On the automorphy of 2-dimensional potentially semi-stable deformation rings of G Q p
aa r X i v : . [ m a t h . N T ] F e b ON THE AUTOMORPHY OF 2-DIMENSIONAL POTENTIALLY SEMI-STABLEDEFORMATION RINGS OF G Q p SHEN-NING TUNG
Abstract.
Using p -adic local Langlands correspondence for GL ( Q p ), we prove that the supportof patched modules M ∞ ( σ )[1 /p ] constructed in [CEG +
16] meet every irreducible component of thepotentially semi-stable deformation ring R ✷ ¯ r ( σ )[1 /p ]. This gives a new proof of the Breuil-M´ezardconjecture for 2-dimensional representations of the absolute Galois group of Q p when p >
2, which isnew in the case p = 3 and ¯ r a twist of an extension of the trivial character by the mod p cyclotomiccharacter. As a consequence, a local restriction in the proof of Fontaine-Mazur conjecture in [Kis09]is removed. Introduction
Let F/ Q p be a finite extension and G F be the absolutely Galois group of F . Fix a finite extension E/ Q p with ring of integers O and residue field k , a continuous representation ¯ r : G F → GL n ( k ) anda continuous character ψ : G F → O × . Denote ε the cyclotomic character and R ✷ ¯ r (resp. R ✷ ,ψ ¯ r ) theuniversal framed deformation ring (resp. universal framed deformation ring with a fixed determinant ψε ) of ¯ r . Under the assumption that p does not divide 2 n , [CEG +
16] constructs an R ∞ [ G ]-module M ∞ by applying Taylor-Wiles-Kisin patching method to algebraic automorphic forms on some definiteunitary group, where R ∞ is a complete local noetherian R ✷ ¯ r -algebra with residue field k and G =GL n ( F ). If y ∈ m-Spec R ∞ [1 /p ], thenΠ y := Hom cont O (cid:0) M ∞ ⊗ R ∞ ,y κ ( y ) , E (cid:1) with κ ( y ) the residue field at y , is an admissible unitary E -Banach space representation of G . Thecomposition R ✷ ¯ r → R ∞ y −→ κ ( y ) defines an κ ( y )-valued point x ∈ m-Spec R ✷ ¯ r [1 /p ] and thus a continuousGalois representation r x : G F → GL n (cid:0) κ ( y ) (cid:1) . It is expected that the Banach space representation Π y depends only on x and that it should be related to r x by the hypothetical p -adic local Langlandscorrespondence; see [CEG +
16] and [CEG +
18] for a detailed discussion.In this paper, we specialize this construction to the case F = Q p and n = 2, in which the p -adiclocal Langlands correspondence is known. The goal of this paper is to prove that every irreduciblecomponent of a potentially semi-stable deformation ring is automorphic. This amounts to showingthat if r x is potentially semi-stable with distinct Hodge-Tate weights, then (a subspace of) locallyalgebraic vectors in Π y can be related to WD( r x ) via the classical local Langlands correspondence,where WD( r x ) is the Weil-Deligne representation associated to r x by Fontaine.One of the ingredients is a result of Emerton and Paˇsk¯unas in [EP18], which shows that the actionof R ∞ on M ∞ is faithful. The statement can also be deduced from the work of Hellmann and Schraen[HS16]. Note that this does not imply that Π y = 0 since M ∞ is not finitely generated over R ∞ . Weovercome this problem by applying Colmez’s Montreal functor ˇ V to M ψ ∞ (a fixed determinant quotientof M ∞ ) and showing that ˇ V ( M ψ ∞ ) is a finitely generated R ψ ∞ -module, where R ψ ∞ is a complete localnoetherian R ✷ ,ψ ¯ r -algebra. This finiteness result is a key idea in this paper. While our paper is written,we were notified that Lue Pan also obtains a similar result independently in [Pan19]. However, wewould like to emphasize that our results are most interesting in the cases not covered by [Paˇs13], whoseresults Pan uses in an essential way.Our strategy runs as follows: we first show that ˇ V ( M ψ ∞ ) is finitely generated over R ψ ∞ J G Q p K andthe action of R ψ ∞ J G Q p K on ˇ V ( M ψ ∞ ) factors through the 2-sided ideal J generated by Cayley-Hamiltonrelations, which imply that ˇ V ( M ψ ∞ ) is a finitely generated module over R ψ ∞ . Using the result ofEmerton and Paˇsk¯unas mentioned above, we show that the action of R ψ ∞ on ˇ V ( M ψ ∞ ) is faithful. Since Date : February 21, 2020. V ( M ψ ∞ ) is a finitely generated R ψ ∞ -module, this implies that the specialization of ˇ V ( M ψ ∞ ) at any y ∈ m-Spec R ψ ∞ [1 /p ] is non-zero, which in turn implies that Π y is nonzero. Combining these, weprove that every irreducible component of a potentially semi-stable deformation ring is automorphicif it contains a point whose associated Galois representation r x is irreducible. So we only have tohandle components in the reducible (thus ordinary) locus, which is known to be automorphic by[BLGG13] except for the case which admits reducible semi-stable non-crystalline components. Thisgives a new proof of the Breuil-M´ezard conjecture outside this particular case by the formalism of[Kis09, GK14, EG14, Paˇs15]. Fortunately, the conjecture for this case can be checked directly bycomputing Hilbert-Samuel multiplicities, thus we prove the conjecture. This is new in the case p = 3and ¯ r a twist of the trivial character by the mod p cyclotomic character.As a consequence, the Fontaine-Mazur conjecture without the local restriction follows immediatelyfrom the original proof of [Kis09]. Theorem 0.1.
Let p > , S a finite set of primes containing { p, ∞} , G Q ,S the Galois group of themaximal extension of Q unramified outside S , and G Q p ⊂ G Q ,S a decomposition group at p . Let ρ : G Q ,S → GL ( O ) be a continuous irreducible odd representation. Suppose that(1) ¯ ρ | Q ( ζ p ) is absolutely irreducible.(2) ρ | G Q p is potentially semi-stable with distinct Hodge-Tate weights.Then (up to twist) ρ comes from a cuspidal eigenforms. We expect that a similar strategy will work in the case when p = 2 and would like to pursue thisproblem in the near future.The paper is organized as follows. In §
1, we state the Breuil-M´ezard conjecture for 2-dimensionalGalois representations of G Q p . In §
2, we recall the p -adic representation theory for GL ( Q p ) and theColmez’s Montreal functor. In §
3, we define patched modules in our setting and show some of theirproperties after applying Colmez’s functor. In §
4, we prove the automorphy of potentially semi-stabledeformation rings and explain how such a result can be applied to the Breuil-M´ezard and Fontaine-Mazur conjectures.
Acknowledgments
Part of this paper is the author’s Ph.D. thesis at the University of Duisburg-Essen. He would like tothank his advisor Vytautas Paˇsk¯unas for suggesting this problem and for his constant encouragementto work on this topic. He also thanks Shu Sasaki for his help on modularity lifting theorem, and JamesNewton, Florian Herzig for pointing out some errors in the earlier draft of this paper.
Notation • p is an odd prime number. • E/ Q p is a sufficiently large finite extension with ring of integers O , uniformizer ̟ and residuefield k . • For a number field F , the completion at a place v is written as F v . • For a local or global field L , G L = Gal( ¯ L/L ). The inertia subgroup for the local field is writtenas I L . • ε : G Q p → Z × p is the p -adic cyclotomic character, whose Hodge-Tate weight is defined to be 1. • ω : G Q p → F × p is the mod p cyclotomic character, and : G Q p → F × p is the trivial character.We also denote other trivial representations by if no confusion arises. • Normalize the local class field theory Art Q p : Q × p → G ab Q p so that p maps to a geometricFrobenius. Then a character of G Q p will also be regarded as a character of Q × p . • For a ring R , m-Spec R denotes the set of maximal ideals of R . • For R a Noetherian ring and M a finite R -module of dimension at most d , let ℓ R p ( M p ) denotethe length of the R p -module M p , and let Z d ( M ) := P p ℓ R p ( M p ) p be a cycle, where the sumis taken over all p ∈ Spec R such that dim R/ p = d . If the support of M is equidimensional ofsome finite dimension d , then we write simply Z ( M ) := Z d ( M ). • For R a Noetherian local ring with maximal ideal m and M a finite R -module, and for an m -primary ideal q of R, let e q ( R, M ) denote the Hilbert-Samuel multiplicity of M with respectto q . We abbreviate e m ( R, R ) = e ( R ). If A is a topological O -module, we write A ∨ := Hom cont O ( A, E/ O ) for the Pontryagin dual of A . If A is a pseudocompact O -torsion free O -module, we write A d := Hom cont O ( A, O ) for itsSchikhof dual. • We write G = GL ( Q p ) and K = GL ( Z p ), and let Z = Z ( G ) ∼ = Q × p denote the center of G .1. The Breuil-M´ezard conjecture
Consider the following data: • a pair of integers λ = ( a, b ) with a > b , • a representation τ : I Q p → GL ( E ) with an open kernel, • a continuous character ψ : G Q p → O × such that ψ | I Q p = ε a + b − det τ .We call such a triple ( λ, τ, ψ ) a p -adic Hodge type. We say a 2-dimensional continuous representation r : G Q p → GL ( E ) is of type ( λ, τ, ψ ) if r is potentially semi-stable (= de Rham) such that its Hodge-Tate weights are ( a, b ), WD( r | I Q p ) ∼ = τ and det r = ψε . Here WD( r ) is the Weil-Deligne representationassociated to r by Fontaine.By a result of Henniart in the appendix of [BM02], there is a unique finite-dimensional smoothirreducible Q p -representation σ ( τ ) (cid:0) resp. σ cr ( τ ) (cid:1) of K , such that for any infinite-dimensional smoothabsolutely irreducible representation π of G and the associated Weil-Deligne representation LL( π )attached to π via the classical local Langlands correspondence, we have Hom K ( σ ( τ ) , π ) = 0 (resp.Hom K ( σ cr ( τ ) , π ) = 0) if and only if LL( π ) | I Q p ∼ = τ (resp. LL( π ) | I Q p ∼ = τ and the monodromy operator N is trivial). We remark that σ ( τ ) and σ cr ( τ ) differ only when τ ∼ = η ⊕ η is scalar, in which case σ ( τ ) ∼ = ˜ st ⊗ η ◦ det , σ cr ( τ ) ∼ = η ◦ detwhere ˜ st is the inflation to GL ( Z p ) of the Steinberg representation of GL ( F p ).Enlarging E if needed, we may assume σ ( τ ) is defined over E . We write σ ( λ ) = Sym a − b − E ⊗ det b and σ ( λ, τ ) = σ ( λ ) ⊗ σ ( τ ). Since σ ( λ, τ ) is a finite-dimensional E -vector space, K = GL ( Z p ) iscompact, and the action of K on σ ( λ, τ ) is continuous, there is a K -stable O -lattice σ ◦ ( λ, τ ) in σ ( λ, τ ).Then σ ◦ ( λ, τ ) / ( ̟ ) is a smooth finite length k -representation of K , we will denote by σ ( λ, τ ) its semi-simplification. One may show that σ ( λ, τ ) does not depends on the choice of a lattice. For each smoothirreducible k -representation ¯ σ of K , we let m ¯ σ ( λ, τ ) be the multiplicity with which ¯ σ occurs in σ ( λ, τ ).Similarly, we let σ cr ( λ, τ ) = σ ( λ ) ⊗ σ cr ( τ ), and we let m cr ¯ σ ( λ, τ ) be the multiplicity with which ¯ σ occursin σ cr ( λ, τ ).Let ¯ r : G Q p → GL ( k ) be a continuous representation. We will write R ✷ ¯ r (resp. R ✷ ,ψ ¯ r ) for theuniversal framed deformation ring of ¯ r (resp. universal framed deformation ring of ¯ r with determinant ψε ) and r ✷ : G Q p → GL ( R ✷ ¯ r ) for the universal framed deformation. If x ∈ m-Spec R ✷ ¯ r [1 /p ], then theresidue field κ ( x ) is a finite extension of E . Let O κ ( x ) be the ring of integers in κ ( x ). By specializingthe universal framed deformation at x , we obtain a continuous representation r x : G Q p → GL ( O κ ( x ) ),which reduces to ¯ r modulo the maximal ideal of O κ ( x ) .Since κ ( x ) is a finite extension of E , r x lies in Fontaine’s p -adic Hodge theory [Fon94]. Kisin hasshown the existence of a reduced, 4-dimensional (if non-trivial), p -torsion free quotient R ✷ ,ψ ¯ r ( λ, τ ) (cid:0) resp. R ✷ ,ψ,cr ¯ r ( λ, τ ) (cid:1) of R ✷ ¯ r such that for all x ∈ m-Spec R ✷ ¯ r [1 /p ], x lies in m-Spec R ✷ ,ψ ¯ r ( λ, τ )[1 /p ](resp. R ✷ ,ψ,cr ¯ r ( λ, τ )[1 /p ]) if and only if r ✷ x is potentially semi-stable (resp. potentially crystalline) oftype ( λ, τ, ψ ).In [BM02], Breuil and M´ezard made an conjecture relating the Hilbert-Samuel multiplicity of R ✷ ,ψ ¯ r ( λ, τ ) /̟ (resp. R ✷ ,ψ,cr ¯ r ( λ, τ ) /̟ ) with the number m ¯ σ ( λ, τ ) (cid:0) resp. m cr ¯ σ ( λ, τ ) (cid:1) defined above. Conjecture 1.1 (Breuil-M´ezard) . For each smooth irreducible k -representation ¯ σ of K , there exists aninteger µ ¯ σ (¯ r ) , independent of λ and τ , such that for all p -adic Hodge types ( λ, τ, ψ ) , we have equalities: e ( R ✷ ,ψ ¯ r ( λ, τ ) /̟ ) = X ¯ σ m ¯ σ ( λ, τ ) µ ¯ σ (¯ r ) e ( R ✷ ,ψ,cr ¯ r ( λ, τ ) /̟ ) = X ¯ σ m cr ¯ σ ( λ, τ ) µ ¯ σ (¯ r ) where the sum is taken over the set of isomorphism classes of smooth irreducible k -representations of K . onjecture 1.1 was proved by [Kis09, Paˇs15, HT15, Paˇs16] in all cases if p ≥ r isnot a twist of an extension of by ω if p = 2 ,
3. We prove the conjecture for p > + p = 3 and ¯ r = ( ω ∗ ) ⊗ χ . Theorem 1.2.
Assume p > . For each smooth irreducible k -representation ¯ σ of K , there existsa 4-dimensional cycle C ¯ σ (¯ r ) of R ✷ ,ψ ¯ r , independent of λ and τ , such that for all p -adic Hodge types ( λ, τ, ψ ) , we have equalities of 4-dimensional cycles: Z ( R ✷ ,ψ ¯ r ( λ, τ ) /̟ ) = X ¯ σ m ¯ σ ( λ, τ ) C ¯ σ (¯ r ) Z ( R ✷ ,ψ,cr ¯ r ( λ, τ ) /̟ ) = X ¯ σ m cr ¯ σ ( λ, τ ) C ¯ σ (¯ r ) where the sum is taken over the set of isomorphism classes of smooth irreducible k -representations of K .Remark . If τ is a 2-dimensional trivial representation of I Q p , then σ cr ( λ, τ ) = Sym a − b − E ⊗ det b .If moreover 1 ≤ a − b ≤ p , then σ cr ( λ, τ ) ∼ = Sym a − b − k ⊗ det b is an irreducible representation of K over k . Thus in this case m ¯ σ = 0 unless ¯ σ ∼ = Sym a − b − k ⊗ det b , in which case the multiplicityis equal to 1. This observation together with Theorem 1.2 implies that if ¯ σ = Sym r k ⊗ det s with0 ≤ r ≤ p − ≤ s ≤ p − λ = ( r + s + 1 , s ) and τ = ⊕ , we have the followingequalities µ ¯ σ (¯ r ) = e ( R ✷ ,ψ,cr ¯ r ( λ, τ ) /̟ ); C ¯ σ (¯ r ) = Z ( R ✷ ,ψ,cr ¯ r ( λ, τ ) /̟ ) . Preliminaries on the representation theory of GL ( Q p )We let G = GL ( Q p ), K = GL ( Z p ) and Z ≃ Q × p be the center of G . Let B be the subgroup ofupper triangular matrices in G . If χ and χ are characters of Q × p , then we write χ ⊗ χ for thecharacter of B which maps ( a b d ) to χ ( a ) χ ( d ).Let Mod sm G ( O ) be the category of smooth G -representation on O -torsion modules. An object π ∈ Mod sm G ( O ) is locally finite if for all v ∈ π , the O [ G ]-submodule generated by v is of finite length. LetMod l . fin G ( O ) to be full subcategory of Mod sm G ( O ) consisting of all locally finite representations and defineMod sm G ( k ) and Mod l . fin G ( k ) in the same way with O replaced by k . Moreover for a continuous character ψ : Z → O × , adding the subscript ψ in any of the above categories indicates the corresponding fullsubcategory of G -representations with central character ψ .An object π of Mod sm G ( O ) is called admissible if π H [ ̟ i ] is a finitely generated O -module for everyopen compact subgroup H of G and every i ≥ π is called locally admissible if for every v ∈ π ,the smallest O [ G ]-submodule of π containing v is admissible. Let Mod l . adm G,ψ ( O ) be a full subcategoryof Mod sm G,ψ ( O ) consisting of locally admissible representations. In [Eme10a], Emerton shows thatMod l . adm G,ψ ( O ) is an abelian category and Mod l . adm G,ψ ( O ) ∼ = Mod l . fin G,ψ ( O ).Every irreducible object π of Mod sm G ( O ) is killed by ̟ , and hence is an object of Mod sm G ( k ). Barthel-Livn´e [BL94, Theorem 33] and Breuil [Bre03a, Theorem 1.6] have classified the absolutely irreduciblesmooth representations π admitting a central character. They fall into four disjoint classes: • characters η ◦ det; • principal series Ind GB ( χ ⊗ χ ), with χ = χ ; • special series Sp ⊗ η ◦ det, where Sp is the Steinberg representation defined by the exact sequence0 → → Ind GB → Sp → • supersingular c-Ind GK σ/ ( T, S − s ), where σ is a smooth irreducible k -representation of K , T, S ∈ End G (c-Ind GK σ ) are certain Hecke operator defined in [BL94, Bre03a], and s ∈ k × .Let Irr G,ψ ( k ) be the set of equivalent classes of smooth irreducible k -representations of G with centralcharacter ψ . We say π, π ′ ∈ Irr
G,ψ ( k ) is in the same block if there exist π , ..., π n ∈ Irr
G,ψ ( k ), suchthat π ∼ = π , π ′ ∼ = π n and either Ext G,ψ ( π i , π i +1 ) or Ext G,ψ ( π i +1 , π i ) is nonzero for 1 ≤ i ≤ n − he classification of blocks can be found in [Paˇs15, Corollary 1.2]. By Proposition 5.34 of [Paˇs13], thecategory Mod l . fin G,ψ ( O ) decomposes into a direct product of subcategoriesMod l . fin G,ψ ( O ) ∼ = Y B Mod l . fin G,ψ ( O ) B where the product is taken over all the blocks B and the objects of Mod l . fin G,ψ ( O ) B are representationswith all the irreducible subquotients lying in B .Let Mod pro G ( O ) be the category of compact O J K K -modules with an action of O [ G ] such that thetwo actions coincide when restricted to O [ K ]. This category is anti-equivalent to Mod sm G ( O ) under thePontryagin dual π π ∨ := Hom O ( π, E/ O ), where the former is equipped with the discrete topologyand the latter is equipped with the compact-open topology. Finally let C ( O ) and C ( k ) be respectivelythe full subcategory of Mod pro G ( O ) anti-equivalent to Mod l . fin G,ψ ( O ) and Mod l . fin G,ψ ( k ).Let Ban adm G,ψ ( E ) be the category of admissible unitary E -Banach space representations on which Z acts by ψ (see [ST02, Section 3]). If Θ is an open bounded G -invariant O -lattice in Π ∈ Ban adm
G,ψ ( E ),then the Schikhof dual Θ d equipped with the weak topology is an object of C ( O ) (see [Paˇs13, Lemmas4.4, 4.6]).2.1. Colmez’s Montreal functor.
In [Col10], Colmez has defined an exact and covariant functor V from the category of smooth, finite-length representations of G on O -torsion modules with a centralcharacter to the category of continuous finite-length representations of G Q p on O -torsion modules.If ψ : Q × p → O × is a continuous character, then we may also consider it as a continuous character ψ : G Q p → O × via class field theory and for all π ∈ Mod sm G,ψ ( O ) of finite length we have V ( π ⊗ ψ ◦ det) ∼ = V ( π ) ⊗ ψ .Moreover, V ( ) = 0, V (Sp) = ω , V (Ind GB χ ⊗ χ ω − ) ∼ = χ , V (c-Ind GK Sym r k / ( T, S − ∼ =ind ω r +12 , where ω is the mod p cyclotomic character, ω : I Q p → k × is Serre’s fundamental characterof level 2, and ind ω r +12 is the unique irreducible representation of G Q p of determinant ω r and suchthat ind ω r +12 | I Q p ∼ = ω r +12 ⊕ ω p ( r +1)2 with 0 ≤ r ≤ p −
1. Note that this determined the imageof supersingular representations under V since every supersingular representation is isomorphic toc-Ind GK Sym r k / ( T, S −
1) for some 0 ≤ r ≤ p − G Q p ( O ) be the category of continuous G Q p -representations on compact O -modules. Following[Paˇs15, Section 3], we define an exact covariant functor ˇ V : C ( O ) → Rep G Q p ( O ) as follows: Let M be in C ( O ), if it is of finite length, we define ˇ V ( M ) := V ( M ∨ ) ∨ ( εψ ) where ∨ denotes the Pontryagindual. For general M ∈ C ( O ), write M ∼ = lim ←− M i , with M i of finite length in C ( O ) and define ˇ V ( M ) :=lim ←− ˇ V ( M i ). With this normalization of ˇ V , we have • ˇ V ( π ∨ ) = 0 if π ∼ = η ◦ det; • ˇ V ( π ∨ ) ∼ = χ if π ∼ = Ind GB χ ⊗ χ ω − , where ω is the mod p cyclotomic character; • ˇ V ( π ∨ ) ∼ = η if π ∼ = Sp ⊗ η ◦ det; • ˇ V ( π ∨ ) ∼ = V ( π ) if π is supersingular.If Π ∈ Ban adm
G,ψ ( E ), we abbreviate ˇ V (Π) = ˇ V (Θ d ) ⊗ O E with Θ is any open bounded G -invariant O -lattice in Π, so that ˇ V is exact and contravariant on Ban adm G,ψ ( E ).2.2. Extension Computations when p = 3 . In this subsection we assume p = 3 and ψ = . For π ′ , π ∈ Mod l . fin G,ψ ( k ), we write Ext G/Z ( π ′ , π ) for the extension group of π ′ by π in Mod l . fin G,ψ ( k ). By [Paˇs10,Thereom 11.4], we have the following table for dim k Ext G/Z ( π ′ , π ): π ′ \ π Sp ω ◦ det Sp ⊗ ω ◦ det ω ◦ det 0 0 0 2Sp ⊗ ω ◦ det 1 0 1 0where Sp is the Steinberg representation defined by 0 → → Ind GB → Sp → B the Borelsubgroup of uppertriangular matrices. t is shown in [Col10, Proposition VII.4.18] that Ext G/Z ( , Sp) ∼ = Hom( Q × p , k ). We denote theextension corresponding to τ ∈ Hom( Q × p , k ) by E τ . We identify Hom( Q × p , k ) with Ext G Q p ( , ) viathe isomorphism Hom( Q × p , k ) ∼ = Hom( G Q p , k ) given by τ τ ◦ Art − Q p composing with isomorphismsHom( G Q p , k ) ∼ = H ( G Q p , k ) ∼ = Ext G Q p ( , ). The cup product defines a non-degenerate pairing H ( G Q p , ) × H ( G Q p , ω ) → H ( G Q p , ω ) ∼ = k. Since Hom( G Q p , k ) is a 2-dimensional k -vector space and the pairing is non-degenerate, the subspaceorthogonal to τ is 1-dimensional, which we denote by kτ ⊥ . We identify Hom( Q × p , k ) with Ext G Q p ( , ω )via τ τ ⊥ . Lemma 2.1.
Ext G/Z (Ind GB ω ⊗ ω, E τ ) is a -dimensional over k .Proof. We claim that every indecomposable representation E with semisimplification ⊕ Sp ⊕ ω ◦ det ⊕ Sp ⊗ ω ◦ det and socle Sp must come from a non-trivial class of Ext G/Z (Ind GB ω ⊗ ω, E τ ). Giventhis, the same proof of [Col10, Proposition VII.4.25] gives that Ext G/Z (Ind GB ω ⊗ ω, E τ ) is 1-dimensional.To prove the claim, we first note that E/ Sp has socle since Sp only allows non-trivial extensionby . This implies that E contains E τ for some τ ∈ Hom( Q × p , k ) as a subrepresentation. Denote D the quotient of E by E τ . We have the exact sequence(1) 0 → E τ → E → D → , which is non-split since the socle of E is Sp. If ω ◦ det lies in the socle of D , then ω ◦ det would liein the socle of E since both Ext G/Z ( ω ◦ det , ) and Ext G/Z ( ω ◦ det , Sp) is zero, which contradicts theassumption. Thus the socle of D is given by Sp ⊗ ω ◦ det and D ∼ = Ind GB ω ⊗ ω because Ext G/Z (Sp ⊗ ω ◦ det , ω ◦ det) is 1-dimensional. This proves the claim and the lemma follows. (cid:3) Proposition 2.2.
For π ′ ∈ { Sp , Sp ⊗ ω ◦ det } , Ext G/Z ( π ′ , E τ ) is 1-dimensional over k . Moreover,the unique non-split extension D τ of π ′ by E τ gives rise to the class of Ext G Q p ( , ˇ V ( π ′∨ )) given by τ ∈ Hom( Q × p , k ) under the Colmez’s functor ˇ V .Proof. We first assume π ′ = Sp. It is proved in [Col10, Proposition VII.4.13 (iii)] that the mapExt G/Z (Ind GB , Ind GB ) → Ext G Q p ( , ) induced by the Colmez’s functor ˇ V is an isomorphism. Thuswe may identify Ext G/Z (Ind GB , Ind GB ) with Hom( Q × p , k ). Using this identification, the image of τ ∈ Hom( Q × p , k ) under the following compositionExt G/Z (Ind GB , Ind GB ) → Ext G/Z (Ind GB , Sp) → Ext G/Z ( , Sp)is E τ , where the first map comes from the functoriality of Ext G/Z (Ind GB , − ) applied to Ind GB ։ Spand the second map comes from the functoriality of Ext G/Z ( − , Sp) applied to ֒ → Ind GB . Considerthe following commutative diagram(2) Ext G/Z (Ind GB , Ind GB ) Ext G/Z (Ind GB , Sp) Ext G/Z ( , Sp)Ext G Q p ( , ) Ext G Q p ( , ) , f ∼ g where vertical maps are induced by the Colmez’s funtor ˇ V . Let(3) 0 → E τ → D τ → Sp → G -socle of D τ , we obtain an extension(4) 0 → → D τ / Sp → Sp → . If this extension is split, then by pulling back Sp, we would deduce that D τ has a subrepresentation π ′′ fitting into an exact sequence 0 → Sp → π ′′ → Sp →
0. Since Ext G/Z (Sp , Sp) = 0, this would imply (3)is split. Hence (4) is non-split. Since Ext G/Z (Sp , ) is 1-dimensional, we deduce that D τ / Sp ∼ = Ind GB and thus D τ gives rise to an element of Ext G/Z (Ind GB , Sp). Moreover, the map g maps this element o the extension class obtained by pulling back ⊂ D τ / Sp, which is equivalent to the extension classof 0 → Sp → E τ → →
0. It follows that the first assertion is equivalent to g is an isomorphism.To show that g is an isomorphism, we consider the exact sequence0 → Ext G/Z (Sp , Sp) → Ext G/Z (Ind GB , Sp) g −→ Ext G/Z ( , Sp) . obtained by applying Hom G/Z ( − , Sp) to the short exact sequence 0 → → Ind GB → Sp → G/Z (Sp , Sp) = 0, the map g is an injection. It is indeed an isomorphism because bothExt G/Z (Ind GB , Sp) and Ext G/Z ( , Sp) are 2-dimensional by [Eme10b, Proposition 4.3.12 (1)] and theabove table.Applying Hom
G/Z (Ind GB , − ) to the short exact sequence 0 → → Ind GB → Sp →
0, we have thefollowing exact sequence0 → Hom
G/Z (Ind GB , Ind GB ) → Hom
G/Z (Ind GB , Sp) → Ext G/Z (Ind GB , ) → Ext G/Z (Ind GB , Ind GB ) f −→ Ext G/Z (Ind GB , Sp) . Using the fact that Ext G/Z (Ind GB , ) = 0 [Eme10b, Proposition 4.3.13 (1)], we see that the map f is aninjection. It is indeed an isomorphism because both the source and target are 2-dimensional [Eme10b,Proposition 4.3.12 (2), Proposition 4.3.15 (3)]. Thus we may identify Ext G/Z (Ind GB , Ind GB ) withExt G/Z ( , Sp) and the second assertion follows from the commutativity of the diagram and [Col10,Proposition VII.4.12], which shows that ˇ V maps the class of Ext G/Z (Ind GB , Ind GB ) having image τ ∈ Ext G/Z ( , Sp) under g ◦ f to the class of Ext G Q p ( , ) given by τ .If π ′ = Sp ⊗ ω ◦ det, then we consider following diagramExt G/Z (Sp ⊗ ω ◦ det , E τ ) Ext G/Z (Ind GB ω ⊗ ω, E τ )Ext G Q p ( , ω ) Ext G Q p ( , ω ) , f where the map f comes from the functoriality of Ext G/Z and the vertical maps are induced by theColmez’s funtor ˇ V . Since Ext G/Z (Ind GB ω ⊗ ω, E τ ) is 1-dimensional by Lemma 2.1, the first asser-tion holds if the map f is an isomorphism. Given this, the second assertion follows immediatelyfrom commutativity of the diagram and [Col10, Proposition VII.4.24], which shows that the image ofExt G/Z (Ind GB ω ⊗ ω, E τ ) in Ext G Q p ( , ω ) is the 1-dimensional k -vector space given by τ ⊥ .To show f is an isomorphism, we consider the long exact sequenceHom G/Z ( ω ◦ det , E τ ) → Ext G/Z (Sp ⊗ ω ◦ det , E τ ) → Ext G/Z (Ind GB ω ⊗ ω, E τ ) → Ext G/Z ( ω ◦ det , E τ )obtained by applying Hom G/Z ( − , E τ ) to the short exact sequence0 → ω ◦ det → Ind GB ω ⊗ ω → Sp ⊗ ω ◦ det → . Note that Ext G/Z ( ω ◦ det , E τ ) = 0 since both Ext G/Z ( ω ◦ det , Sp) and Ext G/Z ( ω ◦ det , ) are zero. Wesee that f is an isomorphism and the proposition follows. (cid:3) Corollary 2.3.
Let π ′ ∈ { Sp , Sp ⊗ ω ◦ det } and E ∈ Mod sm G ( k ) be a representation with socle Sp satisfying ˇ V ( E ) = . Then ˇ V induces an injection Ext G/Z ( π ′ , E ) ֒ → Ext G Q p (cid:0) ˇ V ( E ∨ ) , ˇ V (( π ′ ) ∨ ) (cid:1) . Proof.
We first note that the only absolutely irreducible admissible representation which allows a non-trivial extension by Sp is , and there is no non-trivial extension of (resp. ω ◦ det) by . SinceExt G/Z ( , Sp) is 2-dimensional, we see that E is an extension of ⊕ n by Sp with 0 ≤ n ≤
2. If n = 0,then Ext G/Z ( π ′ , E ) = 0 and the assertion holds trivially. If n = 1, the assertion follows immediatelyfrom Proposition 2.2 since E ∼ = E τ for some τ ∈ Hom( Q × p , k ).Assume n = 2 and c ∈ Ext G/Z ( π ′ , E ) maps to 0. We claim that c = 0. If not, then it induces anon-trivial class c ′ in Ext G/Z ( D, Sp) since Ext G/Z ( π ′ , Sp) = 0, where D is a non-trivial extension of ⊕ by π ′ . Moreover, by the fact that Ext G/Z ( π ′ , ) is 1-dimensional, we have D ∼ = D ′ ⊕ with D ′ the unique non-trivial extension of by π ′ . Consider the commutative diagramExt G/Z ( D, Sp) Ext G/Z ( D ′ , Sp)Ext G Q p (cid:0) , ˇ V (cid:0) ( π ′ ) ∨ ) (cid:1) Ext G Q p (cid:0) , ˇ V (cid:0) ( π ′ ) ∨ ) (cid:1) , f where f is induced by the functoriality of Ext G/Z ( − , Sp) applied to the inclusion D ′ ֒ → D and thevertical maps are induced by ˇ V . We denote c ′′ = f ( c ′ ) ∈ Ext G/Z ( D ′ , Sp). Note that c ′′ is non-trivial and maps to zero under ˇ V . Since Ext G/Z ( π ′ , Sp) = 0, c ′′ determines a unique non-trivial classof Ext G/Z ( π ′ , E ′ ) with the same image in Ext G Q p ( , ˇ V (cid:0) ( π ′ ) ∨ )) under ˇ V , where E ′ is a non-trivialextension of Sp by . This gives a contradiction since its image in Ext G Q p ( , ˇ V (cid:0) ( π ′ ) ∨ )) is nonzero byProposition 2.2. Thus c = 0 and the assertion follows. (cid:3) A finiteness lemma.Lemma 2.4.
Let π, κ ∈ Mod l . fin G,ψ ( k ) B be of finite length such that π SL ( Q p ) = 0 and κ is an irreduciblenon-character, then ˇ V induces: Hom G ( κ, π ) ∼ = Hom G Q p (cid:0) ˇ V ( π ∨ ) , ˇ V ( κ ∨ ) (cid:1) , Ext G,ψ ( κ, π ) ֒ → Ext G Q p (cid:0) ˇ V ( π ∨ ) , ˇ V ( κ ∨ ) (cid:1) . Proof.
In [Paˇs10, Lemma A1], Paˇsk¯unas proved this lemma for supersingular blocks. For other blocks,the argument in his proof reduces the assertion to the case that ˇ V ( π ∨ ) is irreducible. We include theargument here for the sake of completeness. We argue by induction on the length l of ˇ V ( π ∨ ). Assumethat l >
1, then there exists an exact sequence(5) 0 → π → π → π → π SL ( Q p )1 = π SL ( Q p )2 = 0. Since ˇ V is exact, it induces an exact sequence(6) 0 → ˇ V ( π ∨ ) → ˇ V ( π ∨ ) → ˇ V ( π ∨ ) → . Applying Hom G ( κ, − ) to (5) and Hom G Q p ( − , ˇ V ( κ ∨ )) to (6), we obtain two long exact sequences, anda map between then induced by ˇ V . With the obvious notation, we get a commutative diagram0 A B C A B C A B C A B C . By induction hypothesis, the first and third vertical maps are bijections, and the fourth and sixthvertical maps are injections. This implies that the second vertical map is an bijection and the fifthvertical map is an injection.The case l = 1 is dealt in [Col10, Theorem VII.4.7, Proposition VII.4.10, Proposition VII.4.13,Proposition VII.4.24] except when p = 3 and B = { , Sp , ω ◦ det , Sp ⊗ ω ◦ det } ⊗ δ ◦ det, where δ : Q × p → k × is a smooth character and ω : Q × p → k × is the character ω ( x ) = x | x | mod ̟ . Note thatin the exceptional case, we may assume that δ = 1 and the socle of π is Sp by the symmetry of thetable in § (cid:3) Proposition 2.5. If π ∈ Mod l . fin G ( k ) is admissible, then ˇ V ( π ∨ ) is finitely generated as k J G Q p K -module.Proof. Without loss of generality, we can assume π ∈ Mod l . fin G ( O ) B , hence has finitely many irre-ducible subquotients π , ..., π n up to isomorphism. Since ρ i := ˇ V ( π ∨ i ) is a finite-dimensional G Q p -representation over k , Ker ρ i has finite index in G Q p . It follows that K := T i Ker ρ i is of finite index n G Q p and H := G Q p / K is a finite group. Denote P by the maximal pro- p quotient of K and G by thequotient of G defined by the diagram0 K G Q p H P G H G Q p on ˇ V ( π ∨ ) factors through G . Since π can be written as an inductivelimit of finite length smooth representations, it suffices to prove the claim for π of finite length by thedefinition of ˇ V . Set ρ = ˇ V ( π ∨ ) a finite-dimensional G Q p -representation over k . Since K acts triviallyon each irreducible pieces, ρ ( K ) has to be contained in the upper triangular unipotent matrices, henceit’s a p -group. The claim follows because the action of K factors through P .We have the following equivalent conditions:ˇ V ( π ∨ ) is a finitely generated k J G Q p K -module ⇐⇒ ˇ V ( π ∨ ) is a finitely generated k J G K -module(thus a finitely generated k J P K -module) ⇐⇒ ˇ V ( π ∨ ) P is a finitely generated k [ H ]-module(thus a finite-dimensional k -vector space) ⇐⇒ the cosocle of ˇ V ( π ∨ ) is of finite lengthwhere the cosocle is defined to be the maximal semisimple quotient. The first equivalence is due tothe claim and the second equivalence follows from Nakayama lemma for compact modules (see [Bru66,Corollary 1.5]). The last equivalence follows from the fact that the cosocle of ˇ V ( π ∨ ) in the categoryof compact k J G Q p K -modules coincides with the cosocle of ˇ V ( π ∨ ) P in the category of k [ H ]-modules.Since π is admissible, its pro- p Iwahori fixed part is of finite-dimensional. Thus there are onlyfinitely many irreducible representations in the socle of π , which implies that cosocle of ˇ V ( π ∨ ) is offinite length by Lemma 2.4. This proves the proposition. (cid:3) Patched modules
From now on we make the assumption p >
2. Fix a continuous representation ¯ r : G Q p → GL ( k )and enlarge k if necessary. Corollary A.7 of [EG14] (with K = Q p ) provides us with an imaginary CMfield F with maximal totally real subfield F + , and a continuous representation ¯ ρ : G F → GL ( k ) suchthat ¯ ρ is a suitable globalization of ¯ r in the sense of Section 2.1 of [CEG + T = S p ∪ { v } , with S p be the set of places of F + lying above p and v a place prime to p andsatisfying the properties in section 2.3 of [CEG + v ∈ S p , we let ˜ v be a choice of a placeof F lying over v , with the property that ¯ ρ | G F ˜ v ∼ = ¯ r . (Such a choice is possible by our assumption that¯ ρ is a suitable globalization of ¯ r .) We let ˜ T denote the set of places ˜ v , v ∈ T . For each v ∈ T , welet R ✷ ˜ v denote the maximal reduced and p -torsion free quotient of the universal framed deformationring of ¯ ρ | G F ˜ v . We fix a place p ∈ S p . For each v ∈ S p \ { p } , we write ¯ R ✷ ˜ v for an irreducible, reducedand p -torsion free potentially Barsotti-Tate quotient of R ✷ ˜ v (given by an irreducible component of apotentially crystalline deformation ring of ¯ r with Hodge type (1 ,
0) and some inertial type).
Remark . Any ¯ r admits a potentially Barsotti-Tate lift [Sno09, Proposition 7.8.1] and any such liftis potentially diagonalizable [GK14, Lemma 4.4.1].Consider the deformation problem S = ( F/F + , T, ˜ T , O , ε − , { R ✷ ˜ v } ∪ { R ✷ ˜ p } ∪ { ¯ R ✷ ˜ v } v ∈ S p \{ p } ) . With this choice of deformation problem and globalization ¯ ρ for our local Galois representation ¯ r , theTaylor-Wiles-Kisin patching argument carried out in Section 2 of [CEG +
16] produces for some d > O [ G ]-module M ∞ with an arithmetic action of R ∞ = R ✷ ¯ r ˆ ⊗ O ,v ∈ S p −{ p } ¯ R ✷ ˜ v ˆ ⊗ O R ✷ ˜ v J x , ..., x d K (notethat R ✷ ¯ r ∼ = R ✷ ˜ p ) in the sense of [CEG +
18, Section 3], which means the R ∞ [ G ]-module M ∞ satisfies thefollowing properties: AA1 M ∞ is a finitely generated R ∞ J K K -module. A2 M ∞ is projective in the category of pseudocompact O J K K -modules. AA3
For σ = σ ( λ, τ ) or σ cr ( λ, τ ), we define M ∞ ( σ ◦ ) := (cid:16) Hom cont O J K K (cid:0) M ∞ , ( σ ◦ ) d (cid:1)(cid:17) d ∼ = M ∞ ⊗ O J K K σ ◦ , where we are considering continuous homomorphism for the profinite topology on M ∞ and the p -adic topology on ( σ ◦ ) d . This is a finitely generated R ∞ -module by ( AA1 ) and corollary 2.5of [Paˇs15]. The action of R ∞ on M ∞ ( σ ◦ ) factors through R ∞ ( σ ) := R ✷ ¯ r ( σ ) J x , ..., x d K , where R ✷ ¯ r ( σ ) = R ✷ ¯ r ( λ, τ ) (resp. R ✷ ,cr ¯ r ( λ, τ )) if σ = σ ( λ, τ ) (resp. σ cr ( λ, τ )). Furthermore, M ∞ ( σ ◦ )is maximal Cohen-Macaulay over R ∞ ( σ ), and the R ∞ ( σ )[1 /p ]-module M ∞ ( σ ◦ )[1 /p ] is locallyfree of rank 1 over its regular locus. AA4
For such σ , the action of H ( σ ) := End G (c-Ind GK σ ) on M ∞ ( σ ◦ )[1 /p ] is given by the composite H ( σ ) η −→ R ✷ ¯ r ( σ )[1 /p ] → R ∞ ( σ )[1 /p ]where H ( σ ) η −→ R ✷ ¯ r ( σ )[1 /p ] is defined in [CEG +
16, Theorem 4.1] for σ cr ( λ, τ ) and [Pyv18,Theorem 3.6] for σ ( λ, τ ). Definition 3.2.
By [CEG +
16, Lemma 4.18 (2)] and [Pyv18, Proposition 5.5], the support of M ∞ ( σ )is a union of irreducible components of Spec R ✷ ¯ r ( σ ), which we call the set of automorphic componentsof Spec R ✷ ¯ r ( σ ).Let Π ∞ := Hom cont O ( M ∞ , E ). If y ∈ m-Spec R ∞ [1 /p ], thenΠ y := Hom cont O ( M ∞ ⊗ R ∞ ,y O κ ( y ) , E ) = Π ∞ [ m y ]is an admissible unitary E -Banach space representation of G ; see [CEG +
16, Proposition 2.13]. Thecomposition R ✷ ¯ r → R ∞ y −→ O defines an O -valued point x ∈ Spec R ✷ ¯ r and thus a continuous represen-tation r x : G Q p → GL ( O ). We say y is crystalline (resp. semi-stable, resp. potentially crystalline,resp. potentially semi-stable) if r x is crystalline (resp. semi-stable, resp. potentially crystalline, resp.potentially semi-stable).If r : G Q p → GL ( E ) is potentially semi-stable of Hodge type λ , we setBS( r ) := π sm ( r ) ⊗ π alg ( r ) , where π sm ( r ) is the smooth representation of G corresponding via local Langlands correspondence toWD( r ), and π alg ( r ) = det b ⊗ Sym a − b − E is an algebraic representation of G . For y ∈ m-Spec R ∞ [1 /p ]potentially crystalline (resp. potentially semi-stable), it is shown in [CEG +
16, Theorem 4.35] (resp.[Pyv18, Theorem 6.1]) that if π sm ( r x ) is generic irreducible and x lies on an automorphic component ofa potentially crystalline (resp. potentially semi-stable) deformation ring of ¯ r , then the space of locallyalgebraic vectors Π l . alg y in Π y is isomorphic to BS( r x ). Remark . If y ∈ m-Spec R ∞ [1 /p ] lies on an automorphic component and r x is crystabelline, then π sm ( r x ) is a principal series representation and BS( r x ) admits a unique unitary Banach space comple-tion which is topologically irreducible c.f. [BB10, Theorem 4.3.1, Corollary 5.3.1, Corollary 5.3.1] for r x absolutely irreducible and [BE10, Proposition 2.2.1] for r x reducible. Thus the injection BS( r x ) ֒ → Π y gives rise to an embedding of Banach space representations \ BS( r x ) ֒ → Π y .3.1. The action of the center Z . We now describe the action of the center Z of G on M ∞ . Thedeterminant of the universal lifting r ✷ of ¯ r is a character det r ✷ : G ab Q p → ( R ✷ ¯ r ) × lifting det ¯ r . Hence itfactors through εψ univ : G Q p → R × ¯ ψ , where R ¯ ψ is the universal deformation ring of ψ = ω − det ¯ r and ψ univ is the universal deformation of ¯ ψ . Via pullback along the natural homomorphism O [ Z ] → R ¯ ψ [ Z ],the maximal ideal of R ¯ ψ [ Z ] generated by ̟ and the elements z − ψ univ ◦ Art Q p ( z ) gives a maximalideal of O [ Z ]. If we denote by Λ Z the completion of the group algebra O [ Z ] at this maximal ideal,then the character ψ univ ◦ Art Q p ( z ) induces a homomorphism Λ Z → R ¯ ψ ; the corresponding morphismof schemes Spec R ¯ ψ → Spec Λ Z associates to each deformation ψ of ¯ ψ the character ψ ◦ Art Q p of Z .Thus Λ Z is identified with R ¯ ψ .By the local-global compatibility [CEG +
16, Section 4.22], the action of O [ Z ] on M ∞ extends to acontinuous action of Λ Z . Let z : Λ Z → O be an O -algebra homomorphism and ψ be the composition → Λ × Z z −→ O × . We write M ψ ∞ := M ∞ ⊗ Λ Z ,z O , which is an R ψ ∞ = R ✷ ,ψ ¯ r ˆ ⊗ O O J x , ..., x d K -module, andΠ ψ ∞ = Hom cont O ( M ψ ∞ , E ). Proposition 3.4.
The module M ψ ∞ is an O [ G ] -module with an arithmetic action of R ψ ∞ . This meansthat M ψ ∞ satisfies ( AA1 ), (
AA2 ) and (
AA4 ) with R ∞ replaced by R ψ ∞ and ( AA2 ) with pseudocom-pact O J K K -modules replaced by pseudocompact O J K K -modules with central character ψ − .Proof. The proof is same as M ∞ in [CEG + (cid:3) By [CEG +
16, Corollary 4.26], M ψ ∞ lies in C ( O ). Thus we may apply Colmez’s functor to M ψ ∞ andobtain an O [ G Q p ]-module ˇ V ( M ψ ∞ ) with an action of R ψ ∞ , hence an R ψ ∞ J G Q p K -module. Proposition 3.5. ˇ V ( M ψ ∞ ) is finitely generated over R ψ ∞ J G Q p K .Proof. Using Nakayama lemma for compact modules, it is enough to show that ˇ V ( M ψ ∞ ) ⊗ R ψ ∞ k ∼ =ˇ V ( M ψ ∞ ⊗ R ψ ∞ k ) (See [Paˇs13, Lemma 5.50]) is a finitely generated k J G Q p K -module. Note that M ψ ∞ ⊗ R ψ ∞ k is a finitely generated k J K K -module by ( AA1 ), so its Pontryagin dual is an admissible K -representationwith a smooth G -action, and thus an admissible G -representation. The proposition follows from Lemma2.5. (cid:3) Capture.
Let Z ( K ) be the center of K = GL ( Z p ), O J K K be the completed group algebra, andMod pro K ( O ) be the category of compact O J K K -modules. For a continuous character ζ : Z ( K ) → O × ,we let Mod pro K,ζ ( O ) be the full subcategory of Mod pro K ( O ) such that M ∈ Mod pro K ( O ) lies in Mod pro K,ζ ( O )if and only if Z ( K ) acts on M by ζ − . Definition 3.6.
Let { V i } i ∈ I be a set of continuous K -representations on finite-dimensional E -vectorspaces and let M ∈ Mod pro
K,ζ ( O ). We say that { V i } i ∈ I captures M if for any proper quotient M ։ Q ,we have Hom cont O J K K ( M, V i ) = Hom cont O J K K ( Q, V i ) for some i ∈ I .By Corollary 4.26 of [CEG + M ψ ∞ is a nonzero projective object in Mod pro K,ζ ( O ), where ζ = ψ | Z ( K ) . Theorem 3.7.
The action of R ψ ∞ J G Q p K on ˇ V ( M ψ ∞ ) factors through R ψ ∞ J G Q p K /J , where J is a closedtwo-sided ideal generated by g − tr (cid:0) r ∞ ( g ) (cid:1) g +det (cid:0) r ∞ ( g ) (cid:1) for all g ∈ G Q p , where r ∞ : G Q p → GL ( R ψ ∞ ) is the Galois representation lifting ¯ r induced by the natural map R ✷ ,ψ ¯ r → R ψ ∞ .Proof. In [Paˇs16, Proposition 2.7], it is shown that there is a family of K -representations { σ i } i ∈ I ,where σ i is a type for a Bernstein component containing a principal series representation but not aspecial series tensoring with an algebraic representation, which captures every projective object ofMod pro K,ζ ( O ). We have the following commuting diagram: L i ∈ I Hom K ( σ i , Π ψ ∞ ) ⊗ σ i Π ψ ∞ L i ∈ I L y ∈ m-Spec R ψ ∞ ( σ i )[1 /p ] Hom K ( σ i , Π ψ ∞ [ m y ]) ⊗ σ i L i ∈ I L y ∈ m-Spec R ψ ∞ ( σ i )[1 /p ] Π ψ ∞ [ m y ] . ( ⋆ ) ( ∗ ) Since H ( σ i ) acts on Hom K ( σ i , Π ψ ∞ ) via H ( σ i ) → R ✷ ,ψ ¯ r ( σ i )[1 /p ] → R ψ ∞ ( σ i )[1 /p ] by ( AA4 ), it actson Hom K ( σ i , Π ψ ∞ [ m y ]) via H ( σ i ) → R ✷ ,ψ ¯ r ( σ i )[1 /p ] → R ψ ∞ ( σ i )[1 /p ] y −→ κ ( y ). Thus by applying theFrobenius reciprocity to ( ⋆ ), we obtain a mapHom G (c-Ind GK σ i ⊗ H ( σ i ) ,y κ ( y ) , Π ψ ∞ [ m y ]) ⊗ (cid:0) c-Ind GK σ i ⊗ H ( σ i ) ,y κ ( y ) (cid:1) → Π ψ ∞ [ m y ] . Since Π ψ ∞ [ m y ] = Hom cont G (cid:0) M ψ ∞ ⊗ R ψ ∞ ,y κ ( y ) , E (cid:1) , the image of this map is nonzero if and only if y lies inthe support of M ψ ∞ ( σ ), which implies x is potentially crystalline of type σ i (recall x ∈ m-Spec R ✷ ,ψ ¯ r is thepoint induced by y ). We can also deduce from ( AA3 ) that the dimension of Hom G (c-Ind GK σ i ⊗ H ( σ i ) ,y κ ( y ) , Π ψ ∞ [ m y ]) over κ ( y ) is 1 if y lies in the support of M ψ ∞ ( σ i )[1 /p ] and 0 otherwise. In case it isnonzero, we have c-Ind GK σ i ⊗ H ( σ i ) ,y κ ( y ) = BS( r x ) by [Bre03b, Proposition 3.3], and thus induces an njection \ BS( r x ) ֒ → Π ψ ∞ [ m y ] ֒ → Π ψ ∞ by Remark 3.3. Let \ BS( r x ) ◦ := \ BS( r x ) ∩ ( M ψ ∞ ) d be a G -invariant O -lattice of \ BS( r x ). We define N to be the kernel of the R ψ ∞ -algebra homomorphism M ψ ∞ → Y i ∈ I Y y (cid:0) \ BS( r x ) ◦ (cid:1) d (7)induced by ( ∗ ) and \ BS( r x ) ֒ → Π ψ ∞ [ m y ] ֒ → Π ψ ∞ , where y ∈ m-Spec R ψ ∞ ( σ i )[1 /p ] lies in the support of M ψ ∞ ( σ ◦ i )[1 /p ], and define M by the exact sequence0 → N → M ψ ∞ → M → C ( O ) with a compatible action of R ψ ∞ .Tensoring (8) with σ ◦ i over O J K K , we obtain a surjection M ψ ∞ ( σ ◦ i ) ։ M ( σ ◦ i ). By the definition of M , we see that \ BS( r x ) ֒ → M d if y lies in the support of M ψ ∞ ( σ ◦ i )[1 /p ] for some i ∈ I , thus M ( σ ◦ i )[1 /p ]is supported at each point of R ψ ∞ ( σ ◦ i )[1 /p ] at which M ψ ∞ ( σ ◦ i )[1 /p ] is supported. Since M ψ ∞ ( σ ◦ i )[1 /p ] islocally free of rank one over its support and R ψ ∞ ( σ ◦ i ) is p -torsion free, we deduce that M ( σ ◦ i ) ∼ = M ψ ∞ ( σ ◦ i )for all i , which implies that M = M ψ ∞ by capture and thus N = 0. Note that C ( O ) is abelian andclosed under products. Thus the target of (4) lies in the domain of ˇ V and we have an injectionˇ V ( M ψ ∞ ) ∼ = ˇ V ( M ) ֒ → Y i ∈ I Y y ˇ V (cid:18)(cid:16) \ BS( r x ) ◦ (cid:17) d (cid:19) . (9)We claim that the action of R ψ ∞ J G Q p K on ˇ V (cid:16)(cid:0) \ BS( r x ) ◦ (cid:1) d (cid:17) factors through O κ ( x ) J G Q p K /J x , where J x is the closed two-sided ideal generated by g − tr (cid:0) r x ( g ) (cid:1) g + det (cid:0) r x ( g ) (cid:1) for all g ∈ G Q p . Given theclaim, we see that g − tr( r ∞ ( g )) g + det( r ∞ ( g )) acts by 0 on the right hand side of (9), and thus onˇ V ( M ψ ∞ ). This proves the proposition.To prove the claim, we note that BS( r x ) is the locally algebraic vectors of the unitary Banachrepresentation B ( r x ) constructed in [BB10]. Hence we have r x ∼ = ˇ V (cid:0) B ( r x ) (cid:1) ։ ˇ V (cid:0) \ BS( r x ) (cid:1) , and theclaim follows. (cid:3) Corollary 3.8. ˇ V ( M ψ ∞ ) is finitely generated over R ψ ∞ .Proof. By Proposition 3.5 and Theorem 3.7, ˇ V ( M ψ ∞ ) is a finitely generated R ψ ∞ J G Q p K /J -module, soit suffices to show that R ψ ∞ J G Q p K /J is finitely generated over R ψ ∞ . We note that R ψ ∞ J G Q p K /J is aCayley-Hamilton algebra with residual pseudorepresentation associated to ¯ r in the sense of [WE18,Section 2.2], hence it is finitely generated over R ψ ∞ by [WE18, Proposition 3.6]. (cid:3) Proposition 3.9. R ψ ∞ acts on ˇ V ( M ψ ∞ ) faithfully.Proof. Identify Λ Z with the universal deformation ring of the trivial character, we obtain an isomor-phism R ✷ ¯ r ∼ = R ✷ ,ψ ¯ r ˆ ⊗ O Λ Z via ( r, χ ) r ⊗ χ / [CEG +
18, Section 6.1], where χ / is a square root of χ lifting , and thus an isomorphism R ∞ ∼ = R ψ ∞ ˆ ⊗ O Λ Z . Consider the R ∞ -module M := M ψ ∞ ˆ ⊗ O Λ Z . Notethat M carries an arithmetic action of R ∞ by the proof of [CEG +
18, Proposition 6.10, Proposition6.14, Proposition 6.17]. By applying [EP18, Theorem 6.3] to M , we see that R ∞ acts faithfully on M since Spec R ✷ ¯ r is irreducible and reduced (formally smooth except for ¯ r ∼ ( ∗ ω ) ⊗ χ or p = 3 and¯ r | I Q p ∼ ( ω ∗ ω ) ⊗ χ , where ω is a fundamental character of level 2; see [Paˇs13, Corollary B.5] and[B¨10, Theorem 4.2, Theorem 5.2]).Consider V := ˇ V ( M ψ ∞ ) ˆ ⊗ O univ . Note that each point y ∈ Spec R ∞ gives a pair ( w, z ) with w ∈ Spec R ✷ ,ψ ¯ r and z ∈ Spec Λ Z via R ∞ ∼ = R ψ ∞ ˆ ⊗ O Λ Z , which satisfies M ⊗ R ∞ ,y κ ( y ) ∼ = (cid:0) M ψ ∞ ⊗ R ψ ∞ ,w κ ( w ) (cid:1) ⊗ ψ z . For y ∈ m-Spec R ∞ [1 /p ] crystabelline with a principal series type, we have \ BS( r x ) ֒ → Π y by Remark3.3 and thus 0 = ˇ V ( \ BS( r x )) ֒ → ˇ V (Π y ) = 0. Since such points are dense in the support of M [EP18,Theorem 5.2] and M is faithful over R ∞ , we deduce that V is faithful as R ∞ -module. This impliesthat ˇ V ( M ψ ∞ ) is faithful as R ψ ∞ -module and the proposition follows. (cid:3) Corollary 3.10.
For all y ∈ m-Spec R ∞ [1 /p ] , we have Π y = 0 . roof. Let ψε be the character given by det r x . Since R ψ ∞ acts on ˇ V ( M ψ ∞ ) faithfully by Proposition3.9 and ˇ V ( M ψ ∞ ) is finitely generated over R ψ ∞ by Corollary 3.8, Nakayama’s lemma implies thatˇ V ( M ψ ∞ ) ⊗ R ψ ∞ ,y κ ( y ) = 0. On the other hand, since ˇ V (Π y ) ∼ = ˇ V (cid:0) M ψ ∞ ⊗ R ψ ∞ ,y κ ( y ) (cid:1) ∼ = ˇ V ( M ψ ∞ ) ⊗ R ψ ∞ ,y κ ( y ),it follows that ˇ V (Π y ) = 0 and thus Π y = 0. (cid:3) Main results
Theorem 4.1.
For all y ∈ m-Spec R ∞ [1 /p ] such that the associated Galois representation r x is abso-lutely irreducible, we have ˇ V (Π y ) ∼ = r x . In particular, M ∞ ( σ ◦ )[1 /p ] is supported on every non-ordinary(at p ) component of Spec R ∞ ( σ ) for each locally algebraic type σ for G .Proof. Let x ∈ m-Spec R ✷ ¯ r [1 /p ] be the image of y ∈ m-Spec R ∞ [1 /p ] and let r x : G Q p → GL (cid:0) κ ( y ) (cid:1) bethe corresponding Galois representation. If r x is absolutely irreducible, then the action of R ∞ J G Q p K ⊗ R ∞ ,y κ ( y ) on ˇ V (Π y ) factors through g − tr (cid:0) r x ( g ) (cid:1) g + det (cid:0) r x ( g ) (cid:1) by Theorem 3.7, which implies ˇ V (Π y ) ∼ =( r x ) ⊕ n (see [BLR91, Theorem 1]) for some integer n . As Π y = 0 by Proposition 3.10, we get n ≥ y is an admissible Banach space representation, it contains an irreducible subrepresentationΠ, which can be assumed to be absolutely irreducible after extending scalars. If Π SL ( Q p ) = 0, thenˇ V (Π) is nonzero and dim E ˇ V (Π) ≤ r x ) ⊕ n ∼ = ˇ V (Π y ) ։ ˇ V (Π), itfollows that ˇ V (Π) ∼ = r x . Moreover, by [Col10, Theorem VI.6.50] and [Eme11, Theorem 3.3.22], thelocally algebraic vectors inside Π is nonzero if and only if r x is de Rham with distinct Hodge-Tateweights, in which case Π l . alg ∼ = π l . alg ( r x ). Suppose that r x is potentially semi-stable of type σ with π sm ( r x ) generic. Then we obtain that n = 1 and y lies in the support of M ψ ∞ ( σ ◦ ) since1 = dim E Hom K ( σ, Π) ≤ Hom K ( σ, Π y ) ≤ , where the last inequality is by AA3 .If Π SL ( Q p ) = 0, then Π ∼ = Ψ ◦ det, where Ψ : Q × p → κ ( y ) × is a continuous character. Supposethat r x is potentially semi-stable of Hodge type λ = ( a, b ) and inertial type τ , then det r x = ψε with ψ | I Q p = ε a + b − det τ . Since the central character of Π y is identified with ψ via Art Q p , it follows that ψ = Ψ . In case a + b − = Hom K (det c ⊗ η ◦ det , Π SL ( Q p ) ) ֒ → Hom K (det c ⊗ η ◦ det , Π y ) , where c = ( a + b − / η = p det τ ◦ Art Q p is a smooth character. Thus y lies in the supportof M ∞ (det c ⊗ η ◦ det) with λ = ( c + 1 , c ) and τ = η ⊕ η . If furthermore π sm ( r x ) is generic, thenΠ ′ := \ BS( r x ) is also a subrepresentation of Π y by [CEG +
16, Theorem 5.3]. Thus we have0 = Hom K (det c ⊗ η ◦ det , Π SL ( Q p ) ⊕ Π ′ ) ֒ → Hom K (det c ⊗ η ◦ det , Π y ) , which implies that M ∞ (det c ⊗ η ◦ det) ⊗ R ∞ κ ( y ) has dimension greater or equal to 2. This contradicts AA3 and thus Π SL ( Q p ) = 0 for such points. In case a + b − = Hom K (det c ⊗ η ◦ det , Π SL ( Q p ) ⊗ χ ֒ → Hom K (det c ⊗ η ◦ det , Π y ⊗ χ ) , where c = ( a + b ) / η is the smooth character p ε pr( ε ) − det τ ◦ Art Q p and χ = p pr( ε ) with pr : O × → ̟ O given by projection. Consider M = M ψ ∞ ˆ ⊗ O Λ Z as in the proof of Proposition 3.9.Since M also carries an arithmetic action of R ∞ ∼ = R ψ ∞ ˆ ⊗ O Λ Z , the pair ( y, z ) with z ∈ Spec Λ Z givenby χ lies in the support of M (det c ⊗ η ◦ det). In particular, r x ⊗ χ is potentially crystalline of Hodgetype ( c + 1 , c ) and a scalar inertial type. If furthermore π sm ( r x ⊗ χ ) is generic, then we deduce that(Π ⊗ χ ◦ det) SL ( Q p ) = 0 by the same argument as in the even case. Thus Π SL ( Q p ) = 0.Note that the set of crystabelline points with a fixed Hodge type λ = ( a, b ) are dense in the supportof M ∞ [EP18, Theorem 5.2] and ˇ V (Π y ) ∼ = r x if furthermore r x is irreducible, ( a, b ) = ( b +1 , b ) and a + b is odd. This implies the first assertion since the reducible locus of Spec R ✷ ¯ r is a closed subset of lessdimension. Since the any potentially semi-stable deformation ring is equi-dimensional after inverting p and its non-generic locus (i.e. π sm ( r x ) non-generic) has strictly less dimension [All16, Theorem 1.2.7],the second assertion follows immediately from the above discussion. (cid:3) Proposition 4.2.
For σ = σ ( λ, τ ) or σ cr ( λ, τ ) , every irreducible component of R ✷ ¯ r ( σ )[1 /p ] is au-tomorphic, except possibly for the semi-stable non-crystalline component in the case ¯ r ∼ ( ω ∗ ) ⊗ χ , λ = ( a + 1 , a ) and τ = η ⊕ η . roof. By Theorem 4.1, we only have to handle components in the reducible locus. If y ∈ Spec R ψ ∞ ( σ )[1 /p ]is reducible, then r x is a de Rham representation which is an extension of de Rham characters χ by χ . Write V = χ χ − , which is a de Rham character, hence of the form ε k µ , where k is an integer and µ is a character such that µ | I Q p has a finite image. By [Nek93, Proposition 1.24], we may compute thedimension of crystalline (resp. de Rham) extensions of Q p by V using the following formulas: h f ( V ) = h ( V ) + dim Q p D dR ( V ) / Fil D dR ( V )= ( k < k, µ ) = (0 , )0 otherwise h g ( V ) = h f ( V ) + dim Q p D cris ( V ∗ (1)) ϕ =1 = k, µ ) = ( − , )1 if k < , ( k, µ ) = ( − , ) or ( k, µ ) = (0 , )0 otherwiseThis shows that for all such y , r x is potentially crystalline ordinary (and thus potentially diagonal-izable) in the sense of [BLGGT14, section 1.4] except the case specified. By applying Theorem A.4.1of [BLGG13] (see also [CEG +
16, Corollary 5.4]), we may construct a global automorphic Galois repre-sentation corresponding to a point on the same component as y . This implies that all such componentsare automorphic, which completes the proof. (cid:3) Remark . In the case ¯ r ∼ ( ω ∗ ) ⊗ χ , λ = ( a + 1 , a ) and τ = η ⊕ η , then Z (cid:0) R ✷ ¯ r (cid:0) σ ( λ, τ ) (cid:1) /ω (cid:1) isirreducible if ¯ r is tr´es ramifi´e, and is the sum of two irreducible components if ¯ r is peu ramifi´e (includesplit) with one of which is Z (cid:16) R ✷ ¯ r (cid:0) σ cr ( λ, τ ) (cid:1) /ω (cid:17) and the other of which is the closure of the semi-stablenon-crystalline points; see Proposition 3.3.1 of [EG14]. Proof of Theorem 1.2.
We follow the strategy in [EG14] and [Paˇs15]. For every smooth irreducible k -representation ¯ σ of K , we define C ¯ σ (¯ r ) to be the cycle Z (cid:0) R ✷ ¯ r (˜ σ ) /̟ (cid:1) , where ˜ σ is a lift of ¯ σ of theform σ cr ( λ, ⊕ ) (see Remark 1.3). Assuming σ as in Proposition 4.2, we have the following equalityof cycles: Z (cid:0) R ✷ ¯ r ( σ ) /̟ (cid:1) × Z ( k J x , ..., x d K ) = Z (cid:0) R ∞ ( σ ) /̟ (cid:1) = Z (cid:0) M ∞ ( σ ◦ ) /̟ (cid:1) = X ¯ σ m ¯ σ ( λ, τ ) Z (cid:0) M ∞ (¯ σ ) (cid:1) = X ¯ σ m ¯ σ ( λ, τ ) Z (cid:0) R ∞ (˜ σ ) /̟ (cid:1) = X ¯ σ m ¯ σ ( λ, τ ) C ¯ σ (¯ r ) × Z ( k J x , ..., x d K ) , where M ∞ (¯ σ ) is a R ∞ (˜ σ ) /̟ -module defined by Hom cont O J K K (cid:0) M ∞ , ¯ σ ∨ (cid:1) ∨ ∼ = M ∞ (˜ σ ) /̟M ∞ (˜ σ ). Notethat the first and the last equalities follow from the definition of R ∞ ( σ ), the second and fourthequalities follows from Proposition 4.2 and [EG14, Lemma 5.5.1], and the third equality follows fromthe projectivity of M ∞ ; see [Paˇs15, Lemma 2.3]. This proves the theorem but the exceptional case.For the exceptional case (see Remark 4.3), we may check the conjecture by comparing this cycle with Z (cid:16) R ✷ ¯ r (cid:0) ˜ σ (cid:1) /ω (cid:17) with ¯ σ = ˜ st ⊗ η directly; see [BM02, Theorem 5.3.1] and [KW09, Section 3.2.7]. (cid:3) Corollary 4.4.
Every irreducible component of R ✷ ¯ r ( σ )[1 /p ] is automorphic.Proof. Given Theorem 1.2, this follows from Theorem 5.5.2 of [EG14]. (cid:3)
Following from the method of [Kis09, Theorem 2.2.18] and [HT15, Theorem 6.3], we obtain theFontaine-Mazur conjecture below.
Theorem 4.5.
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Fakult¨at f¨ur Mathematik, Universit¨at Duisburg-Essen, 45127 Essen, Germany
E-mail address : [email protected]@stud.uni-due.de