Local deformation rings and a Breuil-Mézard conjecture when l\neq p
aa r X i v : . [ m a t h . N T ] A ug LOCAL DEFORMATION RINGS FOR GL AND ABREUIL–M´EZARD CONJECTURE WHEN l = p JACK SHOTTON
Abstract.
We compute the deformation rings of two dimensional mod l rep-resentations of Gal( F /F ) with fixed inertial type, for l an odd prime, p aprime distinct from l , and F/ Q p a finite extension. We show that in this set-ting an analogue of the Breuil–M´ezard conjecture holds, relating the specialfibres of these deformation rings to the mod l reduction of certain irreduciblerepresentations of GL ( O F ). Introduction
Let p be a prime, and let F be a finite extension of Q p with absolute Galoisgroup G F . We study the (framed) deformation rings for two-dimensional mod l representations of G F , where l is an odd prime distinct from p . More specifically,let E be a finite extension of Q l , with ring of integers O , uniformiser λ , and residuefield F . Let ρ : G F → GL ( F )be a continuous representation. Then there is a universal lifting (or framed defor-mation) ring R (cid:3) ( ρ ) parametrising lifts of ρ . Our main result relates congruencesbetween irreducible components of Spec R (cid:3) ( ρ ) to congruences between certain rep-resentations of GL ( O F ), where O F is the ring of integers of F . Our methodis to give explicit equations for the components of Spec R (cid:3) ( ρ ), which may be ofindependent use.If τ : I F → GL ( E ) is a continuous representation that extends to a representa-tion of G F (an inertial type ), then we say that a representation ρ : G F → GL ( E )has type τ if its restriction to I F is isomorphic to τ . Say that an irreducible compo-nent of Spec R (cid:3) ( ρ ) has type τ if a Zariski dense subset of its E -points correspondto representations of type τ . We define (definition 4.1) a formal sum C ( ρ, τ ) ofirreducible components of the special fibre Spec R (cid:3) ( ρ ) ⊗ O F . For semisimple τ ,this is obtained as the intersection with the special fibre of those components ofSpec R (cid:3) ( ρ ) having type τ ; for non-semisimple τ this must be slightly modified.To an inertial type τ we also associate an irreducible E -representation σ ( τ ) of GL ( O F ), by a slight variant on the definition of [Hen02] (see section 3.3). For anirreducible F -representation θ of GL ( O F ), define m ( θ, σ ( τ )) to be the multiplicityof θ as a Jordan–H¨older factor of the mod λ reduction of σ ( τ ). Then we can stateour main theorem (theorem 4.2): Theorem.
Let ρ : G F → GL ( F ) be a continuous representation. For each irre-ducible F -representation θ of GL ( O F ) , there is a formal sum C ( ρ, θ ) of irreduciblecomponents of Spec R (cid:3) ( ρ ) ⊗ F such that, for each inertial type τ , we have the equality C ( ρ, τ ) = X θ m ( θ, σ ( τ )) C ( ρ, θ ) . In fact the C ( ρ, θ ) are uniquely determined (at least for those θ which actuallyoccur in some σ ( τ )).This theorem is an analogue for mod l representations of G F of the Breuil–M´ezard conjecture [BM02], which pertains to mod p representations of G Q p . Ourstatement is not in the language of Hilbert–Samuel multiplicities used in [BM02],but rather in the geometric language of [EG14]. The original conjecture of Breuil–M´ezard was proved in most cases by Kisin [Kis09a]; further cases were provedby Paskunas [Paˇs15] by local methods, and the full conjecture was proved when p > n -dimensional representationsof G F in [EG14]; the only case known, outside of those just mentioned, is that oftwo-dimensional potentially Barsotti–Tate representations (see [GK14]).In the l = p setting, a comparison of special fibres of (very particular) localdeformation rings was used by Taylor in [Tay08] to prove the change of level re-sults needed to obtain non-minimal automorphy lifting theorems; this is anothermotivation for our result.Our method of proof is to completely explicitly determine equations for deforma-tion rings of fixed type, and indeed obtaining these explicit descriptions is anothergoal of this paper. We reduce to the tamely ramified case, in which we use therelation φσφ − = σ q for φ ∈ G F a lift of Frobenius and σ ∈ I F a generator of tame inertia. Since weare considering lifts ρ of fixed type, and so with fixed characteristic polynomialof ρ ( σ ), we may use the Cayley–Hamilton theorem to reduce this equation to oneof degree at most two in the entries of ρ ( φ ) and ρ ( σ ). These explicit descriptionsshow that the irreducible components of Spec R (cid:3) ( ρ ) ⊗ E are always smooth (whichis also proved in [Pil08]), and that the reduced deformation rings in which thesemisimplification of the restriction to inertia is fixed are always Cohen–Macaulay(see 5.5). It is natural to ask whether these properties persist beyond the caseof two dimensional representations. We note that the generic fibres of our localdeformation rings have been studied in [Pil08] and [Red], but their methods saylittle about the integral structure.In a forthcoming paper, we will extend theorem 4.2 to the case of n -dimensionalrepresentations using global methods.The structure of this paper is as follows. In section 2 we define the universaldeformation rings and show how to reduce their study to the case when ρ is tamelyramified. We also prove some lemmas that will be useful in the calculations thatfollow. In section 3 we define the deformation rings with fixed inertial type that wewill need, and discuss the construction of the representations σ ( τ ). In section 4 westate and prove the main theorem, modulo the calculations of section 5 and resultsof section 6. Section 5 contains the calculations of explicit equations for localdeformation rings, divided into cases according to the value of q mod l . Finally, insection 6 we prove the results on the mod l reduction of the σ ( τ ) that are statedin section 3.4 (and used in the proof of theorem 4.2). OCAL DEFORMATION RINGS FOR GL Acknowledgements.
This work forms part of my Imperial College, LondonPhD thesis, and I am grateful to my supervisor Toby Gee for suggesting this topic,for comments on drafts of this paper, and for answering many questions. I alsothank Kevin Buzzard, David Helm and Tristan Kalloniatis for helpful commentsand corrections, and Jack Thorne for encouraging me to investigate the Cohen–Macaulay property of these deformation rings.This research was supported by the Engineering and Physical Sciences ResearchCouncil and the Philip Leverhulme Trust, and part of this work was done during avisit to the University of Chicago supported by the London Mathematical Societyand the Cecil King Foundation.2.
Preliminaries
Fields and Galois groups.
Suppose that l = p are primes with l > F/ Q p be a finite extension with ring of integers O F , maximal ideal p F ,uniformiser ̟ F and residue field k F of order q . Let F have absolute Galois group G F , inertia group I F , and wild inertia group P F . Let I F ։ I F / ˜ P F ∼ = Z l be themaximal pro- l quotient of I F , so that ˜ P F /P F ∼ = Q l ′ = l,p Z l ′ . Note that ˜ P F is normalin G F and write T F = G F / ˜ P F . The short exact sequence 1 → I F / ˜ P F → T F → G F /I F → T F ∼ = Z l ⋊ ˆ Z . We fix topological generators σ of this Z l and φ of this ˆ Z such that φ is a lift of arithmetic Frobenius. Then the action of ˆ Z on Z l is given by(1) φσφ − = σ q . Let
L/F be an unramified quadratic extension, with residue field k L .Now let E/ Q l be a finite extension with ring of integers O , residue field F anduniformiser λ . Let ǫ : G F → Z × l be the l -adic cyclotomic character, and let : G F → Z × l be the trivial character. If A is any O -algebra then we will regard theseas maps to A × via the structure maps Z l → O → A .Define two integers a and b by a = v l ( q −
1) and b = v l ( q + 1), where v l is the l -adic valuation; at most one of a and b is non-zero, since l is odd.2.2. Deformation rings.
Suppose that M is an n -dimensional F -vector space andthat ρ : G F → GL ( M ) is a continuous representation. Let ( e i ) ni =1 be a basis for M , so that ρ gives a map ρ : G F → GL n ( F ).Let C O denote the category of artinian local O -algebras with residue field F , and C ∧O the category of complete noetherian local O -algebras with residue field F . If A is an object of C O or C ∧O , let m A be its maximal ideal. Define two functors D ( ρ ) , D (cid:3) ( ρ ) : C O → Set as follows: • D ( ρ )( A ) is the set of equivalence classes of ( M, ι ) where: M is a freerank n A -module, ρ : G F → Aut A ( M ) is a continuous homomorphism,and ι : M ⊗ A F ∼ −→ M is an isomorphism commuting with the actions of G F ; • D (cid:3) ( ρ )( A ) is the set of equivalence classes of ( M, ρ, ( e i ) ni =1 ) where: M is afree rank n A -module, ρ : G F → Aut A ( M ) is a continuous homomorphismand ( e i ) ni =1 is a basis of M as an A -module, such that the isomorphism JACK SHOTTON ι : M ⊗ A F ∼ −→ M defined by ι : e i ⊗ e i commutes with the actions of G F .In the first case, ( M, ρ, ι ) and ( M ′ , ρ ′ , ι ′ ) are equivalent if there is an isomorphism α : M → M ′ , commuting with the actions of G F , such that ι = ι ′ ◦ α ; in the secondcase, ( M, ρ, ( e i ) i ) and ( M ′ , ρ ′ , ( e ′ i ) i ) are isomorphic if the map M → M ′ defined by e i e ′ i commutes with the actions of G F . There is a natural transformation offunctors D (cid:3) ( ρ ) → D ( ρ ) given by forgetting the basis.Alternatively, when ρ is regarded as a homomorphism to GL n ( F ), we have theequivalent definitions D (cid:3) ( ρ )( A ) = { continuous ρ : G F → GL n ( A ) lifting ρ } and D ( ρ )( A ) = { continuous ρ : G F → GL n ( A ) lifting ρ } / conjugacy by 1 + M n ( m A ) . The functor D ( ρ ) is not usually pro-representable, but the functor D (cid:3) ( ρ ) alwaysis (see, for example, [Kis09b] (2.3.4)): Definition 2.1.
The universal lifting ring (or universal framed deformation ring)of ρ is the object R (cid:3) ( ρ ) of C ∧O that pro-represents the functor D (cid:3) ( ρ ). The universallift is denoted ρ (cid:3) : G F → GL n ( R (cid:3) ( ρ )).Recall the following calculation (see e.g. [BLGGT14] section 1.2): Lemma 2.2.
The ring R (cid:3) ( ρ )[1 /l ] is generically formally smooth of dimension n . The next lemma enables us to reduce to the case where the residual represen-tation is trivial on ˜ P F . Suppose that θ is an irreducible F -representation of ˜ P F .Then by [CHT08], lemma 2.4.11, there is a lift of θ to an O -representation of ˜ P F ,which may be extended to an O -representation ˜ θ of G θ , where G θ is the group { g ∈ G F : gθg − ∼ = θ } . For each irreducible representation θ of ˜ P F , we pick sucha ˜ θ and a finite free O -module N ( θ ) on which ˜ P F acts as ˜ θ . If M is a set-finite O -module with a continuous action ρ of G F , then define M θ = Hom ˜ P F (˜ θ, M ) . The module M θ has a natural continuous action ρ θ of G θ given by ( gf )( v ) = gf ( g − v ); the subgroup ˜ P F of G θ acts trivially. Lemma 2.3. (Tame reduction) (1)
Let M be a set-finite O -module with a continuous action of G F . Then thereis a natural isomorphism M = M [ θ ] Ind G F G θ ( N ( θ ) ⊗ O M θ ) , where [ θ ] runs over G F -conjugacy classes of irreducible representations of ˜ P F . (2) The isomorphism of part (1) induces a natural isomorphism of functors: D ( ρ ) ∼ −→ Y [ θ ] D ( ρ θ ) where θ runs through a set of representatives for the G F -conjugacy classesof irreducible representations of ˜ P F . OCAL DEFORMATION RINGS FOR GL (3) If R (cid:3) ( ρ θ ) is the universal framed deformation ring for the representation ρ θ of G θ / ˜ P F , then R (cid:3) ( ρ ) ∼ = (cid:18)dO [ θ ] R (cid:3) ( ρ θ ) (cid:19) [[ X , . . . , X n − P n θ ]] where n θ = dim ρ θ . This isomorphism lies above the isomorphism D ( ρ ) ∼ −→ Q [ θ ] D ( ρ θ ) of part (2).Proof. The first two parts are in [CHT08]: part (1) is lemma 2.4.12 and part (2)is corollary 2.4.13. Part (3) is the refinement to framed deformations obtainedby keeping track of a basis in the construction of part (1) of the proposition, asin [Cho09], proposition 2.0.5.As [Cho09] is not easily available, we sketch the argument for part (3): let[ θ ] , [ θ ] , . . . be the G F -conjugacy classes of irreducible ˜ P F -representations. Pickleft coset representatives ( g ij ) j for G θ i in G F . Write N i for N ( θ i ), and choose an O -basis ( f ik ) k of N i .Let A be an object of C O , M be a free rank n A -module with a continuousaction of G F , and M θ i be as above. Given (for each i ) a basis ( e il ) n θi l =1 of M θ i , wecan produce a basis ( e ijkl ) j,k,l of M θ i = A [ G F ] ⊗ A [ G θ ] ( N i ⊗ O M θ i )defined by e ijkl = g ij ⊗ f ik ⊗ e il . Then ( e ijkl ) i,j,k,l is a basis of M .Let F ( A ) be the set of Y = ( Y ijkl,i ′ j ′ k ′ l ′ ) which are n × n matrices of elementsof m A such that Y ijkl,i ′ j ′ k ′ l ′ = 0 if i = i ′ and j = j ′ = k = k ′ = 1(so that n − P n θ i ‘free’ entries of Y remain). Then F defines a functor on C O pro-represented by O [[ X , . . . , X n − P n θ ]] (the variables X being simply an enumerationof those Y ijkl,i ′ j ′ k ′ l ′ which can be non-zero).We then have a natural transformation of functors F × Y [ θ ] D (cid:3) ( ρ θ ) → D (cid:3) ( ρ )taking the tuple ( Y , ( M θ i , ρ θ i , e il ) i ) to the tuple M i Ind G F G θi ( N i ⊗ O M θ i ) , M i Ind G F G θi (˜ θ i ⊗ O ρ θ i ) , ( I n + Y )( e ijkl ) i,j,k,l ! . Then one can check (and this is what is done in [Cho09], proposition 2.0.5) thatthis is in fact an isomorphism, and so we get the claimed isomorphism of pro-representing objects. (cid:3)
Twisting.Lemma 2.4.
Suppose that χ : G F → O × is any character. Then there is a naturalisomorphism R (cid:3) ( ρ ) ∼ −→ R (cid:3) ( ρ ⊗ χ ) . Moreover, if χ and χ satisfy χ = χ then they induce the same maps R (cid:3) ( ρ ) ⊗ F ∼ −→ R (cid:3) ( ρ ⊗ χ i ) ⊗ F . JACK SHOTTON
Proof.
This follows easily from the isomorphism of functors D (cid:3) ( ρ ) → D (cid:3) ( ρ ⊗ χ )given by tensoring with χ (remembering that we are considering O -algebras). Forthe last statement, observe that if the functors are restricted to F -algebras then theisomorphism only depends on χ . (cid:3) Since every F -valued character lifts to O (using the Teichm¨uller lift) this showsthat R (cid:3) ( ρ ) ∼ = R (cid:3) ( ρ ⊗ χ ) for every χ : G F → F × .We also need the calculation of the universal deformation ring of a character, towhich some of our calculations reduce. This is completely standard, but we includeit as a simple illustration of the method. Lemma 2.5.
Let χ : G F → F × be a continuous character. Then R (cid:3) ( χ ) = O [[ X, Y ]]((1 + X ) l a − has l a irreducible components, indexed by the l a th roots of unity. They are formallysmooth of relative dimension one over O .Proof. By lemma 2.4, we may take χ to be trivial. If χ is any lift of χ to an object A of C O , then for g ∈ ˜ P F we must have χ ( g ) n = 1 for some n coprime to l , andtherefore χ ( g ) = 1, so that we are reduced to considering characters of T F . We musthave that χ ( σ ) q = χ ( σ ) and χ ( σ ) ≡ m A , and therefore that χ ( σ ) l a = 1. Weare then free to choose χ ( φ ). Writing χ ( σ ) = 1 + X and χ ( φ ) = 1 + Y , we haveshown that D (cid:3) ( χ )( A ) = Hom C ∧O (cid:18) O [[ X, Y ]]((1 + X ) l a − , A (cid:19) functorially, and so the universal framed deformation ring is as claimed. (cid:3) Multiplicities and cycles.
Suppose that X is a noetherian scheme and that F is a coherent sheaf on X . Let Y be the scheme-theoretic support of F , and let d ≥ dim Y . Let Z d ( X ) be the free abelian group on the d -dimensional points of X ;elements of Z d ( X ) are called d -dimensional cycles. If a ∈ X is a point of dimension d write [ a ] for the corresponding element of Z d ( X ) and define the multiplicity e ( F , a ) to be the length of F a as an O Y, a -module (this is zero if a Y ). Definition 2.6.
The cycle Z d ( F ) associated to F is the element X a e ( F , a )[ a ] ∈ Z d ( X ) . If X = Spec A is affine and F = f M for a finitely generated A -module M , thenwe will write Z d ( M ) for Z d ( F ).If i : X → X ′ is a closed immersion of X in a noetherian scheme X ′ , then thereis a natural inclusion i ∗ : Z d ( X ) → Z d ( X ′ ) for each d . For a coherent sheaf F on X whose support has dimension at most d , we then have i ∗ ( Z d ( F )) = Z d ( i ∗ ( F )) . We will often use this compatibility without comment.A cycle is effective if it is of the form P n a [ a ] for n a ≥
0. Say that an effectivecycle C is a subcycle of an effective cycle C if C − C is also effective. OCAL DEFORMATION RINGS FOR GL A determinantal ring.
For a , b and c natural numbers, if I is the idealgenerated by the a × a minors of a b × c matrix with independent indeterminantentries over a Cohen–Macaulay ring A , then A/I is always Cohen–Macaulay (see[Eis95] theorem 18.18). We include a simple proof in the very special case that weneed below.
Proposition 2.7.
Let k ≥ be an integer and let A be either a field or a dis-crete valuation ring. Let R = A [ X , . . . , X k , Y , . . . , Y k ] and let I ⊳ R be the idealgenerated by the × minors of: (cid:18) X X . . . X k Y Y . . . Y k (cid:19) . Let S = R/I . Then S is a Cohen–Macaulay domain and is flat over A . It isGorenstein if and only if k = 2 .The same is true if we replace S by its completion S ∧ at the ‘irrelevant’ ideal ( X , . . . , X k , Y , . . . , Y k ) .Proof. Note that R and S are naturally graded A -algebras.Suppose that A is a field. It is easy to see that Proj( S ) is a smooth irreducibleprojective variety over A of dimension k + 1 — it is covered by the open sets { X i = 0 } and { Y i = 0 } , each of which is isomorphic to ( A A \ { } ) × A kA . Thus S is a domain. We may extend A so that its cardinality is at least k + 1, and choosepairwise distinct α , . . . , α k ∈ A × .I claim that ( X − α Y , . . . , X k − α k Y k , Y + . . . + Y k ) is a regular sequence in S .To see this, observe that Proj ( S/ ( X − α Y , . . . , X i − α i Y i )) is reduced (we maycheck this on the affine pieces) and that its irreducible components are all of theform Proj (cid:18) R ( X j − α i Y j ) ≤ j ≤ k + ( X j , Y j ) ≤ j ≤ i,j = i (cid:19) for 1 ≤ i ≤ i or of the formProj( S/ ( X , . . . , X i , Y , . . . , Y i )) . Now it is easy to check that X i +1 − α i +1 Y i +1 (if i < k ) or Y + . . . + Y k (if i = k )is a non-zerodivisor on each of these components, and so is a non-zerodivisor on S/ ( X − α Y , . . . , X i − α i Y i ) as required.Now S/ (( X i − α i Y i ) i , Y + . . . + Y k ) ∼ = A [ Y , . . . , Y k ] / ( Y , . . . , Y k ) is Gorenstein if and only if k = 2, as required.If A is a DVR then the following easy lemma (a specialisation of [Sno11] propo-sition 2.2.1) gives the result. Lemma 2.8. If A is a DVR and S is a finitely generated A -algebra such that S ⊗ A/ m A and S ⊗ Frac A are domains of the same dimension, then S is flat over A (that is, a uniformiser of A is a regular parameter in S ). The final statement of the proposition follows from the facts that both localisa-tion and completion preserve the properties of being Gorenstein, Cohen–Macaulay,or A -flat; S ∧ is a domain because its associated graded ring is S , which is a do-main. (cid:3) JACK SHOTTON Types
Inertial types.Definition 3.1. An inertial type τ (of dimension n ) is an equivalence class of pairs( r τ , N τ ) such that: • r τ : I F → GL n ( E ) is a representation with open kernel; • N τ is a nilpotent n × n matrix over E ; • ( r τ , N τ ) extends to a Weil–Deligne representation of G F .In particular, N τ commutes with the image of r τ . Two such pairs are equivalent ifthey are conjugate by an element of GL n ( E ).We say that a continuous representation ρ : G F → GL n ( E ) has inertial type τ ifthe restriction to inertia of the associated Weil–Deligne representation is equivalentto τ .We define some particular two-dimensional types which will often arise. Theywill all be of the form ( r, N ) with r | ˜ P F trivial, and are therefore determined by r ( σ )and N . Define: • τ ζ,s by r ( σ ) = (cid:18) ζ ζ (cid:19) and N = 0, where ζ is an l a th root of unity ( s isfor ‘split’); • τ ζ,ns by r ( σ ) = (cid:18) ζ ζ (cid:19) and N = (cid:18) (cid:19) , where ζ is an l a th root of unity( ns is for ‘non-split’); • τ ζ ,ζ by r ( σ ) = (cid:18) ζ ζ (cid:19) and N = 0 where, ζ and ζ are distinct l a throots of unity; • τ ξ by r ( σ ) = (cid:18) ξ ξ − (cid:19) and N = 0 where, ξ is a non-trivial l b th root ofunity.To see that τ ξ is a type, note that if L/F is the unramified quadratic extension,then there is a character of G L / ˜ P F mapping σ to ξ , which when induced to G F gives a representation of type τ ξ .3.2. Deformation rings with fixed type.Definition 3.2.
Let τ be an inertial type. Then R (cid:3) ( ρ, τ ) is the maximal reduced, l -torsion free quotient of R (cid:3) ( ρ ) with the following property: if x : R (cid:3) ( ρ ) → GL n ( E )is a continuous homomorphism such that the associated representation ρ x : G F → GL n ( E ) has type τ , then x factors through R (cid:3) ( ρ, τ ).The rings R (cid:3) ( ρ ) ⊗ F and R (cid:3) ( ρ, τ ) ⊗ F will occur very often, and so we denotethem respectively by R (cid:3) ( ρ ) and R (cid:3) ( ρ, τ ). From now on suppose that n = 2 . Write τ = ( r τ , N τ ) and assume that E is largeenough that all of the roots of the characteristic polynomial of r τ lie in E . Let R (cid:3) ( ρ, τ ) ◦ be the maximal quotient of R (cid:3) ( ρ ) on which: • if r τ is not scalar then, for all g ∈ I F , the characteristic polynomial of ρ (cid:3) ( g )agrees with that of r τ ; • if r τ is scalar and N τ = 0 then, for all g ∈ I F , ρ (cid:3) ( g ) is scalar and agreeswith r τ ; OCAL DEFORMATION RINGS FOR GL • if r τ is scalar and N τ = 0 then, for all g ∈ I F , the characteristic polynomialof ρ (cid:3) ( g ) agrees with that of r τ . Moreover, we have(2) q (tr ρ (cid:3) ( φ )) = ( q + 1) det( ρ (cid:3) ( φ )) . It is clear that these quotients exist and that the conditions imposed are defor-mation problems for ρ . Lemma 3.3.
The ring R (cid:3) ( ρ, τ ) is a reduced l -torsion free quotient of R (cid:3) ( ρ, τ ) ◦ .If N τ = 0 , then we have that R (cid:3) ( ρ, τ ) is equal to the maximal reduced l -torsionfree quotient of R (cid:3) ( ρ, τ ) ◦ .Proof. The first part is clear unless r τ is scalar and N τ = 0. In this case, we mustshow that any representation ρ : G F → GL ( E ) of type τ satisfies equation (2).The Weil–Deligne representation ( r, N ) corresponding to such a ρ satisfies r | I F = r τ and N = 0. Then r ( φ ) N = qN r ( φ ) implies that r ( φ ) preserves the line ker N andthe quotient E / ker N . If it acts as α on the former and β on the latter then wemust have α = qβ ; as α and β are the eigenvalues of ρ ( φ ) the equation (2) is easilyverified.The final claim follows from the simple observation that any E -point of R (cid:3) ( ρ, τ ) ◦ has associated Galois representation of type τ , except perhaps if r τ is scalar and N τ = 0. (cid:3) Remark 3.4. If R is a reduced, l -torsion free quotient of R (cid:3) ( ρ ) such that R (cid:3) ( ρ, τ )is a quotient of R , then R = R (cid:3) ( ρ, τ ) if and only if the closed points of type τ areZariski dense in Spec R [1 /l ]. In our calculations, when this is true it will always beclear by inspection.3.3. K -Types. Let G = GL ( F ), K = GL ( O F ), and for N ≥ K ( N ) =1 + M ( p NF ) and K ( N ) = (cid:26)(cid:18) a bc d (cid:19) : c ∈ p NF (cid:27) . Let U = O × F and for N ≥ U N = 1 + p NF . The exponent of a character χ of O × F is the smallest N ≥ χ is trivial on U N . If π is an irreducible admissible representation of GL m ( F ) (weonly need m = 1 and m = 2) over E , let rec( π ) be the continuous representation of W F over E associated to π under the local Langlands correspondence (normalisedso as to be preserved by automorphisms of E ).For each two-dimensional inertial type τ = ( r τ , N τ ), we define an irreduciblerepresentation σ ( τ ) by the following recipe: • If τ = τ ,s , then σ ( τ ) is the trivial representation of K . • If τ = τ ,ns , then σ ( τ ) is the inflation to K of the Steinberg representationSt of GL ( k F ). • If τ = ( ⊕ rec( ǫ ) | I F ,
0) for a non-trivial character ǫ of F × of exponent N ,then σ ( τ ) = Ind KK ( N ) ǫ, where ǫ (cid:18)(cid:18) a bc d (cid:19)(cid:19) = ǫ ( a ). • If τ = (rec( π ) | I F ,
0) for a cuspidal representation π of GL ( F ), then by[BH06], 15.5 Theorem, there is a certain subgroup J ⊂ G , containing thecenter of G and compact modulo center, and a representation Λ of J suchthat π = c-Ind GJ Λ . By conjugating, we may suppose that the maximal compact subgroup J of J is contained in K . We then have σ ( τ ) = Ind KJ (Λ | J ) . • If τ = τ ′ ⊗ rec( χ ) | I F , then σ ( τ ) = σ ( τ ′ ) ⊗ ( χ | U ◦ det).This is a slightly modified version of the construction in [Hen02] — the construc-tion there only depends on r τ , and agrees with ours whenever r τ is not scalar. Thefollowing is an easy consequence of [Hen02]: Proposition 3.5. If σ ( τ ) is contained in an irreducible admissible representation π of GL ( F ) and rec( π ) = ( r, N ) then r | I F ∼ = r τ and either N ∼ = N τ or N τ = 0 and N = 0 .If π is infinite-dimensional, then the converse is true. Reduction of types.
Suppose that r : I F → GL ( F ) is such that r extendsto G F . Definition 3.6.
The set L ( r ) is the set of types τ such that there exists a repre-sentation ρ : G F → GL ( O E ) of type τ satisfying ρ | I F ∼ = r. If r | ˜ P F is non-scalar then we abuse notation and also write L ( r ) for the set of r such that ( r, ∈ L ( r ), as in this case every element of L ( r ) is of this form. Lemma 3.7.
Suppose that r is trivial on ˜ P F . Then each element of L ( r ) is one ofthe types τ ζ,s , τ ζ,ns , τ ζ ,ζ , τ ξ defined in section 3.1.Proof. Suppose that ρ : G F → GL ( O E ) is of type τ and is such that ρ | I F ∼ = r .As r | ˜ P F is trivial, ρ must also be trivial on ˜ P F and its type is determined by theeigenvalues of ρ ( σ ) and by a nilpotent matrix N commuting with ρ ( σ ). Now, thefundamental relation φσφ − = σ q shows that the eigenvalues of ρ ( σ ) are the same(but perhaps in a different order) as those of ρ ( σ ) q , and this implies that they are( q − O E , and so must in fact be either l a th or l b th roots of unity (recall thatat most one of a and b is non-zero, since l = 2). If they are distinct l a th roots ofunity, then N must be zero and τ = τ ζ ,ζ ; if they are equal l a th roots of unity then τ = τ ζ,s or τ ζ,ns ; if they are l b th roots of unity then they must be ξ and ξ q = ξ − for an l b th root of unity ξ . Moreover the case ξ = 1 has already been dealt withand so we may assume that ξ = 1, in which case N = 0 and τ = τ ξ . (cid:3) Lemma 3.8. (1)
Suppose that r | ˜ P F is irreducible. There is a lift r of r to GL ( E ) , which we fix. Then L ( r ) = { r ⊗ χ } χ as χ runs over the set ofcharacters χ : I F → E × which extend to G F and reduce to the trivialcharacter. (2) Suppose that r | ˜ P F ∼ = ( r ⊕ r ) | ˜ P F where r and r are distinct charactersof G F . There are lifts r and r of r and r to E × , which we fix. Then L ( r ) = { ( r | I F ⊗ χ ) ⊕ ( r | I F ⊗ χ ) } χ ,χ where χ , χ run over all pairs ofcharacters I F → E × which extend to G F and reduce to the trivial character. (3) Suppose that r | ˜ P F ∼ = ( r ⊕ r c ) | ˜ P F where r and r c are distinct charactersof G L which are conjugate by an element of G F (recall that L/F is theunramified quadratic extension). There is a lift r of r to E × . Then OCAL DEFORMATION RINGS FOR GL L ( r ) = { ( r | I F ⊗ χ ) ⊕ ( r c | I F ⊗ χ c ) } χ as χ runs over all characters I F → E × which extend to G L and reduce to the trivial character.Proof. This follows from proposition 5.1 below; the ingredients in the proof of thatproposition are lemma 2.3 (reduction to the tame case) and lemma 2.4 (lifting ringof a character). (cid:3)
Lemma 3.9. If τ = ( r, is an inertial type with r | ˜ P F non-scalar, then σ ( τ ) isirreducible. If τ ′ is any other inertial type, then σ ( τ ′ ) contains σ ( τ ) if and only if τ ′ ∈ L ( r ) (in which case σ ( τ ) ∼ = σ ( τ ′ ) ).Proof. These are the results of propositions 6.4 and 6.5. (cid:3) If τ = ( r, N ) with r | ˜ P F scalar, then σ ( τ ) need not be irreducible. We give the(well-known) analysis of these σ ( τ ) in section 6.1. For now, we just give names tothe following representations of GL ( k F ) (and hence, by inflation, of K ) over F : • the trivial representation, ; • the Steinberg representation, St (irreducible if q
6≡ − l ); • if q ≡ − l , the cuspidal (but not supercuspidal) subrepresentation π of St. 4. The ‘Breuil–M´ezard conjecture’
Let ρ : G F → GL ( F ) be a continuous representation, and suppose that E issufficiently large that: • every subrepresentation of ρ ⊗ F is already defined over F ; • E contains all of the ( q − • for every τ ∈ L ( ρ | I F ), σ ( τ ) is defined over E .We state our analogue of the Breuil–M´ezard conjecture when l = p . By lemma 2.2and the fact that R (cid:3) ( ρ, τ ) is defined to be O -flat, we havedim R (cid:3) ( ρ, τ ) ≤ . Definition 4.1.
We associate to each type τ = ( r, N ) a cycle C ( ρ, τ ) ∈ Z ( R (cid:3) ( ρ ))as follows: • if N = 0, set C ( ρ, τ ) = Z ( R (cid:3) ( ρ, τ )); • if N = 0 (in which case r must be scalar) let τ ′ = ( r,
0) and set C ( ρ, τ ) = Z ( R (cid:3) ( ρ, τ )) + Z ( R (cid:3) ( ρ, τ ′ )) . Then we have
Theorem 4.2.
For each irreducible F -representation θ of GL ( O F ) , there is aneffective cycle C ( ρ, θ ) ∈ Z ( R (cid:3) ( ρ )) such that, for any inertial type τ , we have anequality of cycles (3) C ( ρ, τ ) = X θ m ( θ, σ ( τ )) C ( ρ, θ ) where m ( θ, σ ( τ )) is the multiplicity of θ as a Jordan–H¨older factor of σ ( τ ) and thesum runs over all θ . Proof.
We proceed case by case, using the results of section 3.4 and of sections 5and 6.1 below.Suppose that ρ | ˜ P F is non-scalar. Then by lemma 3.9, the representations σ ( τ )for τ ∈ L ( ρ | I F ) are all irreducible and isomorphic to a common irreducible repre-sentation, which we call θ . By corollary 5.2, R (cid:3) ( ρ ) has a unique minimal prime,denoted a , which has dimension 4. So we have Z (Spec( R (cid:3) ( ρ ))) = Z · [ a ] . Define C ( ρ, θ ) = [ a ], and C ( ρ, θ ) = 0 for θ = θ . By corollary 5.2, C ( ρ, τ ) = [ a ] = C ( ρ, θ )if τ ∈ L ( ρ | ˜ P F ) and C ( ρ, τ ) = 0otherwise. In other words, for all τ we have C ( ρ, τ ) = X θ m ( θ, σ ( τ )) C ( ρ, θ )as required.If ρ | ˜ P F is scalar, then we may twist ρ by a character of G F and apply lemma 2.4and so suppose for the rest of the proof that ρ | ˜ P F is trivial .If q = ± l , then L ( ρ | I F ) ⊂ { τ ,s , τ ,ns } . By the discussion of section 6.1,we have that σ ( τ ,s ) = and σ ( τ ,ns ) = Stare irreducible and non-isomorphic, and that neither is a Jordan–H¨older factor ofany other σ ( τ ). So the fact that we can define the C ( ρ, θ ) so as to satisfy equation(3) is a triviality, as there are no relations amongst the σ ( τ ) for different τ . Wework out what the C ( ρ, θ ) are explicitly: for θ = or St we define C ( ρ, θ ) = 0.Otherwise, there are four cases to consider: • if ρ ( φ ) has eigenvalues with ratio not in { , ± q } then by proposition 5.3there is a unique minimal prime a nr of R (cid:3) ( ρ ). In this case, define C ( ρ, ) = [ a nr ] C ( ρ, St) = [ a nr ]; • if ρ is an extension of the trivial character by itself then by proposition 5.5part 1 there is a unique minimal prime a nr of R (cid:3) ( ρ ). In this case, define C ( ρ, ) = [ a nr ] C ( ρ, St) = [ a nr ]; • if ρ is a non-split extension of the trivial character by the cyclotomic char-acter then by proposition 5.5 part 2 there is a unique minimal prime a N of R (cid:3) ( ρ ). In this case, define C ( ρ, ) = 0 C ( ρ, St) = [ a N ]; OCAL DEFORMATION RINGS FOR GL • if ρ is the direct sum of the trivial character and the cyclotomic characterthen by proposition 5.5 part 2 there are two minimal primes of R (cid:3) ( ρ ),denoted there by a nr and a N . In this case, define C ( ρ, ) = [ a nr ] C ( ρ, St) = [ a nr ] + [ a N ] . It is then easy to verify that equation (3) holds; we just do the last case. We seefrom proposition 5.5 part 2 that C ( ρ, τ ,s ) = [ a nr ] = C ( ρ, ) C ( ρ, τ ,ns ) = [ a nr ] + [ a N ]= C ( ρ, St)and C ( ρ, τ ) = 0 for all other τ , exactly as required by equation (3).If q = − l , then L ( ρ | I F ) ⊂ S ξ { τ ,s , τ ,ns , τ ξ } for ξ a non-trivial l b th rootof unity. By the discussion of section 6.1, we have that σ ( τ ,s ) = ,σ ( τ ξ ) = π , and σ ( τ ,ns ) ss = ⊕ π where and π are irreducible and non-isomorphic, and are not Jordan–H¨olderfactors of any other σ ( τ ). For θ = or π we define C ( ρ, θ ) = 0. Otherwise, thereare four cases to consider: • if ρ ( φ ) has eigenvalues with ratio not in {± } then by proposition 5.3 thereis a unique minimal prime a nr of R (cid:3) ( ρ ). In this case, define C ( ρ, ) = [ a nr ] C ( ρ, π ) = 0; • if ρ is an extension of the trivial character by itself then by proposition 5.6part 1 there is a unique minimal prime a nr of R (cid:3) ( ρ ). In this case, define C ( ρ, ) = [ a nr ] C ( ρ, π ) = 0; • if ρ is a non-split extension of the trivial character by the cyclotomic char-acter then by proposition 5.6 part 2a there is a unique minimal prime, de-noted a N in that proposition, of R (cid:3) ( ρ, τ ,ns ), which we regard as a primeof R (cid:3) ( ρ ). In this case, define C ( ρ, ) = 0 C ( ρ, π ) = [ a N ]; • if ρ is the direct sum of the trivial character by the cyclotomic characterthen in proposition 5.6 part 2b three four-dimensional primes of R (cid:3) ( ρ ) aredefined, denoted there a nr , a N and a N ′ . In this case, define C ( ρ, ) = [ a nr ] C ( ρ, π ) = [ a N ] + [ a N ′ ] . It is then easy to verify that equation (3) holds using proposition 5.3 in the firstcase and proposition 5.6 parts 1, 2a, and 2b in the second, third, and fourth cases;again we just do the fourth case, which is the most complicated. Equation (3) isequivalent to the equations: C ( ρ, τ ,s ) = C ( ρ, ) =[ a nr ] C ( ρ, τ ,ns ) = C ( ρ, ) + C ( ρ, π )=[ a nr ] + [ a N ] + [ a N ′ ] C ( ρ, τ ξ ) = C ( ρ, π ) =[ a N ] + [ a N ′ ]and C ( ρ, τ ) = 0if τ S ξ { τ ,s , τ ,ns , τ ξ } . But by proposition 5.6 part 2b we have: C ( ρ, τ ,s ) = Z ( R ( ρ, τ ,s )) =[ a nr ] C ( ρ, τ ,ns ) = Z ( R ( ρ, τ ,s )) + Z ( R ( ρ, τ ,ns ))=[ a nr ] + [ a N ] + [ a N ′ ] C ( ρ, τ ξ ) = Z ( R ( ρ, τ ξ )) =[ a N ] + [ a N ′ ]and C ( ρ, τ ) = 0if τ S ξ { τ ,s , τ ,ns , τ ξ } , as required.If q = 1 mod l , then L ( ρ | I F ) ⊂ S ζ,ζ ,ζ { τ ζ,s , τ ζ,ns , τ ζ ,ζ } for ζ , ζ and ζ (possiblytrivial) l a th roots of unity with ζ = ζ . By the discussion of section 6.1, we havethat σ ( τ ζ,s ) = ,σ ( τ ζ,ns ) = St , and σ ( τ ζ ,ζ ) = ⊕ Stwhere and St are irreducible and non-isomorphic, and are not Jordan–H¨olderfactors of any other σ ( τ ). For θ = or St we define C ( ρ, θ ) = 0. Otherwise, thereare four cases to consider: • if ρ ( φ ) has eigenvalues with ratio not in {± } then by proposition 5.3 thereis a unique minimal prime a nr of R (cid:3) ( ρ ). In this case, define C ( ρ, ) = [ a nr ] C ( ρ, St) = [ a nr ]; • if ρ is a ramified extension of the trivial character by itself then by propo-sition 5.8 part 1 there is a unique minimal prime a N of R (cid:3) ( ρ, τ ,ns ) whichwe regard as a four-dimensional prime of R (cid:3) ( ρ ). In this case, define C ( ρ, ) = 0 C ( ρ, St) = [ a N ]; OCAL DEFORMATION RINGS FOR GL • if ρ is a unramified extension of the trivial character by itself then byproposition 5.8 parts 2 and 3 there are four-dimensional primes of R (cid:3) ( ρ )which are denoted there by [ a nr ] and [ a N ]. In this case, define C ( ρ, ) = [ a nr ] C ( ρ, St) = [ a nr ] + [ a N ] . It is then easy to verify that equation (3) holds using proposition 5.3 in the firstcase, proposition 5.8 part 1 in the second case, and proposition 5.8 parts 2 and 3in the third case (according as ρ is split or not); again we just do the third case,which is the most complicated. Equation (3) is equivalent to the equations: C ( ρ, τ ζ,s ) = C ( ρ, ) =[ a nr ] C ( ρ, τ ζ,ns ) = C ( ρ, St) =[ a nr ] + [ a N ] C ( ρ, τ ζ ,ζ ) = C ( ρ, ) + C ( ρ, St)=[ a nr ] + [ a nr ] + [ a N ]and C ( ρ, τ ) = 0if τ S ζ,ζ ,ζ { τ ζ,s , τ ζ,ns , τ ζ ,ζ } . But by proposition 5.8 parts 2 and 3 we have: C ( ρ, τ ζ,s ) = Z ( R ( ρ, τ ζ,s )) =[ a nr ] C ( ρ, τ ζ,ns ) = Z ( R ( ρ, τ ζ,s )) + Z ( R ( ρ, τ ,ns ))=[ a nr ] + [ a N ] C ( ρ, τ ζ ,ζ ) = Z ( R ( ρ, τ ζ ,ζ )) =2[ a nr ] + [ a N ]and C ( ρ, τ ) = 0if τ S ζ,ζ ,ζ { τ ζ,s , τ ζ,ns , τ ζ ,ζ } , as required. (cid:3) Remark 4.3.
Although the definition of C ( ρ, τ ) may seem ad-hoc, it in fact hasthe following natural interpretation: it is the reduction modulo λ of the cyclein Z ( R (cid:3) ( ρ )) obtained by taking the Zariski closure of the closed points x ∈ Spec R (cid:3) ( ρ )[1 /l ] such that rec − ( ρ x ) | K contains σ ( τ ). Remark 4.4.
We conjecture that the theorem remains true when l = 2.5. Calculations
Let ρ : G F → GL ( F ) be a continuous representation. The aims of this sectionare to give explicit presentations for the rings R (cid:3) ( ρ, τ ) and to compute the cycles Z ( R (cid:3) ( ρ, τ )) ∈ Z (Spec R (cid:3) ( ρ )). We continue to assume that E is sufficiently large,as defined at the start of the previous section.5.1. Simple cases.
When ρ | ˜ P F is not scalar, then lemma 2.3 allows us to determinethe universal framed deformation rings. Recall that if r : I F → GL ( F ) is arepresentation that extends to G F then we have defined the set L ( r ) of types thatlift r . Proposition 5.1. If ρ | ˜ P F is irreducible, then R (cid:3) ( ρ ) ∼ = O [[ X, Y, Z , Z , Z ]] / ((1 + X ) l a − . The l a irreducible components of Spec R (cid:3) ( ρ ) are precisely the Spec R (cid:3) ( ρ, τ ) for τ ∈ L ( ρ | I F ) .If ρ | ˜ P F is a sum of distinct characters which extend to G F , then R (cid:3) ( ρ ) ∼ = O [[ X , X , Y , Y , Z , Z ]] / ((1 + X ) l a − , (1 + X ) l a − . The l a irreducible components of Spec R (cid:3) ( ρ ) are precisely the Spec R (cid:3) ( ρ, τ ) for τ ∈ L ( ρ | I F ) .If ρ | ˜ P F is a sum of distinct characters which are conjugate by the non-trivialelement of G L \ G F , then R (cid:3) ( ρ ) ∼ = O [[ X, Y, Z , Z , Z ]] / ((1 + X ) l b − . The l b irreducible components of Spec R (cid:3) ( ρ ) are precisely the Spec R (cid:3) ( ρ, τ ) for τ ∈ L ( ρ | I F ) .Proof. This follows straightforwardly from lemma 2.3. Suppose first that ρ | ˜ P F isirreducible. Then there is a unique irreducible representation θ of ˜ P F such that ρ θ (in the notation of lemma 2.3) is non-zero. For that θ , ρ θ is an unramifiedone-dimensional representation of G F . So by lemmas 2.3 and 2.5: R (cid:3) ( ρ ) ∼ = R (cid:3) ( ρ θ )[[ Z , Z , Z ]] ∼ = O [[ X, Y, Z , Z , Z ]] / ((1 + X ) l a − . We have ρ (cid:3) ∼ = ˜ θ ⊗ χ (cid:3) where χ (cid:3) is the universal character G F → R (cid:3) ( ρ θ ) × .Suppose now that ρ | ˜ P F = θ ⊕ θ for distinct characters θ and θ . Suppose firstthat the θ i are not G F -conjugate. As in lemma 2.3, we pick O -characters ˜ θ and˜ θ of G F lifting and extending θ and θ . Then (in the notation of lemma 2.3) ρ θ and ρ θ are both unramified characters. By lemmas 2.3 and 2.5: R (cid:3) ( ρ ) ∼ = (cid:16) R (cid:3) ( ρ θ ) ˆ ⊗ R (cid:3) ( ρ θ ) (cid:17) [[ Z , Z ]] ∼ = O [[ X , X , Y , Y , Z , Z ]] / ((1 + X ) l a − , (1 + X ) l a − . We have ρ (cid:3) ∼ = ˜ θ ⊗ χ (cid:3) ⊕ ˜ θ ⊗ χ (cid:3) where each χ (cid:3) i is the universal character over R (cid:3) ( ρ θ i ).Suppose finally that θ and θ are G F -conjugate. We take θ = θ ; then G θ = G L where L is a quadratic extension of F . In fact, since ˜ P F ⊂ G L and l is odd, we musthave that G L is the unramified quadratic extension of F . As in lemma 2.3, pick an O -character ˜ θ of G L lifting and extending θ . Then (in the notation of lemma 2.3) ρ θ is an unramified character of G L . By lemmas 2.3 and 2.5: R (cid:3) ( ρ ) ∼ = R (cid:3) ( ρ θ )[[ Z , Z , Z ]] ∼ = O [[ X, Y, Z , Z , Z ]] / ((1 + X ) l b − , since v l ( q −
1) = l b . We have ρ (cid:3) ∼ = Ind G F G L (cid:16) ˜ θ ⊗ χ (cid:3) (cid:17) where χ (cid:3) is the universal character over R (cid:3) ( ρ θ ). OCAL DEFORMATION RINGS FOR GL We show that f : Spec( R (cid:3) ( ρ, τ )) τ is a bijection from the set of irreduciblecomponents of Spec( R (cid:3) ( ρ )) to L ( ρ | I F ). It is easy to see that f is an injection(from our explicit expressions for ρ (cid:3) ). The type of the E -points of Spec( R (cid:3) ( ρ, τ ))is constant on irreducible components, so to show that a particular τ is in theimage of f it suffices to produce a lift of ρ to E of type τ . Each τ ∈ L ( ρ | I F ) is, bydefinition, the type of a lift of some ρ ′ with ρ ′ | I F ∼ = ρ | I F . But it is clear from thecalculations above that the image of f only depends on ρ | I F , and so f is surjectiveas required. (cid:3) Corollary 5.2. If ρ | ˜ P F is not scalar, then R (cid:3) ( ρ ) has a unique minimal prime a ,which has dimension 4. For τ an inertial type we have that Z ( R (cid:3) ( ρ, τ )) = [ a ] if τ ∈ L ( ρ | ˜ P F ) and Z ( R (cid:3) ( ρ, τ )) = 0 otherwise. We may now assume that ρ | ˜ P F is scalar; after a twist (invoking [CHT08] lemma2.4.11 to extend the character occurring in ρ | ˜ P F to the whole Galois group), wemay assume that ρ | ˜ P F is trivial , so that any lift of ρ | ˜ P F is also trivial. In this case,then, ρ | I F is inflated from a representation of the (procyclic) pro- l group I F / ˜ P F over a field of characteristic l . Any irreducible representation in characteristic l ofan l -group is trivial, and so ρ | I F must be an extension of the trivial representationby the trivial representation. Now, because φσφ − = σ q , ρ ( φ ) maps the subspaceof fixed vectors of ρ ( σ ) to itself; therefore ρ must be an extension of unramifiedcharacters. That is, there is a short exact sequence0 → χ → ρ → χ → χ and χ . Such an extension corresponds to an element of H ( G F , χ χ − ); by a simple calculation with the local Euler characteristic formulaand local Tate duality, this cohomology group is non-zero if and only if χ = χ or χ = χ ǫ . So we can easily deal with the case where neither of these two possibilitiescan occur. Proposition 5.3.
Suppose that ρ | ˜ P F is trivial and that ρ ( φ ) has eigenvalues α, β ∈ F with α/β
6∈ { , q, q − } . Then R (cid:3) ( ρ ) ∼ = O [[ A, B, P, Q, X, Y ]]((1 + P ) l a − , (1 + Q ) l a − , and ρ (cid:3) ( σ ) is diagonalizable with eigenvalues P and Q .For ζ an l a th root of unity (possibly equal to 1), we have that R (cid:3) ( ρ, τ ζ,s ) = O [[ A, B, P, Q, X, Y ]] / (1 + P − ζ, Q − ζ ) ∼ = O [[ A, B, X, Y ]] is formally smooth of relative dimension 4 over O , and R (cid:3) ( ρ, τ ζ,ns ) = 0 . If q = 1mod l and ζ , ζ are distinct l a th roots of unity, then R (cid:3) ( ρ, ψ, τ ζ ,ζ ) = O [[ A, B, P, Q, X, Y ]](2 + P + Q − ζ − ζ , P Q − ( ζ − ζ − ∼ = O [[ A, B, P, X, Y ]] / (1 + P − ζ )(1 + P − ζ ) . For all other τ , R (cid:3) ( ρ, ψ, τ ) = 0 . The ideal a nr defining R (cid:3) ( ρ, τ ,s ) is the unique minimal prime of R (cid:3) ( ρ ) . Wehave: Z ( R (cid:3) ( ρ, τ )) = [ a nr ] if τ = τ ζ,s a nr ] if τ = τ ζ ,ζ if τ = τ ζ,ns . Proof.
First note that, by the above cohomology calculation, ρ ( σ ) must be trivial.Let α and β be lifts of α and β to O . Suppose that A is an object of C O andthat M is a free A -module of rank 2 with a continuous action of G F given by ρ : G F → Aut A ( M ), reducing to ρ modulo m A . Suppose that the characteristicpolynomial of ρ ( φ ) is ( X − α − A )( X − β − B ), where A, B ∈ m A – note that byHensel’s lemma the characteristic polynomial does have roots in A reducing to α and β . Then there is a decomposition M = ( ρ ( φ ) − α − A ) M ⊕ ( ρ ( φ ) − β − B ) M. Here it is crucial that α + A , β + B and α − β + A − B are all invertible in A . If v α , v β is a basis of eigenvectors of ρ ( φ ) in M ⊗ F and v α , v β is a basis of M lifting v α , v β then there are unique X, Y ∈ m A such that v α + Xv β , v β + Y v α are eigenvectors of ρ ( φ ). Moreover, replacing ( v α , v β ) by ( µv α , µv β ) for µ ∈ m A does not change X and Y .Therefore we may assume that ρ ( φ ) = (cid:18) α β (cid:19) and that ρ ( φ ) = (cid:18) XY (cid:19) − (cid:18) α + A β + B (cid:19) (cid:18) XY (cid:19) ρ ( σ ) = (cid:18) XY (cid:19) − (cid:18) P RS Q (cid:19) (cid:18) XY (cid:19) where X, Y, P, R, S, Q ∈ m A are uniquely determined by ρ . The equation φσφ − = σ q implies that (cid:18) α + A β + B (cid:19) (cid:18) P RS Q (cid:19) (cid:18) α + A β + B (cid:19) − = (cid:18) P RS Q (cid:19) q . Looking at the top right and bottom left entries gives that R = S = 0. Thenlooking at the diagonal entries gives that (1 + P ) q − = (1 + Q ) q − = 1, which isequivalent to (1 + P ) l a = (1 + Q ) l a = 1. Thus R (cid:3) ( ρ ) ∼ = O [[ A, B, P, Q, X, Y ]]((1 + P ) l a − , (1 + Q ) l a − . The possible inertial types are τ ζ,s and τ ζ ,ζ ( τ ζ,ns cannot occur since all lifts arediagonalisable). Clearly R (cid:3) ( ρ, τ ζ,s ) is defined by the equations 1 + P = 1 + Q = ζ . The ring R (cid:3) ( ρ, τ ζ ,ζ ) ◦ is cut out by the equations 2 + P + Q = ζ + ζ ,(1 + P )(1 + Q ) = ζ ζ and the redundant equations (1 + P ) l a = (1 + Q ) l a = 1. But R (cid:3) ( ρ, τ ζ ,ζ ) ◦ ∼ = O [[ A, B, P, X, Y ]] / ((1 + P − ζ )(1 + P − ζ ))is reduced and λ -torsion free and so is equal to R (cid:3) ( ρ, τ ζ ,ζ ) . For the reduction modulo λ , simply note that: R (cid:3) ( ρ ) = F [[ A, B, P, Q, X, Y ]] / ( P l a , Q l a ) R (cid:3) ( ρ, τ ζ,s ) = F [[ A, B, P, Q, X, Y ]] / ( P, Q ) OCAL DEFORMATION RINGS FOR GL and R (cid:3) ( ρ, τ ζ ,ζ ) = F [[ A, B, P, Q, X, Y ]] / ( P , Q , P + Q ) . So a nr = ( P, Q ) is the unique minimal prime of R (cid:3) ( ρ ) and the multiplicities are asclaimed. (cid:3) We extract one part of the proof of this proposition for future use:
Lemma 5.4. If ρ ( φ ) has distinct eigenvalues, we may assume that it is diagonal.In that case, there exists a unique matrix (cid:18) XY (cid:19) ∈ GL ( R (cid:3) ( ρ )) , reducing to theidentity modulo the maximal ideal, such that ρ (cid:3) ( φ ) = (cid:18) XY (cid:19) − Φ (cid:18) XY (cid:19) fora diagonal matrix Φ .Proof. This is simply the first half of the proof of the previous proposition. (cid:3) q = ± l . Suppose that q = ± l . By lemma 5.3, we have alreadydealt with the cases in which the eigenvalues of ρ ( φ ) are not in the ratio 1 or q ± .All other cases are dealt with by the following (after twisting and conjugating ρ ).Note that, by lemma 3.7, the only possible types when ρ | ˜ P F is trivial are τ ,s and τ ,ns . Proposition 5.5.
Suppose that q = ± l , and that ρ | ˜ P F is trivial. Then (1) Suppose that ρ ( σ ) is trivial, and that ρ ( φ ) = (cid:18) y (cid:19) for y ∈ F . Then R (cid:3) ( ρ, τ ,s ) = R (cid:3) ( ρ ) is formally smooth of relative dimension 4 over O ,while R (cid:3) ( ρ, τ ,ns ) = 0 . (2) Suppose that ρ ( σ ) = (cid:18) x (cid:19) and ρ ( φ ) = (cid:18) q
00 1 (cid:19) .If x = 0 , then R (cid:3) ( ρ, τ ,ns ) = R (cid:3) ( ρ ) is formally smooth of relative di-mension 4 over O , while R (cid:3) ( ρ, τ ,s ) = 0 .If x = 0 then R (cid:3) ( ρ ) ∼ = O [[ X , . . . , X ]] / ( X X ) . The quotients by the two minimal primes are R (cid:3) ( ρ, τ ,s ) and R (cid:3) ( ρ, τ ,ns ) ,so that both are formally smooth of relative dimension 4 over O . The mini-mal primes a nr and a N of R (cid:3) ( ρ ) which respectively define R (cid:3) ( ρ, τ ,s ) and R (cid:3) ( ρ, τ ,ns ) are distinct.Proof. For the first part, write ρ (cid:3) ( σ ) = (cid:18) A BC D (cid:19) ρ (cid:3) ( φ ) = (cid:18) P y + RS Q (cid:19) where y is a lift of y (taken to be zero if y = 0) and A, B, C, D, P, Q, R, S ∈ m .Let I = ( A, B, C, D ). Considering the equation ρ (cid:3) ( φ ) ρ (cid:3) ( σ ) = ρ (cid:3) ( σ ) q ρ (cid:3) ( φ )modulo the ideal I m gives equations Cy ≡ ( q − A , B + Dy ≡ qAy + qB , C ≡ qC and ( q − D + qCy ≡
0, all modulo I m . As q = 1 mod l we find that I = I m . Therefore, by Nakayama’s lemma, I = 0 and ρ (cid:3) is unramified. So R (cid:3) ( ρ ) = R (cid:3) ( ρ, τ ,s ) ∼ = O [[ P, Q, R, S ]] as claimed. Note that this proof is still validif q = − l .The proof of the second part is similar. By lemma 5.4, we may write ρ (cid:3) ( σ ) = (cid:18) XY (cid:19) − (cid:18) A x + BC D (cid:19) (cid:18) XY (cid:19) ρ (cid:3) ( φ ) = (cid:18) XY (cid:19) − (cid:18) q (1 + P ) 00 1 + Q (cid:19) (cid:18) XY (cid:19) with x a lift of x (taken to be zero if x = 0) and A, B, C, D, X, Y, P, Q ∈ m .Let I = ( A, C, D ). Considering the relation φσφ − = σ q modulo I m and apply-ing Nakayama’s lemma as before now yields A = C = D = 0 (using that q = 1mod l ). The relation (not modulo any ideal) gives that ( x + B )( P − Q ) = 0, and itis easy to see if this equality holds then the given formulae for ρ (cid:3) do indeed definea representation so that R (cid:3) ( ρ ) = O [[ B, P, Q, X, Y ]](( x + B )( P − Q )) . If x = 0 then this implies that P = Q . Then R (cid:3) ( ρ ) = O [[ B, P, X, Y ]]. It is clearthat R (cid:3) ( ρ ) = R (cid:3) ( ρ, τ ,ns ), and the proposition follows.If x = 0 then, writing U = P − Q , we have R (cid:3) ( ρ ) = O [[ B, P, U, X, Y ]] / ( BU ). Inthese coordinates, it is clear from the description of ρ (cid:3) that R (cid:3) ( ρ, τ ,s ) = R (cid:3) ( ρ ) / ( B )and R (cid:3) ( ρ, τ ,ns ) = R (cid:3) ( ρ ) / ( U ) . The proposition follows. (cid:3) q = − l . Suppose that q = − l . By proposition 5.3, we havealready dealt with the cases in which the eigenvalues of ρ ( φ ) are not in the ratio 1or −
1. All other cases are dealt with by the following (after twisting and conjugating ρ ). By lemma 3.7, the only possible types when ρ | ˜ P F is trivial are τ ,s , τ ,ns and τ ξ for ξ a non-trivial l b th root of unity. Proposition 5.6.
Suppose that q = − l and that ρ | ˜ P F is trivial. (1) Suppose that ρ ( σ ) = (cid:18) (cid:19) and ρ ( φ ) = (cid:18) y (cid:19) for y ∈ F . Then R (cid:3) ( ρ, τ ,s ) = R (cid:3) ( ρ ) is formally smooth of relative dimension 4 over O , while R (cid:3) ( ρ, τ ,ns ) = R (cid:3) ( ρ, τ ξ ) = 0 . If a nr is the unique minimal prime of R (cid:3) ( ρ ) , then we have Z ( R (cid:3) ( ρ, τ )) = [ a nr ] if τ = τ ,s if τ = τ ,ns if τ = τ ξ . (2) Suppose that ρ ( σ ) = (cid:18) x (cid:19) and ρ ( φ ) = (cid:18) q
00 1 (cid:19) for x ∈ F . OCAL DEFORMATION RINGS FOR GL (a) If x = 0 , then R (cid:3) ( ρ, τ ,ns ) and R (cid:3) ( ρ, τ ξ ) are formally smooth of rel-ative dimension 4 over O , while R (cid:3) ( ρ, τ ,s ) = 0 . If a N is the primeideal of R (cid:3) ( ρ ) cutting out R (cid:3) ( ρ, τ ,ns ) then we have (4) Z ( R (cid:3) ( ρ, τ )) = if τ = τ ,s [ a N ] if τ = τ ,ns [ a N ] if τ = τ ξ . (b) If x = 0 , then R (cid:3) ( ρ, τ ,s ) is formally smooth of relative dimension 4over O and R (cid:3) ( ρ, τ ,ns ) ∼ = O [[ X , . . . , X ]](( X , X ) ∩ ( X , X − ( q + 1))) is a non-Cohen–Macaulay ring of relative dimension 4 over O . Itsspectrum is the scheme theoretic union of two formally smooth compo-nents that do not intersect in the generic fibre. Lastly, R (cid:3) ( ρ, τ ξ ) ∼ = O [[ X , . . . , X ]]( X X − ( ξ − ξ − ) ) is a complete intersection domain of relative dimension 4 over O withformally smooth generic fibre. If a nr is the prime of R (cid:3) ( ρ ) corre-sponding to R (cid:3) ( ρ, τ ,s ) and a N , a ′ N are the prime ideals of R (cid:3) ( ρ ) cor-responding to the two minimal primes of R (cid:3) ( ρ, τ ,ns ) , then we have (5) Z ( R (cid:3) ( ρ, τ )) = [ a nr ] if τ = τ ,s [ a N ] + [ a N ′ ] if τ = τ ,ns [ a N ] + [ a N ′ ] if τ = τ ξ .Proof. The proof of the first part is identical to that of proposition 5.5, part 1.For the second part, by lemma 5.4 we may write ρ (cid:3) ( σ ) = (cid:18) XY (cid:19) (cid:18) A x + BC D (cid:19) (cid:18) XY (cid:19) ρ (cid:3) ( φ ) = (cid:18) XY (cid:19) (cid:18) − (1 + P ) 00 1 + Q (cid:19) (cid:18) XY (cid:19) with x a lift of x (taken to be zero if x = 0) and A, B, C, D, X, Y, P, Q ∈ m .Firstly, it is clear that R (cid:3) ( ρ, τ ,s ) = 0 if x = 0 and R (cid:3) ( ρ, τ ,s ) ∼ = O [[ P, Q, X, Y ]]if x = 0.Next we deal with τ ,ns . On R (cid:3) ( ρ, τ ,ns ) we have the equationstr( ρ (cid:3) ( σ )) = 2det( ρ (cid:3) ( σ )) = 1 q tr( ρ ( φ )) = ( q + 1) det( ρ ( φ ))and ρ (cid:3) ( φ ) ρ (cid:3) ( σ ) ρ (cid:3) ( φ ) − = ρ (cid:3) ( σ ) q . The first two of these may be rewritten as A = − D and A + ( x + B )( C ) = 0and the third can be written as( q + 1 + P + qQ )( q + 1 + Q + qP ) = 0 . By the Cayley–Hamilton theorem, ( ρ (cid:3) ( σ ) − = 0 on R (cid:3) ( ρ, τ ,ns ) ◦ ; it followsthat ρ (cid:3) ( σ ) q − q ( ρ (cid:3) ( σ ) −
1) on R (cid:3) ( ρ, τ ,ns ) ◦ and so the relation φσφ − = σ q together with D = − A yields the equation: A − ( x + B ) P Q − C Q P − A ! = (cid:18) qA q ( x + B ) qC − qA (cid:19) . Equating coefficients and using that 2 and q − A = D = 0 and that ( x + B )( q + 1 + qQ + P ) = 0(6) C ( q + 1 + Q + qP ) = 0(7) ( x + B ) C = 0(8) ( q + 1 + Q + qP )( q + 1 + qQ + P ) = 0(9)is a complete set of equations cutting out R (cid:3) ( ρ, τ ,ns ) ◦ (the last two equationsbeing, respectively, the conditions on det( ρ (cid:3) ( σ )) and on ρ (cid:3) ( φ )).If x = 0 then these equations are equivalent to q + 1 + qQ + P = 0 and C = 0and so we see that R (cid:3) ( ρ, τ ,ns ) ∼ = O [[ B, P, X, Y ]] . If x = 0 then the left hand sides of the four equations given generate the ideal I = ( B, q + 1 + Q + qP ) ∩ ( C, q + 1 + qQ + P )in O [[ B, C, P, Q, X, Y ]]. Since O [[ B, C, P, Q, X, Y ]] /I is reduced and λ -torsion freeand a Zariski dense set of its E -points have type τ ,ns , it is equal to R (cid:3) ( ρ, τ ,ns ). Af-ter the change of variables X = q ( q +1+ Q + qP )( q − P ) , ( X , X , X , X , X ) = ( B, C, P, X, Y )we get the presentation given in the proposition.Let S = O [[ X , X , X ]]( X , X ) ∩ ( X , X − ( q + 1)) . Then S has dimension two. We show that S is not Cohen–Macaulay; the same isthen true for R (cid:3) ( ρ, τ ,ns ). Now, λ is a non-zerodivisor in S , and S /λ = F [[ X , X , X ]]( X X , X X , X X , X ) . The maximal ideal of S /λ is annihilated by X , and X = 0 in S /λ . So S /λ , andhence S , is not Cohen–Macaulay. The remaining statements about R (cid:3) ( ρ, τ ,ns ) areclear.Now suppose that τ = τ ξ . On R (cid:3) ( ρ, τ ξ ) we havetr( ρ (cid:3) ( σ )) = ξ + ξ − det( ρ (cid:3) ( σ )) = 1 OCAL DEFORMATION RINGS FOR GL and ρ (cid:3) ( φ ) ρ (cid:3) ( σ ) ρ (cid:3) ( φ ) − = ρ (cid:3) ( σ ) q . The first two of these may be rewritten as A + D = ξ + ξ − − AD − ( x + B ) C = 2 − ξ − ξ − . By the Cayley–Hamilton theorem, ( ρ (cid:3) ( σ ) − ξ )( ρ (cid:3) ( σ ) − ξ − ) = 0. As T q ≡ ξ + ξ − − T mod ( T − ξ )( T − ξ − )in Z [ T ], the relation φσφ − = σ q yields A − ( x + B ) P Q − C Q P D ! = (cid:18) ξ + ξ − − − A − ( x + B ) − C ξ + ξ − − − D (cid:19) . Equating coefficients and combining with the equation det( ρ (cid:3) ( σ )) = 1 we get: A = D = ξ + ξ − − x + B )( P − Q ) = 0(11) C ( P − Q ) = 0(12) 4( x + B ) C = ( ξ − ξ − ) . (13)If x = 0 then these equations are equivalent to P = Q and C = ( ξ − ξ − ) x + B ) , so that R (cid:3) ( ρ, τ ξ ) ∼ = O [[ X, Y, B, P ]] . If x = 0, then the equations imply that0 = BC ( P − Q ) = (cid:18) ξ − ξ − (cid:19) ( P − Q )and hence that P = Q , as R (cid:3) ( ρ, τ ξ ) is λ -torsion free by definition. Thus R (cid:3) ( ρ, τ ξ ) ∼ = O [[ X, Y, B, C, P ]](4 BC − ( ξ − ξ − ) ) . The remaining statements about R (cid:3) ( ρ, τ ξ ) are clear.Now we calculate the various Z ( R (cid:3) ( ρ, τ )). For part 1, this is trivial. For part 2,we have computed each R (cid:3) ( ρ, τ ) as a quotient of the ring F [[ A, B, C, D, P, Q, X, Y ]]by an ideal which we call I ( τ ). We see that if x = 0 then I ( τ ,ns ) = I ( τ ξ ), and R (cid:3) ( ρ, τ ,s ) = 0, from which equation 4 follows. If x = 0 then I ( τ ,s ) = ( A, B, C, D ) I ( τ ,ns ) = ( A, D, BC, B ( Q − P ) , C ( Q − P ) , ( Q − P ) )and I ( τ ξ ) = ( A, D, BC, Q − P ) . The minimal primes above these I ( τ ) in F [[ A, . . . , Y ]] are a nr = ( A, B, C, D ), a N = ( A, C, D, Q − P ) and a N ′ = ( A, B, D, Q − P ); the multiplicities in equation 5are then easily verified. (cid:3) Remark 5.7.
When ρ is unramified and ρ ( φ ) = (cid:18) q
00 1 (cid:19) , the ring R (cid:3) ( ρ, τ ,ns )is not Cohen–Macaulay. However the ring R (cid:3) ( ρ, unip), defined to be the maximalreduced quotient of R (cid:3) ( ρ ) on which ρ (cid:3) ( σ ) is unipotent (so that Spec R (cid:3) ( ρ, unip) isthe scheme-theoretic union of Spec R (cid:3) ( ρ, τ ,s ) and Spec R (cid:3) ( ρ, τ ,ns ) in Spec R (cid:3) ( ρ )), is Cohen–Macaulay. Indeed it is easy to see from the above proof that R (cid:3) ( ρ, unip) ∼ = O [[ X , . . . , X ]]( X X , X ( X − ( q + 1)) , X X )which is Cohen–Macaulay (( λ, X + X + X , X , X , X ) is a regular sequence).5.4. q = 1 mod l . Suppose that q = 1 mod l . By proposition 5.3, we have alreadydealt with the cases in which the eigenvalues of ρ ( φ ) are distinct. All other casesare dealt with by the following (after twisting and conjugating ρ ). Note that bylemma 3.7, the only possible types when ρ | ˜ P F is trivial are τ ζ,s , τ ζ,ns and τ ζ ,ζ for ζ any l a th root of unity and ζ , ζ any distinct l a th roots of unity. Proposition 5.8.
Suppose that q = 1 mod l and that ρ | ˜ P F is trivial. Suppose that ρ ( σ ) = (cid:18) x (cid:19) and ρ ( φ ) = (cid:18) y (cid:19) for x, y ∈ F . (1) If x = 0 then R (cid:3) ( ρ, τ ζ,s ) = 0 , while R (cid:3) ( ρ, τ ζ,ns ) and R (cid:3) ( ρ, τ ζ ,ζ ) areformally smooth over O of relative dimension 4.If a N is the four-dimensional prime of R (cid:3) ( ρ ) corresponding to R (cid:3) ( ρ, τ ,ns ) then we have: (14) Z ( R (cid:3) ( ρ, τ )) = if τ = τ ζ,s [ a N ] if τ = τ ζ,ns [ a N ] if τ = τ ζ ,ζ . (2) If x = 0 and y = 0 , then R (cid:3) ( ρ, τ ζ,s ) and R (cid:3) ( ρ, τ ζ,ns ) are formally smoothover O of relative dimension 4 while R (cid:3) ( ρ, τ ζ ,ζ ) ∼ = O [[ X , . . . , X ]] / ( X X − ( ζ − ζ ) ) is a complete intersection domain of relative dimension 4 over O .If a nr and a N are the prime ideals of R (cid:3) ( ρ ) corresponding to R (cid:3) ( ρ, τ ,s ) and R (cid:3) ( ρ, τ ,ns ) respectively, then (15) Z ( R (cid:3) ( ρ, τ )) = [ a nr ] if τ = τ ζ,s [ a N ] if τ = τ ζ,ns a nr ] + [ a N ] if τ = τ ζ ,ζ . (3) If x = y = 0 , then R (cid:3) ( ρ, τ ζ,s ) is formally smooth over O of relative dimen-sion 4, R (cid:3) ( ρ, τ ζ,ns ) is a non-Gorenstein Cohen–Macaulay domain of rela-tive dimension 4 over O , while R (cid:3) ( ρ, τ ζ ,ζ ) is a non-Gorenstein Cohen–Macaulay domain of relative dimension 4 over O .Both R (cid:3) ( ρ, τ ζ,s ) and R (cid:3) ( ρ, τ ζ,ns ) are domains; let the correspondingprimes of R (cid:3) ( ρ ) be a nr and a N respectively. Then (16) Z ( R (cid:3) ( ρ, τ )) = [ a nr ] if τ = τ ζ,s [ a N ] if τ = τ ζ,ns a nr ] + [ a N ] if τ = τ ζ ,ζ . OCAL DEFORMATION RINGS FOR GL Proof.
Write ρ (cid:3) ( σ ) = (cid:18) A x + BC D (cid:19) ρ (cid:3) ( φ ) = (cid:18) P y + RS Q (cid:19) with A, B, C, D, P, Q, R, S ∈ m and x, y lifts of x, y (taken to be zero if x or y = 0).First, we have that R (cid:3) ( ρ, τ ζ,s ) = 0 if x = 0 and R (cid:3) ( ρ, τ ζ,s ) ∼ = O [[ P, Q, R, S ]]otherwise.Next, we look at R (cid:3) ( ρ, τ ζ ,ζ ) for ζ and ζ distinct l a th roots of unity. Thecondition that ρ (cid:3) ( σ ) has characteristic polynomial ( t − ζ )( t − ζ ) is equivalent tothe equations A + D = ζ + ζ − AD − ( x + B ) C = ( ζ − ζ − . Since ( t − ζ )( t − ζ ) | t q − −
1, by the Cayley–Hamilton theorem we have ρ (cid:3) ( σ ) q = ρ (cid:3) ( σ )on R (cid:3) ( ρ, τ ζ ,ζ ) ◦ . So the relation φσφ − = σ q yields: (cid:18) A x + BC D (cid:19) (cid:18) P y + RS Q (cid:19) = (cid:18) P y + RS Q (cid:19) (cid:18) A x + BC D (cid:19) . Equating coefficients, eliminating D and writing U = P − Q and F = A − D = 2 A − ( ζ + ζ − R (cid:3) ( ρ, τ ζ ,ζ ) is the reduced, l -torsion–free quotient of O [[ B, C, F, P, R, S, U ]]by the relations: ( x + B ) S = ( y + R ) C (17) F ( y + R ) = U ( x + B )(18) F S = U C (19) ( ζ − ζ ) = F + 4( x + B ) C. (20)If x = 0 then these equations are equivalent to U = F ( y + R )( x + B ) − , C = (cid:0) ( ζ − ζ ) − F (cid:1) ( x + B ) − and S = C ( y + R )( x + B ) − , so that R (cid:3) ( ρ, τ ζ ,ζ ) ∼ = O [[ B, F, P, R ]] . If x = 0 and y = 0, then F = BU ( y + R ) − and C = BS ( y + R ) − will be asolution to the equations (17) to (20) provided that( ζ − ζ ) = (cid:18) By + R (cid:19) ( U + 4( y + R ) S );writing ( X , . . . , X ) = ( B ( y + R ) − , U + 4( y + R ) S, P, R, U ) we get R (cid:3) ( ρ, τ ζ ,ζ ) ∼ = O [[ X , . . . , X ]] X X − ( ζ − ζ )
26 JACK SHOTTON as claimed. The other statements about R (cid:3) ( ρ, τ ζ ,ζ ) follow easily.If x = y = 0, then let A = O [[ B, C, F, P, R, S, U ]] and I ⊳ A be the ideal: I = (cid:0) ( ζ − ζ ) − F − BC, BS − CR, F R − BU, F S − CU (cid:1) . Note that the ideal J = ( BS − CR, F R − BU, F S − CU )is generated by the 2 × (cid:18) B C FR S U (cid:19) . So, by proposition 2.7, A /J isa Cohen–Macaulay, non-Gorenstein domain. Since F − BC is not zero in thedomain A /J ⊗ F , ( λ, F − BC ) is a regular sequence in A /J . Hence ( F − BC − ( ζ − ζ ) , λ ) is a regular sequence in A /J , and therefore A /I is O -flat, Cohen–Macaulay and non–Gorenstein. It is reduced because it is Cohen–Macaulay and,as we shall show in the next paragraph, generically reduced.To show that A /I is irreducible, it suffices to show that X = Spec( A /I ⊗ E ) isirreducible. This follows if we can show that X is formally smooth and connected.As F − BC = 0 on X , it is covered by the affine open subsets U B = { B = 0 } and U F = { F = 0 } . By the argument used in the x = 0 case, U B is formally smooth. Asimilar argument works for U F : the projection map p : X →
Spec (cid:18) O [[ F, B, C, U, P ]]( F + 4 BC − ( ζ − ζ ) ) ⊗ E (cid:19) is an isomorphism from U F onto an open subscheme; but the right hand side iseasily seen to be formally smooth. Hence X is formally smooth. Note that thecomposition of the map p with the projection away from U is a continuous mapwith connected fibres and connected image, which admits a continuous section(obtained by taking R = S = U = 0); it follows that X is connected, as required.Since X is formally smooth it is certainly reduced; therefore A /I is genericallyreduced (as it is O -flat), just as we claimed above.Now we turn to R (cid:3) ( ρ, τ ζ,ns ). By lemma 2.4 we may assume that ζ = 1. Thecondition that the characteristic polynomial of ρ (cid:3) ( σ ) be ( t − is equivalent tothe equations: A + D = 0 AD − ( x + B ) C = 0 . Writing T = P + Q and U = P − Q , the condition that q tr( ρ (cid:3) ( φ )) = ( q + 1) det( ρ (cid:3) ( φ ))becomes ( q − ( T + 2) = ( q + 1) ( U + 4( y + R ) S ) . Since t q − ≡ q ( t −
1) mod ( t − , the Cayley–Hamilton theorem shows that ρ (cid:3) ( σ ) q − q ( ρ (cid:3) ( σ ) − R (cid:3) ( ρ, τ ,ns ). From φσφ − = σ q we therefore get the equation( φ − σ − − ( σ − φ −
1) = ( q − σ − φ OCAL DEFORMATION RINGS FOR GL on R (cid:3) ( ρ, τ ,ns ). Equating coefficients and substituting D = − A we get the equa-tions A + ( x + B ) C = 0(21) ( q − ( T + 2) = ( q + 1) ( U + 4( y + R ) S )(22) C ( y + R ) − S ( x + B ) = ( q − A (1 + P ) + ( x + B ) S )(23) U ( x + B ) − A ( y + R ) = ( q − A ( y + R ) + ( x + B )(1 + Q ))(24) 2 AS − CU = ( q − C (1 + P ) − AS )(25) S ( x + B ) − C ( y + R ) = ( q − C ( y + R ) − A (1 + Q )) . (26)After replacing P with T + U and Q with T − U , this is a complete set of equationsfor R (cid:3) ( ρ, τ ,ns ) in O [[ A, B, C, R, S, T, U ]].We replace equations (23) and (26) by their sum and difference:( q − AU + ( x + B ) S + C ( y + R )) = 0(27) ( q + 1)( C ( y + R ) − ( x + B ) S ) = ( q − A (2 + T ) . (28)As R (cid:3) ( ρ, τ ,ns ) is λ -torsion free, equation (27) implies that(29) AU + ( x + B ) S + C ( y + R ) = 0 . We could also write this equation as tr(( σ − φ ) = 0.Putting α ( T ) = ( q − T ) q +1 , we find that equations (21), (22), (24),(25) and [(28)and (29)] may respectively be rewritten: A + ( x + B ) C = 04( y + R ) S + ( U − α ( T ))( U + α ( T )) = 02 A ( y + R ) − ( x + B )( U − α ( T )) = 02 AS − C ( U + α ( T )) = 02 C ( y + R ) + A ( U − α ( T )) = 02( x + B ) S + A ( U + α ( T )) = 0 . Let I be the ideal of O [[ A, B, C, R, S, T, U ]] generated by these equations and let R ′ = O [[ A, B, C, R, S, T, U ]] /I , so that R (cid:3) ( ρ, τ ,ns ) is the maximal reduced l -torsionfree quotient of R ′ .If x = 0 then C , U and S are uniquely determined by A , B , R and T so that R (cid:3) ( ρ, τ ,ns ) ∼ = O [[ A, B, R, T ]] . If y = 0, then S , C and A are uniquely determined by B , R , T and U so that R (cid:3) ( ρ, τ ,ns ) ∼ = O [[ B, R, T, U ]] . If x = y = 0, so that x = y = 0, observe that R ′ ∼ = B J + J where B = O [[ X , . . . , X , Y , . . . , Y , T ]] , the ideal J is generated by the 2 × (cid:18) X X X X Y Y Y Y (cid:19) and J = ( X + Y , X − Y + 2 q − q +1 ). (The change of variables is X = A , X = B , Y = C , Y = − A , X = − R/ (2 + T ), Y = 2 S (2 + T ), Y = ( U − α ( T )) / (2 + T ),and X = ( U + α ( T )) / (2+ T ).) Then by proposition 2.7, B /J is a Cohen–Macaulay,non-Gorenstein domain. Moreover, ( λ, X + Y , X − Y ) may be checked to be aregular sequence on B /J . Therefore ( X + Y , X + Y + 2 q − q +1 , λ ) is also regular,and so B / ( J + J ) is Cohen–Macaulay, O -flat and not Gorenstein. The same isthen true for R ′ .We show that R ′ ⊗ F is a domain, which implies that R ′ is a domain. Let I bethe image of I in F [[ A, B, C, R, S, T, U ]]. Then I is homogeneous so gr( R ′ ⊗ F ) = F [ A, B, C, R, S, T, U ] /I and it suffices to check that this is a domain (by [Eis95]corollary 5.5). It is therefore sufficient to check that Proj(gr( R ′ ⊗ F )) is reducedand irreducible. But it is easy to check this on the usual seven affine pieces. Thisargument is from [Tay09].Next we show that R (cid:3) ( ρ, τ ,ns ) is reduced. In fact, we show that Y = Spec( R (cid:3) ( ρ, τ ,ns ) ⊗ E )is formally smooth, which implies that R (cid:3) ( ρ, τ ,ns ) is reduced because it is Cohen–Macaulay and O -flat. For ⋆ = B , C , R , S , U − α ( T ) or U + α ( T ) let U ⋆ = { ⋆ =0 } ⊂ Y be the corresponding affine open subscheme. Then the U ⋆ are an affineopen cover of Y . For ⋆ = B , C , R or S we see that U ⋆ is formally smooth by thesame argument as for the cases x = 0 and y = 0 above. For U U ± α ( T ) , the projectionmorphism p : U U − α ( T ) → Spec (cid:18) O [[ C, R, S, T ]]4 RS − ( U + α ( T ))( U − α ( T )) ⊗ E (cid:19) is an isomorphism onto an open subscheme. But the right hand scheme is easilyseen to be formally smooth as required.Finally we calculate the Z ( R ( ρ, τ )). We do this when x = y = 0, as the othercases are similar but easier. We have written each R (cid:3) ( ρ, τ ) as the quotient of F [[ A, B, C, R, S, T, U ]] by an ideal which we call I ( τ ). Let us recall the presenta-tions: I ( τ ζ,s ) = ( A, B, C ) I ( τ ζ,ns ) = ( A + BC, RS + U , CR + AU, BS + AU, AR − BU, AS − CU ) I ( τ ζ ,ζ ) = ( A + BC, BS − CR, AR − BU, AS − CU )(using that A + D = 0 in R (cid:3) ( ρ, τ ) for each τ , we have eliminated D and written F = A − D = 2 A ). We have already shown that I ( τ ζ,s ) and I ( τ ζ,ns ) are prime —they are the ideals denoted a nr and a N in the statement of the theorem. It is clearthat Z ( R (cid:3) ( ρ, τ ζ,s )) = [ a nr ]and Z ( R (cid:3) ( ρ, τ ζ,ns )) = [ a N ] . Suppose that p is a prime ideal of F [[ A, B, C, R, S, T, U ]] containing I ( τ ζ ,ζ ). Weshow that p contains a nr or a N . If B, C ∈ p then A ∈ p as A + BC ∈ I ( τ ζ ,ζ ),and we have a nr ⊂ p . Otherwise, suppose that B p . As A + BC ∈ p , either both A and C are in p or neither is. If A, C ∈ p then from 2 AR − BU ∈ p we deducethat U ∈ p , while from BS − CR ∈ p we deduce that S ∈ p . It is then easy to seethat a N ⊂ p . If A, B, C p then because B (2 CR + AU ) and C (2 BS + AU ) are in OCAL DEFORMATION RINGS FOR GL I ( τ ζ ,ζ ) we see that 2 CR + AU, BS + AU ∈ p . This implies that A (4 RS + U ) ∈ p ,and so 4 RS + U ∈ p and hence a N ⊂ p as required.To finish, it is easy to check that e ( R (cid:3) ( ρ, τ ζ ,ζ ) , a nr ) = 2and that e ( R (cid:3) ( ρ, τ ζ ,ζ ) , a N ) = 1 , and so we get equation 16. (cid:3) Cohen–Macaulayness. If τ is a semisimple representation of I F over E , let R ( ρ, τ ) ′ be the maximal reduced and l -torsion–free quotient of R ( ρ ) all of whose E -points give rise to representations ρ of G F with ρ | ssI F ∼ = τ . Then I claim that R ( ρ, τ ) ′ is always Cohen–Macaulay. Indeed, if τ is non-scalar then we have provedthis above. If τ is scalar and q = − l , then we can deduce the claim fromthe above calculations together with exercise 18.13 of [Eis95], which says that if R/I and
R/J are d -dimensional Cohen–Macaulay quotients of a noetherian localring R , and dim R/ ( I + J ) = d −
1, then R/ ( I ∩ J ) is Cohen–Macaulay. When q = − l the claim follows from the above calculations unless ρ is the directsum of the trivial and cyclotomic characters, in which case we use remark 5.7.For n -dimensional representations the unrestricted framed deformation ring R (cid:3) ( ρ )is always Cohen–Macaulay (in fact, a complete intersection; this is due to DavidHelm, building on work of Choi [Cho09]). It is natural to wonder whether therings obtained by fixing the semisimplified restriction to inertia are always Cohen–Macaulay. Note that they are not always Gorenstein.For a discussion of how the Cohen–Macaulay property of local deformation ringscan be used to show that certain global Galois deformation rings are flat over O ,see section 5 of [Sno11].6. Reduction of types – proofs.
The aim of this section is to analyse the reduction modulo l of the K -types σ ( τ )defined in section 3, and in particular to prove lemma 3.9.6.1. The essentially tame case.
Suppose that τ = ( r τ , N τ ) where r τ is a tamelyramified, semisimple representation of I F . Then σ ( τ ) is inflated from a represen-tation of GL ( k F ). We will always use the same notation for a representation of GL ( k F ) and its inflaton to GL ( O F ). For this subsection let G = GL ( k F ), let B be the subgroup of upper-triangular matrices, let U be the subgroup of unipotentelements of B , let Z be the center of G and fix an embedding α : k × L ֒ → G . Fix anon-trivial additive character ψ of U . Then we have (see e.g. [BH06] chapter 6): • If r τ = (rec( ˜ χ ) ⊕ rec( ˜ χ )) | I F and N τ = 0, where ˜ χ | O × F is inflated from acharacter χ of k × F , then σ ( τ ) = ( χ ◦ det) ⊗ St , where St is the Steinberg representation of G ; • If r τ = (rec( ˜ χ ) ⊕ rec( ˜ χ )) | I F and N τ = 0, where ˜ χ | O × F is inflated from acharacter χ of k × F , then σ ( τ ) = χ ◦ det; • If r τ = (rec( ˜ χ ) ⊕ rec( ˜ χ )) | I F , where ˜ χ | O × F and ˜ χ | O × F are inflated from distinct characters χ and χ of k × F , then σ ( τ ) = µ ( χ , χ )where µ ( χ , χ ) = Ind GB ( χ ⊗ χ ); • If r τ = (Ind G F G L rec(˜ θ )) | I F where ˜ θ | O × L is inflated from a character θ of k × L which is not equal to its Gal( k L /k F ) conjugate θ c , then σ ( τ ) = π θ where π θ = Ind GZU ( θ | Z ψ ) − Ind Gα ( k × L ) θ (this virtual representation is a gen-uine irreducible representation that is independent of the choice of ψ ).The only isomorphisms between these representations are of the form µ ( χ , χ ) ∼ = µ ( χ , χ ) and π θ ∼ = π θ c .We want to understand the reductions of these representations modulo l , andfor this see [Hel10]. We will use analagous notation for representations of G incharacteristic zero and in characteristic l ; hopefully this will not cause confusion.If q = ± l , then reduction modulo l is a bijection between irreducible F l -representations of G and irreducible E -representations of G , as G has order q ( q + 1)( q − which is coprime to l .If q = 1 mod l , then the distinct irreducible representations of GL ( k F ) over F are χ ◦ det and St ⊗ ( χ ◦ det) for χ : k × F → F × , µ ( χ , χ ) for χ , χ : k × F → F × a pairof distinct characters, and π θ for θ : k × L → F × character which is not isomorphic toits conjugate. The notation is all entirely analagous to the characteristic zero case.Once again, the only isomorphisms are µ ( χ , χ ) ∼ = µ ( χ , χ ) and π θ ∼ = π θ c . Thereductions of the characteristic zero representations are: • χ ◦ det = χ ◦ det; • St ⊗ χ ◦ det = St ⊗ ( χ ◦ det); • µ ( χ , χ ) = µ ( χ , χ ) if χ = χ ; • µ ( χ , χ ) = ( χ ◦ det) ⊕ St ⊗ ( χ ◦ det) if χ = χ = χ ; • π θ = π θ .For the last of these, we must observe that θ/θ c is a character of k × L /k × F , a groupwhich has order q + 1 and so coprime to l (as l > θ = θ c then θ = θ c .If q ≡ − l , then the distinct irreducible representations are: χ ◦ det for χ : k × F → F × , µ ( χ , χ ) for χ , χ : k × F → F × unordered pair of distinct characters, π θ for θ : k × L → F × a character which is not isomorphic to its conjugate, and( χ ◦ det) ⊗ π for χ : k × F → F × a character. This last needs some explanation: π isthe reduction modulo l of π θ for any character θ : k × L /k × F → E × which is not equalto θ c but whose reduction modulo l is trivial. Once again, the only isomorphismsare µ ( χ , χ ) ∼ = µ ( χ , χ ) and π θ ∼ = π θ c . The reductions of the characteristic 0representations are: • χ ◦ det = χ ◦ det; • µ ( χ , χ ) = µ ( χ , χ ); • π θ = π θ if θ = θ c ; • π θ = π ⊗ ( θ | k × F ◦ det) if θ = θ c ; OCAL DEFORMATION RINGS FOR GL • St ⊗ ( χ ◦ det) has π ⊗ ( χ ◦ det) as a submodule with quotient χ ◦ det.In particular, comparing this analysis with lemma 3.8 shows that: Lemma 6.1. If τ = ( r, and τ ′ = ( r ′ , are scalar on P F but not on ˜ P F , then σ ( τ ) and σ ( τ ′ ) are irreducible and are isomorphic if and only if r ≡ r ′ mod l . The wild case. If τ = ( r,
0) and all twists of r are wildly ramified (we saythat τ is ‘essentially wildly ramified’), then the following lemma will allow us toshow that σ ( τ ) is irreducible. If ρ is a Z l -representation of a group H , we write ρ for ρ ⊗ F l . Lemma 6.2.
Suppose that H ⊳ J ⊂ K are profinite groups such that H is open in K , H has pro-order coprime to l , and J/H is an abelian l -group. Suppose that λ is a Z l -representation of J , and write η for the restriction of λ to H . Suppose that η (and hence λ ) is irreducible. Suppose that if g ∈ K intertwines η , then g ∈ J .Then (1) The representations of J extending η are precisely λ i = λ ⊗ ν i as ν i runthrough the characters of J/H . There is an isomorphism
Ind JH η ⊗ E ∼ = L i λ i . The unique F l -representation extending η is λ , and all of the Jordan–H¨older factors of Ind JH η are isomorphic to λ . (2) A F l -representation ρ of J contains λ as a subrepresentation if and only ifit contains λ as a quotient. (3) The representations
Ind KJ λ i and Ind KJ λ are irreducible.Proof. (1) In characteristic 0 we argue as follows. First note that the rep-resentations λ i are distinct, otherwise λ | H would have a non-scalar en-domorphism, contradicting Schur’s lemma. By Frobenius reciprocity, the λ i are distinct irreducible constituents of Ind JH η . Since the sum of theirdimensions is dim Ind JH η , they are the only irreducible constituents. ByFrobenius reciprocity, any representation extending η must occur in Ind JH η and so must be one of the λ i , as required. In characteristic l , first note that λ is irreducible since the pro-order of H is coprime to l . It follows fromthis and the fact that ν i is trivial for all i that the Jordan–H¨older factorsof Ind JH η are isomorphic to λ . Frobenius reciprocity then implies that λ isthe unique irreducible representation of J extending H .(2) It follows from part 1 that Hom J ( λ, ρ ) = 0 if and only if Hom J (Ind JH η, ρ ) =0. By Frobenius reciprocity, this is equivalent to Hom H ( η, ρ ) = 0. But bythe assumption on the pro-order of H , F l -representations of H are semisim-ple, and so this is equivalent to Hom H ( ρ, η ) = 0, which by the same argu-ment is equivalent to Hom J ( ρ, Ind JH η ) = 0.(3) First, note that dim Hom K (Ind KJ λ, Ind KJ λ ) = 1, by Mackey’s decomposi-tion formula and the assumption that elements of K \ J do not intertwine η . Now suppose that ρ is an irreducible subrepresentation of Ind KJ λ . ByFrobenius reciprocity and part 2 we may deduce that ρ is also an irreduciblequotient of Ind KJ λ . The compositionInd KJ λ ։ ρ ֒ → Ind KJ λ is then a non-zero element of Hom K (Ind KJ λ, Ind KJ λ ), and is therefore scalar.But this is only possible if ρ = Ind KJ λ , as required. The statement aboutInd KJ λ i follows. (cid:3) Proposition 6.3.
Let τ = ( r, be an essentially wildly ramified inertial type.Then there exists a subgroup J ⊂ K , an irreducible representation λ of J , anda subgroup ˜ J ⊳ J , such that ( ˜ J , J, K, λ ) satisfy the hypotheses on ( H, J, K, λ ) inlemma 6.2 and such that σ ( τ ) = Ind KJ λ .In particular, σ ( τ ) is irreducible.Proof. Suppose first that r is the restriction to I F of a reducible representation of G F . Then σ ( τ ) = Ind KK ( N ) ǫ ⊗ ( χ ◦ det) for a character ǫ of O × F of exponent N ≥ χ of O × F . Let J = K ( N ), and let˜ J = (cid:26)(cid:18) a bc d (cid:19) ∈ J : a has order coprime to l modulo p F (cid:27) . Then ˜ J , J and ǫ satisfy all the required hypotheses — the only one to check is that ǫ | ˜ J is not intertwined by any element of K \ J . We deduce this (in somewhat circularfashion) from the irreducibility of Ind KJ ( ǫ ), since this is shorter than a direct proof.If g ∈ K intertwines ǫ | ˜ J , then Hom ˜ J ∩ g ˜ Jg − ( ǫ, ǫ g ) = 0. By Mackey’s formula,dim Hom ˜ J ( ǫ, Ind K ˜ J ǫ ) = X g ∈ ˜ J \ K/ ˜ J dim Hom ˜ J ∩ g ˜ Jg − ( ǫ, ǫ g ) . The left hand side is in turn equal to dim Hom K (Ind K ˜ J ǫ, Ind K ˜ J ǫ ). But Ind K ˜ J ǫ = L i Ind KJ ǫ i where ǫ i are the characters of J extending ǫ | ˜ J , and by the appendixto [BM02], these Ind KJ ǫ i are irreducible and distinct. Therefore the left hand sideis equal to ( J : ˜ J ). The right hand side has a contribution of 1 from each g ∈ J/ ˜ J ,and therefore from no other g , as required.Now suppose that r is the restriction to I F of an irreducible representationof G F . Then σ ( τ ) = Ind KJ λ for an irreducible representation λ of J extendingan irreducible representation η of a pro- p normal subgroup J of J (see [BH06],sections 15.5, 15.6 and 15.7 — note that our J is the maximal compact subgroupof their J α , but our J agrees with their J α ). We have J/J = k × , where k is theresidue field of a quadratic extension of F , and so J has a normal subgroup ˜ J ofpro-order coprime to l such that J/ ˜ J is an l -group. Then ( ˜ J, J, K, λ ) satisfy allthe required hypotheses — the intertwining statement follows from [BH06], 15.6Proposition 2. (cid:3)
Proposition 6.4.
Let τ = ( r, and τ ′ = ( r ′ , be inertial types that are not scalaron ˜ P F . If r ≡ r ′ mod l , then σ ( τ ) and σ ( τ ′ ) are isomorphic.Proof. If either of r and r ′ is (after to a twist) tamely ramified, then so is the otherand this is contained in lemma 6.1. Otherwise, by lemma 3.8, we are in one of thefollowing cases:(1) r = ( χ ⊕ χ ) | I F for characters χ and χ of G F that are distinct on P F ,and r ′ = ( χ ′ ⊕ χ ′ ) | I F for characters χ ′ and χ ′ of G F with χ i ≡ χ ′ i for i = 1 , r = (Ind G F G L ξ ) I F and r ′ = (Ind G F G L ξ ′ ) I F for wildly ramified characters ξ and ξ ′ of G L such that ξ ≡ ξ ′ , and such that ξ | ˜ P F does not extend to G F .(3) r | ˜ P F is irreducible and r ′ = r ⊗ χ for a character χ of I F that extends to G F and such that χ ≡ l . OCAL DEFORMATION RINGS FOR GL In the first case, we may write χ i = rec( ǫ i ) and χ ′ i = rec( ǫ ′ i ) with ǫ i and ǫ ′ i characters of F × such that ǫ i ≡ ǫ ′ i mod l and such that ǫ = ǫ /ǫ has exponent N ≥
1. Since ǫ ′ = ǫ ′ /ǫ ′ also has exponent N , we have σ ( τ ) = ǫ ⊗ Ind KK ( N ) ǫ ≡ ǫ ′ ⊗ Ind KK ( N ) ǫ ′ mod l = σ ( τ ′ ) . In the second case, by twisting we may reduce to the case where (
L/F, rec − ( ξ ))is an unramified minimal admissible pair ( [BH06] paragraph 19.6). Then, followingthrough the explicit construction of [BH06] paragraphs 19.3 and 19.4, we see thatthere are:(1) a simple stratum ( A , n, α ) with associated compact open subgroups J ⊂ J ⊂ K , with J pro- p and J/J ∼ = k × L ;(2) a representation η of J and extensions λ and λ ′ of η to J such thatInd KJ ( λ ) = σ ( τ ) and Ind KJ ( λ ′ ) = σ ( τ ′ ).Indeed, up to conjugacy ( A , n, α ), J and η are determined by rec − ( ξ ) | U L =rec − ( ξ ′ ) | U L . The representations λ and λ ′ are defined in terms of rec − ( ξ ) andrec − ( ξ ′ ) by the formulae of [BH06] 19.3.1 and corollary 19.4 (together with the cor-rection factor of paragraph 34.4, an unramified twist ∆ ξ , that makes no differenceto the argument). It is clear from these that if ξ ≡ ξ ′ then λ ≡ λ ′ as required.In the final case, r ′ = r ⊗ χ for a character χ of I F that extends to G F . Bycompatibility of τ σ ( τ ) with twisting, σ ( τ ′ ) = σ ( τ ) ⊗ rec − ( χ ) ◦ det ≡ σ ( τ ) mod l as required. (cid:3) Proposition 6.5.
Let τ = ( r, and τ ′ = ( r ′ , be inertial types that are not scalaron ˜ P F . If σ ( τ ) and σ ( τ ′ ) are isomorphic, then r ≡ r ′ mod l .Proof. If one of r and r ′ has a twist which is trivial on P F , then so does the otherand in this case the proposition follows from 6.1.Otherwise may, by twisting, assume that σ ( τ ) and σ ( τ ′ ) satisfy l ( σ ) ≤ l ( σ ⊗ χ )for all characters χ of O × F (the definition of l ( σ ) is the as in [BH06] paragraph 12.6).In this case σ ( τ ) and σ ( τ ′ ) contain the same, non-empty, sets of fundamental strata(because this only depends on the restriction to pro- p subgroups).If one of σ ( τ ) and σ ( τ ′ ) contains a split fundamental stratum ( [BH06] 13.2) thenso does the other. In this case, [BH06] corollary 13.3 implies that they cannot becuspidal types and so we must have σ ( τ ) = Ind KK ( N ) ( ǫ ) and σ ( τ ′ ) = Ind KK ( N ′ ) ( ǫ ′ )for some ǫ and ǫ ′ of exponents N and N ′ . It is easy to see that in fact we musthave N = N ′ . From lemma 6.2 we deduce that ǫ ≡ ǫ ′ mod l , and so τ ≡ τ ′ mod l as required.Otherwise, σ ( τ ) = Ind KJ λ and σ ( τ ′ ) = Ind KJ λ ′ for a simple stratum ( A , n, α )with associated groups J ⊂ J and representations λ and λ ′ extending the repre-sentation η of J . From lemma 6.2 we deduce that λ ′ = λ ⊗ η for a character η of J/J with η ≡ l . If A is unramified, then by the reverse of the argument in the second case of theprevious proposition we see that τ = (Ind G F G L ξ ) | I F and τ = (Ind G F G L ξ ) | I F for ξ and ξ ′ characters of G L with ξ | I L ≡ ξ ′ | I L , whence the result.If A is ramified, then η can be regarded as a character of J/J ∼ = k × M = k × F with η ≡ l for some ramified quadratic extension M/F . I claim that there is acharacter χ of O × F with η = χ ◦ det and χ ≡ l . Indeed, as l > O × F of the character χ of k × F satisfying χ ≡ l and χ = η .Then σ ( τ ) = σ ( τ ′ ) ⊗ ( χ ◦ det) and so τ = τ ′ ⊗ rec( χ ) ≡ τ ′ mod l as required. (cid:3) References [BH06] Colin J. Bushnell and Guy Henniart,
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