreducible component C of Spec R ∞ , or Spec R ∞ ,χ is the same as giving an irreduciblecomponent C v of each v ∈ S . However, C v is unique if v ∈ S p by Condition 6(b) and Corollary2.8 or v = v (noting R ¯ (cid:3) ρ e v is formally smooth over O E ). So giving C (as above) is the same asgiving an irreducible component C v of each v ∈ Ω, and we denote by C = ⊗ v ∈ Ω C v . By (5.25),Proposition 4.12 (and the fact m ∞ ( B ) = 0), there is an irreducible component C = ⊗ v ∈ Ω C v ofSpec R ∞ , contained in the support of m ∞ ( B ) . Denote by C the modulo ̟ E reduction of C . We44ee C is contained in the support of m ∞ ( B ) /̟ E . Using the isomorphism in (5.34), we deduce that C is contained in the support of m ∞ ( B ) χ /̟ E . Thus Supp R ∞ ,χ m ∞ ( B ) χ contains an irreduciblecomponent C ′ = ⊗ v ∈ Ω C ′ v . Since R ¯ (cid:3) ρ e v ,χ e v is irreducible for v ∈ Ω (cf. [35, Prop. 3.1 (1)]), we see C ′ v = R ¯ (cid:3) ρ e v ,χ e v for v ∈ Ω. Using again the isomorphism in (5.34), Supp R ∞ , /̟ E m ∞ ( B ) /̟ E containsthe modulo ̟ E reduction C ′ of C ′ . However, by [35, Prop. 3.1 (2)], we deduce that the modulo ̟ E reduction of any irreducible component of Spec R ∞ , is contained in C ′ . Together with [35,Prop. 3.1 (3)], we deduce any irreducible component Spec R ∞ , is contained in Supp R ∞ , m ∞ ( B ) .In particular x ∞ ∈ Supp R ∞ , [1 /p ] m ∞ ( B ) [1 /p ]. And we have thus m ∞ ( B ) [1 /p ] / m ∞ x = 0. Using(5.30), we deduce then m ( U p , B )[1 /p ] / m x = 0 (noting m ( U p , B ) = m ( U p , B )). The R P − ord ρ, S , B , -actionon m ( U p , B ) factors through e T ( U p ) P − ord ρ, B = e T ( U p ) P − ord ρ, B , we deduce there exists a maximal idea m Tx of e T ( U p ) P − ord ρ, B [1 /p ] (which is actually equal to the image of m x ) such that m ( U p , B )[1 /p ] / m Tx = 0.By Theorem 4.23, we obtain a closed point z = ( x T , { z e v,i } ) ∈ Spec A [1 /p ] where x T is the pointassociated to m Tx , and z e v,i = tr ρ e v,i (noting that the preimage of m x of the first morphism in (4.33)is the prime ideal corresponding to { tr ρ e v,i } ).Step (3): Suppose the condition 5 (a), then by Proposition 4.15, x T is classical, and the theoremfollows.Suppose now the condition 5 (b). By Proposition 4.12, we have a non-zero map b ⊗ v ∈ S p i =1 , ··· ,k e v ( b π z e v,i ⊗ E ε s e v,i +1 − ◦ det) −→ Ord P (cid:0) b S ( U p , E ) ρ (cid:1) B [ m Tx ] . Since ρ e v,i is crystalline, by the p -adic Langlands correspondence for GL ( Q p ), the irreducible con-stituents of b π an z e v,i are all subquotients of locally analytic principal series (induced from locally alge-braic characters of T ( Q p )). We deduce then there exist locally algebraic characters χ e v of T ( Q p ) for v ∈ S p such that ⊗ v ∈ S p χ e v ֒ −→ J B ∩ L P (cid:0) Ord P (cid:0) b S ( U p , E ) ρ (cid:1) B [ m Tx ] an (cid:1) , (5.36)where J B ∩ L P ( − ) denotes the Jacquet-Emerton functor ([18]). From the locally analytic representa-tion J B ∩ L P (Ord P (cid:0) b S ( U p , E ) ρ (cid:1) an B ), using Emerton’s machinery [19], we can construct an eigenvariety E P − ord B as in [7, § • any point of E P − ord B can be parameterized as ( m z , χ ) where χ is a character of T ( F + ⊗ Q Q p )( ∼ = Q v ∈ S p T ( Q p )) and m z is a maximal ideal of e T ( U p ) P − ord ρ, B [1 /p ]; • ( m z , χ ) ∈ E P − ord B if and only if J B ∩ L P (cid:0) Ord P (cid:0) b S ( U p , E ) ρ (cid:1) an B (cid:1) [ m z , T ( F + ⊗ Q Q p ) = χ ] = 0 . In particular, by (5.36), we see x := ( m Tx , ⊗ v ∈ S p χ e v ) ∈ E P − ord B . By the same argument as for[7, (7.28)], one can show there exists a natural injection ( E P − ord B ) red ֒ → E where E denotes theeigenvariety associated to G with the tame level U p (constructed from J B ( b S ( U p , E ) an ρ )). Hence weget a point x = ( m Tx , ⊗ v ∈ S p χ e v ) ∈ E . Since ρ x, e v is crystalline and generic for all v ∈ S p , by [8, Thm.5.1.3, Rem. 5.1.4], m Tx is classical. This concludes the proof.45 .6. Locally analytic socle We use the (patched) GL ( Q p )-ordinary families to show some results towards Breuil’s locallyanalytic socle conjecture [6] for certain non-trianguline case. We begin with some preliminaries onrepresentations. Lemma 5.8.
Let U be a unitary admissible Banach representation of L P ( Q p ) over E , U an be thesubrepresentation of locally analytic vectors. Then the following diagram commutes U an −−−−→ (Ind G p P ( Q p ) U an ) an y y U −−−−→ (Ind G p P ( Q p ) U ) C , (5.37) where (Ind − ) an (resp. (Ind − ) C ) denotes the locally analytic (resp. the continuous) parabolic in-duction, where the top horizontal map sends u to f u ∈ C la ( N , U an ) ֒ → (Ind G p P ( Q p ) U an ) an via [20,(2.3.7)] (see § N ) with f u the constant function of value u , and where the bottom horizontalmap is given by the composition (see [20, Cor. 4.3.5] for the first isomorphism, and see [20, (3.4.7)]for the second map, which is the canonical lifting map of loc. cit. with respect to N ) U ∼ −−→ Ord P (cid:0) (Ind G p P ( Q p ) U ) C (cid:1) −→ (Ind G p P ( Q p ) U ) C . Proof.
The lemma follows by the same argument as in [7, Lem. 4.20].
Lemma 5.9.
Let V be a unitary admissible Banach representation of G p := G ( F + ⊗ Q Q p ) over E ,and U be a unitary admissible Banach representation of L P ( Q p ) over E . Suppose that we have an L P ( Q p ) -equivariant non-zero map U → Ord P ( V ) , such that the following composition is non-zero: U lalg ֒ −→ U −→ Ord P ( V ) . (5.38) Then the composition (where the second map is induced by the second map of (5.38) by [21, Thm.4.4.6]) (Ind G p P ( Q p ) U lalg ) an ֒ −→ (Ind G p P ( Q p ) U ) C −→ V (5.39) is non-zero. Moreover, the following diagram commutes U lalg −−−−→ (Ind G p P ( Q p ) U lalg ) an( . ) y ( . ) y Ord P ( V ) −−−−→ V (5.40) where the top horizontal map is given as in the horizontal map of (5.37) with U an replaced by U lalg ,and the bottom horizontal map is the canonical lifting with respect to N .Proof. It is sufficient to show (5.40) is commutative. However, by [21, Thm. 4.4.6], the followingdiagram commutes U −−−−→ (Ind G p P ( Q p ) U ) C ( . ) y ( . ) y Ord P ( V ) −−−−→ V . U an replaced by any closed subrepresentation of U an , andin particular holds for U lalg ).For a weight λ = ( λ , · · · , λ n ) of GL n , let L ( λ ) be the unique simple quotient of U( g l n ) ⊗ U( b ) λ with b the Lie algebra of the Borel subgroup B of lower triangular matrices. If λ is integral and isdominant for a parabolic P ⊇ B , then L ( λ ) lies in the BGG category O p alg (cf. [27, § p isthe Lie algebra of P ). Theorem 5.10.
Suppose that all the assumptions, except the assumption 5, of Theorem 5.7 holdand suppose that we are in the situation of Step (1) of the proof of Theorem 5.7. Suppose thefollowings (as a replacement of the assumption 5 of Theorem 5.7) hold • for all v | p , ρ e v is Hodge-Tate of distinct Hodge-Tate weights h e v, > · · · > h e v,n ; • for all v | p , i = 1 , · · · , k e v , ρ e v,i is de Rham and absolutely irreducible.Then there exists a non-zero morphism of locally analytic G p -representations: b ⊗ v | p F GL n P e v ( L ( − s e v · λ e v ) , π ∞ e v ) −→ b S ( U p , E )[ m ρ ] an , (5.41) where • “ − [ m ρ ] ” denotes the subspace annihilated by the maximal ideal m ρ ⊆ R ρ, S [1 /p ] associated to ρ , • F GL n P e v ( − , − ) is the Orlik-Strauch functor ([27]), • λ e v := ( λ e v, , · · · , λ e v,n ) with λ e v,i := h e v,i + i − (so λ e v is dominant for B ), • π ∞ e v = ⊗ k e v i =1 π ∞ e v,i ( s e v,i +1 − with “ − ( s e v,i +1 − ” the twist unr( p − ( s e v,i +1 − ) ◦ det and with π ∞ e v,i the smooth GL n e v,i ( Q p ) representation corresponding to WD( ρ e v,i ) (normalized in the waythat b π ( ρ e v,i ) lalg is isomorphic to the tensor product of π ∞ e v,i with an algebraic representation of GL n e v,i ( Q p ) ), • s e v ∈ S n satisfies that we have an equality of ordered sets (cid:0) h e v,s − e v (1) , · · · , h e v,s − e v ( n ) (cid:1) = (cid:0) h ρ e v, , , h ρ e v, ,n e v, , · · · , h ρ e v,k e v , , h ρ e v,k e v ,n e v,k e v (cid:1) , { h ρ e v,i , , h ρ e v,i ,n e v,i } being the set of the Hodge-Tate weights of ρ e v,i with h ρ e v,i , ≥ h ρ e v,i ,n e v,i (so − s e v · λ e v is dominant for P e v ).Proof. By Step (2) of the proof of Theorem 5.7 (and we use the notation there), we have z =( x T , { z e v,i } ) ∈ Spec A [1 /p ]. By Proposition 4.12, there exists a non-zero L P ( Q p )-equivariant mor-phism (without loss of generality, we assume the residue field at x is equal to E ) b ⊗ v ∈ S p i =1 , ··· ,k e v ( b π z e v,i ⊗ E ε s e v,i +1 − ◦ det) −→ Ord P (cid:0) b S ( U p , E ) ρ (cid:1) B [ m Tx ] . (5.42)Note that Ord P (cid:0) b S ( U p , E ) ρ (cid:1) B [ m Tx ] = Ord P (cid:0) b S ( U p , E ) ρ (cid:1) B [ m ρ ]. Since ρ e v,i is absolutely irreducible,we have b π z e v,i ∼ = b π ( ρ e v,i ). Also by the same argument as in the proof of Proposition 4.15, (5.42)induces a non-zero L P ( Q p )-equivariant morphism ⊗ v ∈ S p i =1 , ··· ,k e v ( b π ( ρ e v,i ) lalg ⊗ E ε s e v,i +1 − ◦ det) −→ Ord P (cid:0) b S ( U p , E ) ρ (cid:1) B [ m ρ ] .
47y Lemma 5.9, we deduce hence a non-zero G p -equivariant morphism b ⊗ v ∈ S p (cid:0) Ind GL n ( Q p ) P e v ⊗ k e v i =1 ( b π ( ρ e v,i ) lalg ⊗ E ε s e v,i +1 − ◦ det) (cid:1) an ∼ = (cid:0) Ind G p P ⊗ v ∈ S p i =1 , ··· ,k e v ( b π ( ρ e v,i ) lalg ⊗ E ε s e v,i +1 − ◦ det) (cid:1) an −→ b S ( U p , E ) ρ [ m ρ ] . For v ∈ S p , we have ⊗ k e v i =1 ( b π ( ρ e v,i ) lalg ⊗ E ε s e v,i +1 − ◦ det) ∼ = π ∞ e v ⊗ E L ( s e v · λ e v ) P e v where L ( s e v · λ e v ) P e v denotes the algebraic representation of the Levi subgroup of P e v (containing thediagonal subgroup) of highest weight s e v · λ e v (with respect to the Borel subgroup of upper triangularmatrices). By [6, Cor. 2.5] and [10, Lem. 2.10], we have b ⊗ v | p F GL n P e v ( L ( − s e v · λ e v ) , π ∞ e v ) ∼ = F G p P ( ⊗ v ∈ S p L ( − s e v · λ e v ) , ⊗ v ∈ S p π ∞ e v ) ∼ = soc G p (cid:0) Ind G p P ⊗ v ∈ S p ( π ∞ e v ⊗ E L ( s e v · λ e v ) P e v ) (cid:1) an −→ b S ( U p , E ) ρ [ m ρ ] an . (5.43)We show the composition is non-zero. By Lemma 5.9, the composition ⊗ v ∈ S p ( π ∞ e v ⊗ E L ( s e v · λ e v ) P e v ) −→ (cid:0) Ind G p P ⊗ v ∈ S p ( π ∞ e v ⊗ E L ( s e v · λ e v ) P e v ) (cid:1) an −→ b S ( U p , E ) ρ [ m ρ ] an is non-zero. By [5, Prop. 3.4 (i)], the first map actually factors through F G p P ( ⊗ v ∈ S p L ( − s e v · λ e v ) , ⊗ v ∈ S p π ∞ e v ). We deduce thus (5.43) is non-zero, and this concludes the proof. Remark 5.11.
Keep the assumptions and notation in Theorem 5.10, and assume ρ is automorphicsuch that b S ( U p , E )[ m ρ ] lalg = 0 . Let Π ∞ e v be the unique generic subquotient of (Ind GL n ( Q p ) P e v ( Q p ) π ∞ e v ) ∞ . Bythe local-global compatibility in classical local Langlands correspondence, we have ⊗ v ∈ S p (cid:0) Π ∞ e v ⊗ E L ( λ e v ) (cid:1) ֒ −→ b S ( U p , E )[ m ρ ] an , (5.44) where L ( λ e v ) denotes the algebraic representation of GL n ( Q p ) of highest weight λ e v (with respect to B ). Let Q e v ⊇ P e v be the maximal parabolic subgroup such that s e v · λ e v is dominant for Q e v , and L Q e v be the Levi subgroup of Q e v containing the torus. Assume (Ind L Q e v ( Q p ) P e v ( Q p ) ∩ L Q e v ( Q p ) π ∞ e v ) ∞ is irreduciblefor v ∈ S p . By [10, Lem. 2.10] [27, Thm. (iv)] (and the fact ⊗ v ∈ S p (Ind L Q e v ( Q p ) P e v ( Q p ) ∩ L Q e v ( Q p ) π ∞ e v ) ∞ is irreducible as smooth Q v ∈ S p L Q e v ( Q p ) -representation), we see b ⊗ v ∈ S p F GL n P e v ( L ( − s e v · λ e v ) , π ∞ e v ) istopologically irreducible. When there exists v ∈ S p such that s e v = 1 , then F GL n P e v ( L ( − s e v · λ e v ) , π ∞ e v ) isnot locally algebraic, and the morphism (5.41) provides an injection (other than (5.44)): b ⊗ v ∈ S p F GL n P e v ( L ( − s e v · λ e v ) , π ∞ e v ) ֒ −→ soc G p b S ( U p , E )[ m ρ ] an . (5.45) This extra constituent appearing in the socle of b S ( U p , E )[ m ρ ] an is predicted by Breuil’s locally ana-lytic socle conjecture ([5, Conj. 5.3]), which was proved when ρ e v is crystalline and generic in [8](see also [5], [17] etc. for partial results on the conjecture). However, all the previous results usedeigenvarieties in an essential way, and hence were limited to the case where ρ e v is trianguline. Bycontrast, (5.45) also applies to the case where π ∞ e v,i is cuspidal for some i (with s e v = 1 , n e v,i = 2 ),which then gives a non-trivial example (probably the first, to the author’s knowledge) towards theconjecture in this case. eferences [1] Thomas Barnet-Lamb, Toby Gee, David Geraghty, and Richard Taylor. Potential automorphyand change of weight. Annals of Mathematics , 179(2):501–609, 2014.[2] Tom Barnet-Lamb, David Geraghty, Michael Harris, and Richard Taylor. A family of calabi-yau varieties and potential automorphy ii.
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