LLifting non-ordinary cohomology classes for SL
Abstract
In this paper, we present a generalisation of a theorem of David and Rob Pol-lack. In [PP09], they give a very general argument for lifting ordinary eigenclasses(with respect to a suitable operator) in the group cohomology of certain arithmeticgroups. With slightly tighter conditions, we prove the same result for non-ordinaryclasses. Pollack and Pollack apply their results to the case of p -ordinary classesin the group cohomology of congruence subgroups for SL , constructing explicitoverconvergent classes in this setting. As an application of our results, we givean extension of their results to the case of non-critical slope classes in the samesetting. Introduction
Background
Modular symbols are cohomological objects that are powerful computational and theo-retical tools in the study of automorphic forms. Classical modular symbols are elementsin the cohomology of a locally symmetric space with coefficients in some polynomialspace, and in many cases, there are ways of viewing such elements in the group coho-mology of certain arithmetic subgroups. For example, to a modular form of weight k andlevel Γ ( N ), one can attach an element of the group cohomology H (Γ ( N ) , V k − ( C )),where V k − ( C ) is the space of homogeneous polynomials in two variables over C of degree k −
2. These cohomology groups are equipped with an action of the Hecke operators,and the association of a modular symbol to an automorphic form respects this action.In [Ste94], Glenn Stevens developed the theory of overconvergent modular symbols byreplacing the space of polynomials with a much larger space, that of p -adic distributions .There is a surjective Hecke-equivariant map from this space to the space of classical mod-ular symbols (with p -adic coefficients). As a map from an infinite dimensional space toa finite dimensional space, this ‘specialisation map’ must necessarily have infinite di-mensional kernel, but in the same preprint, Stevens proved his control theorem , whichsays that upon restriction to the ‘small slope eigenspaces’, this specialisation map infact becomes an isomorphism. This control theorem – an analogue of Coleman’s smallslope classicality theorem – has had important ramifications in number theory, beingused to construct p -adic L -functions (see [PS11] and [PS12]) and Stark-Heegner pointson elliptic curves (see [Dar01] and [DP06]).Such control theorems have now been proved in a variety of other cases, including –but certainly not limited to – for compactly supported cohomology classes attached toHilbert modular forms by Daniel Barrera Salazar in [BS15], for compactly supported Mathematics Subject Classification : 11F75 (primary), 11F85 (secondary) a r X i v : . [ m a t h . N T ] J u l ifting non-ordinary cohomology classes for SL Chris Williams cohomology classes attached to Bianchi modular forms in [Wil17], and for ordinary co-homology classes attached to automorphic forms for SL by David and Robert Pollackin [PP09]. In the latter, Pollack and Pollack gave a very general argument for explicitlylifting group cohomology eigenclasses (of a suitable operator) in the ordinary case, thatis, when the corresponding eigenvalue is a p -adic unit. This general lifting theorem hasbeen used in a variety of other settings, including in the work of Xevi Guitart and MarcMasdeu in the explicit computation of Darmon points (see [GM14]).Whilst control theorems do exist in wide generality – for example, Eric Urban hasproved a control theorem for quite general reductive groups in [Urb11] – these theoremsare rarely constructive when we pass beyond the ordinary case. In this note, we gener-alise the (constructive) lifting theorem of Pollack and Pollack to non-ordinary classes.To do this, we use an idea of Matthew Greenberg in [Gre07], which the author foundinvaluable in developing the theory of overconvergent modular symbols over imaginaryquadratic fields.In the remainder of the paper, we give an application of this theorem. In particular, wegive an extension of the results of Pollack and Pollack over SL to explicitly constructoverconvergent eigenclasses in the non-critical slope case. There are subtleties in thissituation that do not need to be considered in the ordinary case; in particular, whilstPollack and Pollack lift with respect to the operator U p induced by the element π .. = p
00 0 p , we instead consider the two elements π = p
00 0 p , π = p , with π π = π . These induce commuting operators U p, and U p, on the cohomologywith U p, U p, = U p . We then lift twice; once with respect to the operator U p, to amodule of ‘partially’ overconvergent coefficients, then with respect to the operator U p, to the module of fully overconvergent coefficients used by Pollack and Pollack. In eachcase, we get a notion of ‘non-critical slope’, and by combining these two notions we geta larger range of ‘non-criticality’ than if we had just considered the operator U p . This issimilar in spirit to the results of [Wil17], Section 6, where control theorems are provedfor GL over an imaginary quadratic field in which the prime p splits as pp . This is doneby lifting first to a module of half-overconvergent coefficients with respect to U p , thento a module of fully overconvergent coefficients with respect to U p .We give a very brief summary of the results in the case of SL . First, we summarise theset-up: Notation 0.1: (i) Let λ = ( k , k ,
0) be a dominant algebraic weight of the torus T ⊂ GL / Q , and let Γ ⊂ Γ ( p ) be a congruence subgroup of SL .(ii) Let L/ Q p be a finite extension with ring of integers O L .(iii) Let V λ ( O L ) be the (finite-dimensional) space of classical coefficients over O L , tobe defined in Section 4.2.(iv) Let V ?λ denote V λ with a twisted action, as defined in Definition 4.12.2 ifting non-ordinary cohomology classes for SL Chris Williams (v) Let D λ ( O L ) be the (infinite-dimensional) space of overcovergent coefficients over O L , to be defined in Section 4.3.(vi) Let ρ λ : H r (Γ , D λ ( O L )) → H r (Γ , L λ ( O L ) be the specialisation map on the coho-mology at λ , where L λ ( O L ) .. = Im( D λ ( O L )) ⊂ V ?λ ( L ) is the image of specialisationon the coefficients, to be defined in Section 4.4.2.Then, in Theorem 5.13, we prove: Theorem.
Suppose α , α ∈ O L with v p ( α ) < k − k + 1 and v p ( α ) < k + 1 . Thenthe restriction ρ λ : H r (Γ , D λ ( O L )) U p,i = α i ∼ −→ H r (Γ , L λ ( O L ) U p,i = α i of the specialisation map to the simultaneous α i -eigenspaces of the U p,i operators is anisomorphism. Figure 1:
Graphic showing range of lifting for fixed k = k and varying k (with dottedline v p ( α ) + v p ( α ) = k + 2 ) Summary of argument
We give a brief summary of the argument we use to prove the general lifting theorem.The major component in the proof is showing that the specialisation map is surjective,in the process constructing an explicit lift of any element of the target space. Supposewe start with spaces D and V , with actions of a group Γ and an operator U , and supposethat U also acts naturally on the group cohomology of these spaces. Suppose moreoverthat we have a surjection pr : D → V that is equivariant with respect to the actionof Γ and U , inducing a map ρ on the cohomology. We also assume that we can find afiltration D ⊃ F D ⊃ F D ⊃ · · · such that if we define A N D .. = D/F N D , then we have A D = V . We also suppose that, among other conditions, we have D ∼ = lim ← A N D .We then start with a U -eigenclass φ ∈ H (Γ , A D ) with eigenvalue α . Further assumethat α is an algebraic integer (and hence can be thought of as living in the ring ofintegers of a finite extension of Q p ). 3 ifting non-ordinary cohomology classes for SL Chris Williams (i) First suppose that φ is ordinary at p , that is, suppose α is a p -adic unit. Thenwe take a cocycle ϕ representing φ , and lift it to any cochain f ϕ : Γ → D .As α is a unit, we can apply the operator α − U to this cochain. The magic isthat ϕ = f ϕ | α − U (mod F D ) is an A D -valued cocycle that is independent ofchoices, and thus defines a canonical lift of φ to a U -eigensymbol φ ∈ H (Γ , A D ).Continuing in this vein, we get compatible classes φ N ∈ H (Γ , A N D ) for each N ,and thus an eigenclass in the inverse limit Φ ∈ H (Γ , D ) that maps to φ under ρ .(ii) For more general α , we need a subtler argument. We would like to be able toapply the operator α − U , but since α need not be a unit, we must strengthen ourassumptions. In particular, we need the following:(a) A stronger condition on the filtration; namely, if µ ∈ F N D, then µ | U ∈ αF N +1 D. (b) An additional piece of data; namely, a Γ- and U -stable submodule D α of D such that if µ ∈ D α , we have µ | U ∈ αD .The benefit of this is that we can make sense of the operator α − U on cochainsthat have values in D α . We can run morally the same argument as above inthis case. Unfortunately, the details of the argument become considerably moretechnical.It is natural to ask when such conditions are satisfied. Condition (b) is relatively weak,and it seems reasonable to expect that a submodule D α satisfying this condition existsin wider generality; in particular, when D is a module of p -adic distributions on a finitenumber of variables, D α can be defined by imposing a simple condition on the low degreemoments. Condition (a), however, is stronger, and leads to the notion of small slope .To illustrate this, consider the following examples of cases where such filtrations exist: • One can find suitable filtrations in the cases of modular symbols attached to mod-ular forms of weight k + 2 over Q (see [Gre07]). In this case, condition (a) issatisfied only if v p ( α ) < k + 1, that is, if the modular form has small slope at p . • A similar result is given for modular forms over an imaginary quadratic field K in [Wil17]. In the case of weight ( k, k ), and p O K = pp split, the naturalfiltrations for U p and U p satisfy condition (a) (with respect to α p and α p ) onlyif v p ( α p ) , v p ( α p ) < k + 1 . A more detailed description of these results is given inSection 3.
Structure
In the first section, we describe the set-up of the theorem and the precise properties werequire of our filtrations. In the second, we give a proof of our main theorem. In thethird, we summarise the case of GL over an imaginary quadratic field. In the fourth,we set up the case of SL by giving the relevant definitions of the various coefficientspaces and specialisation maps, and finally, in the fifth section, we define the filtrationswe require in this case before stating the results for SL . Acknowledgements
I would like to thank Marc Masdeu for encouraging me to publish the lifting theoremcontained in this paper. I am also indebted to David Loeffler, as ever, for his veryhelpful suggestions as I worked on applying this theorem to the case of non-criticalslope eigenclasses for SL . 4 ifting non-ordinary cohomology classes for SL Chris Williams
The author was supported by an EPSRC DTG doctoral grant at the University ofWarwick.
1. Setup
Notation 1.1:
Suppose that we have:(i) A monoid Σ,(ii) A group Γ ≤ Σ,(iii) A ring R and a right R [Σ]-module D ,(iv) An R [Σ]-stable filtration of D , given by D ⊃ F D ⊃ F D ⊃ · · · , such that if wedefine A N D .. = D/ F N D , then we havelim ←− A N D = D, and where the F N D have trivial intersection, and(v) For some fixed α ∈ R , a right Σ-stable submodule D α of D , with V α .. = Im( D α →A D ) . Note that for each γ ∈ Σ such that Γ and γ − Γ γ are commensurable, and any Γ-module D , we have an operator U γ on the cohomology group H r (Γ , D ) defined in the usual way,that is, by the composition of the mapsH r (Γ , D ) res −−−−−→ H r (Γ ∩ γ − Γ γ, D ) γ −−−−−→ H r (Γ ∩ γ Γ γ − , D ) cores −−−−−→ H r (Γ , D ) . Theorem 1.2.
Suppose that α is a non-zero element of R , that D α and V α and theircorresponding cohomology groups have trivial R -torsion, and that for some π ∈ Σ , wehave(a) If µ ∈ D α , then µ | π ∈ αD , and(b) If µ ∈ F N D, then µ | π ∈ α F N +1 D. Then the restriction of the natural map ρ : H r (Γ , D α ) → H r (Γ , V α ) to the α -eigenspacesof the U π operator is an isomorphism. Remark:
This result is very similar to Theorem 3.1 of [PP09]; their conditions areslightly weaker, but their conclusion requires α to be a unit. In their case, they do notrequire the condition on trivial R -torsion, and then prove that there is a unique eigenliftΦ of an eigensymbol φ that has Ann R (Φ) = Ann R ( φ ). For simplicity, we have imposedthis condition to ensure these annihilators are trivial. In the cases we consider, theseconditions are satisfied.We have natural Σ-equivariant projection mapspr N : D −→ A N D that induce Σ-equivariant maps ρ N : H r (Γ , D ) −→ H r (Γ , A N D ) , ifting non-ordinary cohomology classes for SL Chris Williams (and hence ρ .. = ρ : H r (Γ , D α ) → H r (Γ , V α ) by restriction) as well as maps pr M,N : A M D → A N D for M ≥ N that similarly induce maps ρ M,N . Thus we have an inversesystem, and we have lim ←− H r (Γ , A N D ) = H r (Γ , D ) . First we pass to a filtration where the Σ-action is nicer. Define F N D α = F N D ∩ D α . Thisis a Σ-stable filtration of D α , since D α is Σ-stable. It’s immediate that if µ ∈ F N D α , then µ | π ∈ α F N +1 D α . Define A N D α = D α / F N D α , so that we have the following(where the vertical maps are injections): D π M > A M D π
M,N > A N DD α ∧ π M > A M D α ∧ π M,N > A N D α . ∧ Again, we see that lim ←− H r (Γ , A N D α ) = H r (Γ , D α ) . (1) Notation: (The U operator at the level of cochains). In [PP09], a description of the U = U π operator at the level of cochains is given. In particular, they take an explicitfree resolution · · · δ −−−−−→ F δ −−−−−→ F δ −−−−−→ F d −−−−−→ Z −→ Z by right Z [Γ]-modules; then, for a right Z [Γ]-module D , they use this to explic-itly write down the spaces C r (Γ , D ) .. = Hom Γ ( F r , D ) of cochains, Z r (Γ , D ) .. = Ker( d r : C r (Γ , D ) → C r +1 (Γ , D ) of cocycles, and B r (Γ , D ) .. = d r − (Γ , D ) of coboundaries, where d r is the obvious map induced by δ r . Then the group cohomology is defined as H r (Γ , D ) .. = Z r (Γ , D ) /B r (Γ , D ).Now, F π ∗ → Z → Z [ π − Γ π ]-modules, and as F ∗ → Z → Z [ π − Γ π ]-modules, there is a Z [ π − Γ π ]-complex map τ ∗ from F ∗ to F π ∗ lifting the identity map on Z .Pick a set { π i } of coset representatives for Γ π in Γ π Γ, noting that this is finite bycommensurability. Then define U : Hom( F r , D ) → Hom( F r , D ) at the level of cochainsby ( ϕ | U )( f r ) .. = X i ϕ ( τ r ( f r · π − i )) · ππ i , ϕ ∈ Hom( F r , D ) , f r ∈ F r . Pollack and Pollack prove (in Lemma 3.2) that this induces a map of chain complexesand hence a map of cohomology groups. In fact, this map is nothing other than U π asdefined above. Definition. ( U -eigensymbols of eigenvalue α ). Since A N D α may have non-trivial α -torsion, we should make the statement “ U π -eigensymbol in H r (Γ , A N D α )” more precise.By condition (a) of 1.2, if µ ∈ D α , then µ | π ∈ αD . We can thus consider π as a mapfrom D α to D in a natural way, and define another map V π from D α to D by setting x | V π = y, where x | π = αy. ifting non-ordinary cohomology classes for SL Chris Williams
We see we have a formal equality of maps αV π = π from D α to D . Thus we get anoperator V .. = V π : H r (Γ , D α ) −→ H r (Γ , D )on the cohomology, so that we have an equality of operators αV = U as operators onH r (Γ , D α ). There is also a canonical operator ε : H r (Γ , D α ) −→ H r (Γ , D )induced by the inclusion D α → D . We see that if φ ∈ H r (Γ , D α ) satsifies φ | U = αφ ,then ε ( φ ) = φ | V as elements of H r (Γ , D ). Remark:
The reason we don’t simply just define V = α − U π is that ‘dividing by α ’ isnot in general a well-defined notion on D .It is easy to see that for each N , V gives rise to an operator V N : A N D α → A N D .Denote the canonical map H r (Γ , A N D α ) → H r (Γ , A N D ) by ε N . We say an element ϕ N ∈ H r (Γ , A N D α ) is a U -eigensymbol of eigenvalue α if ε N ( ϕ N ) = ϕ N | V N as elementsof H r (Γ , A N D ). Henceforth, when we talk about U -eigensymbols, it shall be assumedthat the eigenvalue is α .
2. Proof of Theorem 1.2
Proof. (Theorem 1.2). We first prove surjectivity. Take a U -eigensymbol φ of eigen-value α in H r (Γ , V α ) = H r (Γ , A D α ). Suppose there exists a lift φ N ∈ H r (Γ , A N +1 D α )of φ to a U -eigensymbol for some N . We prove that we can canonically lift φ N to some φ N +1 , and thus we will be done by induction and equation (1), as we have constructedan element in the inverse limit. We prove this in a series of claims.Take a cocycle ϕ N representing φ N , and lift it to a cochain ϕ ∈ C Γ ( F r , D α ). Weapply V at the level of cochains, obtaining a cochain ϕ | V : F n → D . Define a cochain τ N +1 : F n −→ A N +1 D by composing this with the reduction map. This is in fact a cocycle; as ϕ N is a cocycle, dϕ takes values in F N D α , and thus as we have d ( ϕ | V ) = ( dϕ ) | V taking values in F N +1 D (by properties of V ), it follows that dτ N +1 = 0. Thus τ N +1 represents some cohomologyclass [ τ N +1 ] D ∈ H r (Γ , A N +1 D ) . Claim 2.1.
The cohomology class [ τ N +1 ] D is independent of choices.Proof. Suppose we take a different cochain e ϕ lifting a different cocycle f ϕ N to a cochaintaking values in D α . Then [ ρ N ( ϕ − e ϕ )] D α = 0, where ρ N is the natural reduction map,as ϕ N and f ϕ N both represent φ N . Thus [ ϕ − e ϕ ] D α ∈ H r (Γ , D α ) is represented by acocycle ψ taking values in F N D α . Therefore [ ϕ − e ϕ ] D α | V is represented by ψ | V , whichby examining the explicit action of U on cochains we see to take values in F N +1 D . Afterreduction (mod F N +1 D ), we see that[ τ N +1 ] D − [ ρ N +1 ( e ϕ | V )] D = [ ρ N +1 ( ψ | V )] D = 0 , which is the result. Claim 2.2.
There exists a cocycle representing [ τ N +1 ] D taking values in the smallerspace A N +1 D α . ifting non-ordinary cohomology classes for SL Chris WilliamsProof. As φ N is a U -eigensymbol, we know that as cocycles, ϕ N and τ N .. = ρ N ( ϕ | V )determine the same cohomology class in H r (Γ , A N D ). Thus there exists some cobound-ary b N ∈ B r (Γ , A N D ) such that ϕ N = τ N + b N . Then by definition b N = d ( c N ) forsome c N ∈ C r − (Γ , A N D ). Lift c N arbitrarily to a cochain c N +1 ∈ C r − (Γ , A N +1 ), anddefine b N +1 .. = d ( c N +1 ) ∈ B r (Γ , A N +1 D ). Then ρ N +1 ,N ( τ N +1 + b N +1 ) = τ N + b N = ϕ N ∈ Z r (Γ , A N D α ) . Therefore it follows that ϕ N +1 .. = τ N +1 + b N +1 takes values in the smaller space A N +1 D α . As τ N +1 + b N +1 ∈ Z r (Γ , A N +1 D ), it follows that ϕ N +1 ∈ Z r (Γ , D α ). Thus ϕ N +1 is the required cocycle to prove the claim.Define φ N +1 .. = [ ϕ N +1 ] D α ∈ H r (Γ , A N +1 D α ) to be the A N +1 D α -valued cohomologyclass determined by ϕ N +1 . Claim 2.3.
The cohomology class φ N +1 is independent of all choices.Proof. Suppose we choose a different preimage f c N of b N under d , leading to a different (cid:93) c N +1 and (cid:93) b N +1 , and thus a different (cid:94) ϕ N +1 . Then ϕ N +1 − (cid:94) ϕ N +1 = b N +1 − (cid:93) b N +1 = d ( c N +1 − (cid:93) c N +1 ) . As ϕ N +1 − (cid:94) ϕ N +1 takes values in A N +1 D α , so must c N +1 − (cid:93) c N +1 ; hence b N +1 − (cid:93) b N +1 ∈ B r (Γ , A N +1 D α ), so that[ ϕ N +1 ] D α = [ (cid:94) ϕ N +1 ] D α ∈ H r (Γ , A N +1 D α ) . Thus they also determine the same cohomology class, namely [ τ N +1 ] D , in H r (Γ , A N +1 D ) . As the cohomology class [ τ N +1 ] D is also uniquely determined by Claim 2.1, we’redone. Claim 2.4. φ N +1 is a U -eigensymbol with eigenvalue α .Proof. It’s clear that the representative ϕ N +1 of φ N +1 is a lift of ϕ N , by definition. Thusany lift ϕ of ϕ N +1 to a cochain taking values in D α is also a lift of ϕ N , and accordingly,it follows that φ N +1 | V N +1 .. = [ ρ N +1 ( ϕ | V )] D = [ τ N +1 ] D . Also by definition, ϕ N +1 and τ N +1 represent the same elements of H r (Γ , A N +1 D ), sothat ε ( φ N +1 ) = [ τ N +1 ] D . Combining the two equalities gives ε ( φ N +1 ) = φ N +1 | V N +1 ,which is the required result.Thus we obtain surjectivity. Take some U -eigensymbol φ ∈ H r (Γ , V α ) = H r (Γ , A D α ),and for each N ∈ N , lift it to a U -eigensymbol φ N using the above method. Then weobtain an element of the inverse limit lim ← H r (Γ , A N D α ) , which we know is isomorphicin a natural way to H r (Γ , D α ). This element is thus a U -eigensymbol that maps to φ under the specialisation map.It remains to prove injectivity. Suppose φ ∈ ker( ρ ); we aim to show that φ = 0.Consider the exact sequence0 −→ F D α −→ D α −→ V α −→ . ifting non-ordinary cohomology classes for SL Chris Williams
This leads to a long exact sequence of cohomology · · · H r (Γ , F D α ) −→ H r (Γ , D α ) ρ −−−−−→ H r (Γ , V α ) −→ · · · , and accordingly any element of ker( ρ ) must lie in the image of H r (Γ , F D α ). This isthe same as saying φ can be represented by a cocycle ϕ taking values in F D α . We nowconclude using: Claim 2.5.
Let φ ∈ H r (Γ , D α ) be represented by a cocycle ϕ taking values in F D α . If φ is a U -eigensymbol, then φ = 0 .Proof. We consider ε ( φ ) = [ ϕ ] D , which is also a U -eigensymbol. It thus makes sense toapply the operator V to [ ϕ ] D , for which it is a fixed point. By condition (b) of Theorem1.2, the V operator takes F N D to F N +1 D ; therefore, as [ ϕ ] D is represented by ϕ | V N for any N (by the eigensymbol property), we see that for each N , the symbol [ ϕ ] D isrepresented by a cocycle taking values in F N D . But the intersection of the F N D istrivial by assumption. Thus ε ( φ ) = [ ϕ ] D is 0.It remains to prove that the map ε is injective. We now know that ϕ is a cobound-ary in C n (Γ , D ), so that there exists some c ∈ C n − (Γ , D ) with ϕ = d ( c ). But as ϕ takes values in D α , it follows that c must also take values in D α . Thus ϕ is also acoboundary in C n (Γ , D α ), and φ = [ ϕ ] D α = 0, as required.This completes the proof of Theorem 1.2.
3. Application to GL × GL × GL , which is conceptually easier to understand than the case of SL . Inparticular, we present the results in a concrete setting in the style of [Wil17], wherethese results were first proved. Recall the set-up: Notation:
Let K be an imaginary quadratic field with ring of integers O K , and let p be a rational prime that splits as pp in K . Let Γ ⊂ Γ ( p ) ⊂ SL ( O K ) be a congruencesubgroup. Let Σ denote the set of complex embeddings of K , and let λ = ( k, k ) ∈ Z [Σ]be a weight, where k is non-negative. Let L/ Q p be a finite extension with ring of integers O L . Definition 3.1.
Let V k ( O L ) .. = Sym k ( O L ) be the space of homogeneous polynomials intwo variables of degree k over O L .We can identify V k ( O L ) ⊗ V k ( O L ) with a space of polynomial functions on O K ⊗ Z Z p ina natural way. Definition 3.2.
Let A k ( O L ) .. = O L h z i be the Tate algebra over O L , that is, the spaceof power series in one variable whose coefficients tend to zero as the degree tends toinfinity. Remark:
For ease of notation, we will henceforth drop O L from the notation. All tensorproducts are over O L . 9 ifting non-ordinary cohomology classes for SL Chris Williams
Let Σ ( p ) ⊂ M ( O L ) ∩ GL ( L ) be the set of matrices that are upper-triangular modulo p . In particular, we have Γ ⊂ Σ ( p ). Then A k has a natural left action of Σ ( p ),depending on k (justifying the notation), given by (cid:18) a bc d (cid:19) · f ( z ) = ( a + cz ) k f (cid:18) b + dza + cz (cid:19) . This action preserves the subspace V k and hence gives rise to component-wise actionsof Σ ( p ) on V k ⊗ V k , V k ⊗ A k and A k ⊗ A k . Accordingly, we get right actions of Σ ( p ) on their corresponding topological duals V ∗ k ⊗ V ∗ k , V ∗ k ⊗ D k and D k b ⊗ D k respectively. Bydualising the inclusions, we get Σ ( p ) -equivariant surjections D k b ⊗ D k pr −−−−−→ V ∗ k ⊗ D k pr −−−−−→ V ∗ k ⊗ V ∗ k , that induce mapsH (Γ , D k b ⊗ D k ) ρ −−−−−→ H (Γ , V ∗ k ⊗ D k ) ρ −−−−−→ H (Γ , V ∗ k ⊗ V ∗ k )on the cohomology.We define filtrations as follows: Definition 3.3. (i) Let N be an integer and define F N D k .. = { µ ∈ D k : µ ( z r ) ∈ π N − rL O L } , where π L is a uniformiser in O L . Then define F N [ V ∗ k ⊗ D k ] .. = V ∗ k ⊗ F N D k . This is Σ ( p )-stable by arguments in [Gre07] and [Wil17]. Now define F N [ V ∗ k ⊗ D k ] .. = F N [ V ∗ k ⊗ D k ] ∩ ker(pr ) , which is also Σ ( p )-stable as pr is Σ ( p )-equivariant.(ii) Similarly, define F N [ D k b ⊗ D k ] .. = ( F N D k b ⊗ D k ) ∩ ker(pr ) , which again is Σ ( p )-stable.Let α ∈ O L and let π p .. = [( ) , (cid:0) p (cid:1) ] and π p .. = [ (cid:0) p (cid:1) , ( )] ∈ Σ ( p ) . First, wehave:
Lemma 3.4.
Suppose v p ( α ) < k + 1 . Then we have(i) If µ ∈ F N [ V ∗ k ⊗ D k ] , then µ | π p ∈ αF N +1 [ V ∗ k ⊗ D k ] . (ii) If µ ∈ F N [ D k b ⊗ D k ] , then µ | π p ∈ αF N +1 [ D k b ⊗ D k ] . We then define the analogue of the module D α as follows: Definition 3.5. (i) Define D αk .. = { µ ∈ D k : µ ( z r ) ∈ αp − r O L } , and then define[ V ∗ k ⊗ D k ] α .. = V ∗ k ⊗ D αk . (ii) Similarly, define [ D k b ⊗ D k ] α .. = D αk b ⊗ D k . Lemma 3.6. (i) If µ ∈ [ V ∗ k ⊗ D k ] α , then µ | π p ∈ αV ∗ k ⊗ D k . ifting non-ordinary cohomology classes for SL Chris Williams(ii) If µ ∈ [ D k b ⊗ D k ] α , then µ | π p ∈ α D k b ⊗ D k . Accordingly, we can lift using Theorem 1.2, first along ρ using the operator U p inducedby π p , and secondly along ρ using the operator U p induced by π p . In particular, wehave: Theorem 3.7.
Let α p , α p ∈ O L with v p ( α ) , v p ( α ) < k + 1 . Then the restriction of themap ρ .. = ρ ρ to the simultaneous α p and α p eigenspaces of the U p and U p operatorsrespectively is an isomorphism. Remarks: (i) In [Wil17], these results are used to construct p -adic L -functions for au-tomorphic forms for GL over an imaginary quadratic field, in the spirit of [PS11].In particular, we associate to such an automorphic form a canonical element inthe overconvergent cohomology, from which we can very naturally build a rayclass distribution that interpolates L -values of the automorphic form. It would beinteresting to know if similar results existed in the case of SL .(ii) In the interests of transition to the case of SL , we can rephrase the above def-initions in a more abstract way. In particular, let G .. = Res K/ Q GL , with Borelsubgroup B and opposite Borel B opp . Define T to be the torus, and note we canview λ as a dominant weight for T , and that V k ⊗ V k is the representation of GL ofhighest weight λ with respect to B opp . Note that for an extension L/ Q p , we have G ( L ) ∼ = GL ( L ) × GL ( L ). Then A k b ⊗ A k is the ring of analytic functions on B ( L )that transform like λ under multiplication by elements of T ( L ), whilst V k ⊗ A k isthe ring of analytic functions on B ( L ) that transform like λ under multiplicationby elements of GL ( L ) × T Q ( L ), where T Q ( L ) is the torus of diagonal matrices inthe algebraic group GL / Q . In particular, the definitions in the following sectionare a natural analogue of the theory described concretely above.
4. Overconvergent modular symbols for SL ofPollack and Pollack in [PP09]. We first recall the setting, and also develop the notionof ‘partially overconvergent’ modular symbols for SL . We recall the setting; where possible, we keep to the notation used by Pollack and Pollackin [PP09] for clarity. For further details, the reader is directed to their paper. Let G bethe algebraic group GL / Q , and denote by B (resp. B opp ) its Borel subgroup of upper-triangular (resp. lower-triangular) matrices, with T and N (resp. N opp ) the subgroupsof B (resp. B opp ) consisting of the diagonal and unipotent matrices respectively. Notethat B = T N . Let p be a prime, let Γ ( p ) be the subgroup of SL ( Z ) of matrices thatare upper-triangular modulo p , and let Γ be a congruence subgroup of SL ( Z ) containedin Γ ( p ). Let λ be a dominant algebraic character of the torus T , which can be seen as an element λ = ( k , k , k ) ∈ Z . Let V λ be the (unique) representation of G with highest weight λ with respect to B opp ; for example, when λ = ( k, , V λ ( A ) is nothing butSym k ( A ), for a suitable coefficient module A .11 ifting non-ordinary cohomology classes for SL Chris Williams
Remark:
We will restrict to the case where λ = ( k , k , λ = ( k + v, k + v, v ), and then V λ ∼ = V λ ⊗ det v , where λ = ( k , k , λ is the same as that for λ scaledby v in each component. We denote by C p the completion of fixed algebraic closure of Q p , and write O C p forits ring of integers. We now define two different overconvergent coefficient modulescorresponding to two different parabolic subgroups of SL . T = SL We first look at the case where we consider the parabolic subgroup T = SL × SL × SL . This identically mirrors the work of Pollack and Pollack in [PP09]. In particular, let I denote the subgroup of G ( O C p ) of matrices that are upper-triangular modulo themaximal ideal of O C p .We consider continuous function f : B ( O C p ) → O C p satisfying the condition f ( tb ) = λ ( t ) f ( b ) , t ∈ T ( O C p ) , b ∈ B ( O C p ) (2)We note that any such function is determined by its restriction to N ( O C p ), and that wecan identify N ( O C p ) with O C p by identifying x y z ←→ ( x, y, z ) ∈ O C p . We write f ( x, y, z ) for the image of this matrix under f .Let L/ Q p be a finite extension with ring of integers O L . We say that such a function f is L -rigid analytic if, for ( x, y, z ) ∈ N ( O C p ), we can write f in the form f ( x, y, z ) = X r,s,t ≥ c rst x r y s z t , where c rst ∈ L tends to 0 as r + s + t → ∞ . Alternatively, this occurs if and only if f ( x, y, z ) ∈ L h x, y, z i , the Tate algebra in three variables over L . Writing O L for thering of integers of L , there is likewise an integral version with c rst ∈ O L . Remark:
Henceforth, we will state all definitions and results in terms of coefficients in O L , since in the sequel we use this integrality in an essential way to define filtrations.We could easily instead state the definitions using L in place of O L . Definition 4.1. (i) Write A λ ( O L ) for the space of O L -rigid analytic functions on B ( O C p ) that satisfy equation (2).(ii) Let D λ ( O L ) denote the topological dual D λ ( O L ) .. = Hom cts ( A λ ( O L ) , O L )(resp. Hom cts ( A λ ( O L ) , O L )), the space of rigid analytic distributions on B ( O C p ) of weight λ . 12 ifting non-ordinary cohomology classes for SL Chris Williams
In an abuse of notation, we write x r y s z t for the unique extension to B ( O C p ) of thefunction on N ( O C p ) that sends x y z x r y s z t , and note that any µ ∈ D λ ( O L ) is uniquely determined by its values at x r y s z t for r, s, t ≥
0. Pollack and Pollack call this function f rst . P .. = SL × SL We now define a different module of overconvergent coefficients. This is, in a sense, a smaller module of coefficients, and will play the role of ‘half-overconvergent’ coefficientsin the following.Let P = SL × SL ⊂ SL . If λ = ( k , k ,
0) with k ≥ k , we get an associatedrepresentation W λ ( A ) .. = det k ⊗ Sym k ( A )of P ( A ) = SL ( A ) × SL ( A ), for suitable A . We can replace B with the larger subgroup B of matrices that are block lower-triangular with respect to this parabolic subgroup– that is, matrices that are zero in the (2 ,
1) and (3 ,
1) entries – and consider the spaceof functions f : B ( O C p ) −→ W λ ( O C p ) satisfying the condition f ( tg ) = λ ( t ) f ( g ) ∀ t ∈ P ( O C p ) , g ∈ B ( O C p ) , where λ ( t ) ∈ GL( W λ ) . Note that any such function is entirely determined by its restriction to B ( O C p ), andindeed by its values on the subgroup x y ∈ B ( O C p ) , by a similar argument to before. We say such a function is O L -rigid analytic if it is anelement of O L h x, y i ⊗ L W λ ( L ). Definition 4.2.
Write A Pλ ( O L ) for the space of O L -rigid analytic functions on B ( O C p )that transform like λ under elements of P . Proposition 4.3.
Let f ∈ A Pλ ( O L ) . For g ∈ B , let P g ( X, Y ) .. = f ( g ) ∈ W λ ( O L ) , wherewe consider elements of W λ as homogeneous polynomials of degree k in two variablesover O L . Define a function f : B ( O C p ) −→ O C p by f ( g ) = P g (0 , . Then f ∈ A λ ( O L ) . Moreover, the association f f gives anisomorphism A Pλ ( O L ) ∼ −→ (cid:26) f ( x, y, z ) = X r,s,t ≥ α r,s,t x r y s z t ∈ A λ ( O L ) : α r,s,t = 0 for t > k (cid:27) . Proof.
Firstly, note that f is rigid analytic in three variables. In particular, let g .. = x y and P g ( X, Y ) = k X t =1 X r,s ≥ α r,s,t x r y s X t Y k − t , ifting non-ordinary cohomology classes for SL Chris Williams using rigidity of f . Then consider g .. = x y z = z x y . Recall that GL ( L ) acts on W λ ( L ) by w | (cid:0) a bc d (cid:1) ( X, Y ) = w ( bY + dX, aY + cX ) , so that f ( x, y, z ) = P g (0 ,
1) = P g ( X + z, Y ) (cid:12)(cid:12)(cid:12)(cid:12) X =0 ,Y =1 = P g ( z,
1) = k X t =1 X r,s ≥ α r,s,t x r y s z t . (3)The rigidity follows. Now we show that f transforms under T as λ . Let g ∈ B ( O C p )and t = ( t , t , t ) ∈ T ( O C p ). Then compute P tg ( X, Y ) = f ( tg )( X, Y ) = t k f ( g )( t X, t Y ) = t k P g ( t X, t Y ) . Accordingly, we have f ( tg ) = P tg (0 ,
1) = t k P g (0 , t ) = t k t k P g (0 ,
1) = λ ( t ) f ( g ) , as required.Finally, it remains to show that the map induces the stated isomorphism. From equation(3), it is clear that f = 0 if and only if f = 0, so that the association f f is injective.It is also clear that the image is the right-hand side of the isomorphism. This completesthe proof. Definition 4.4. (i) From now on, in an abuse of notation using this isomorphism,we write A Pλ ( O L ) for this subspace of A λ ( O L ).(ii) Let D Pλ ( O L ) denote the topological dual D Pλ ( O L ) .. = Hom cts ( A Pλ ( O L ) , O L ) , the space of rigid analytic distributions on B ( O C p ) of weight λ over O L .Note that by dualising the inclusion A Pλ ( O L ) ⊂ A λ ( O L ), we get a surjective mappr λ : D λ ( O L ) −→ D Pλ ( O L ) , where the notation will become clear in the sequel. Remark 4.5:
Note that D Pλ ( O L ) is, in a sense, ‘partially’ overconvergent, in the sensethat it is overconvergent in the variables x, y and classical in z . In the next section, wewill introduce operators π = p
00 0 p and π = p , whose product is the element π considered by Pollack and Pollack in [PP09]. We willultimately lift a classical modular symbol to one that takes values in D Pλ ( O L ) using π ,and then lift this further to a symbol that takes values in the space D λ ( O L ) of fullyoverconvergent coefficients using π . 14 ifting non-ordinary cohomology classes for SL Chris Williams Σ and specialisation λ action Let X denote the image of the Iwahori group I in N opp ( O C p ) \ G ( O C p ) under the naturalembedding, and note that we can identify X with B ( O C p ) in a natural way. Let I .. = I ∩ SL ( Z ) . (Note that I = Γ ( p ) in this setting, though we retain the notation for ease of comparisonwith Pollack and Pollack.) We also define π and π as in Remark 4.5, and let Σ be thesemigroup generated by I , π and π .Note that I acts on N opp ( O C p ) \ G ( O C p ) by right multiplication, and as π normalises N opp , we also have a right action of π on this space by N opp ( O C p ) g | π = N opp ( O C p ) π − gπ. Thus we have an action of Σ on this space. This action preserves X and hence givesrise to a right action of Σ on B ( O C p ). This in turn gives a left action of Σ on A λ ( O L )by γ · f ( b ) = f ( b | γ ) , and dually a right action of Σ on D λ ( O L ) by µ | γ ( f ) = µ ( γ · f ) . In [PP09], Lemma 2.1, Pollack and Pollack give an explicit description of this action.We recap their results:
Lemma 4.6. (i) Let λ = ( k , k , . For γ ∈ I , the weight λ action of γ on f ∈ A λ ( O L ) is given by ( γf ))( x, y, z ) =( a + a x + a y ) k − k ( m − m y − m z + m xz ) k × f (cid:18) a + a x + a ya + a x + a y , a + a x + a ya + a x + a y , − m + m y + m z − m xzm − m y − m z + m xz (cid:19) , where γ = ( a ij ) and m ij is the ( i, j ) th minor of γ .(ii) We have π · f ( x, y, z ) = f ( px, py, z ) and π · f ( x, y, z ) = f ( x, py, pz ) . Proof.
For part (i), see [PP09], Lemma 2.1. For part (ii), this is easily checked bycomputing π − x y z π = px py z . The case of π is done similarly. Proposition 4.7.
The action of Σ preserves the subspace A Pλ ( O L ) of A λ ( O L ) .Proof. The space A Pλ ( O L ) is the span of the functions x r y s z t with t ≤ k (under suitablerestrictions on the coefficients). So it suffices to show that γ · x r y s z t lies in this span.But from Lemma 4.6 above, this is clear. Corollary 4.8.
The map D λ ( O L ) → D Pλ ( O L ) given by dualising the inclusion is equiv-ariant with respect to the action of Σ . ifting non-ordinary cohomology classes for SL Chris Williams λ We want to exhibit a map from overconvergent to classical coefficients, which we’ll call specialisation to weight λ . To this end, let v λ be a highest weight vector in V λ ( O L )(which we take to be a right representation of G ). More precisely, this is an elementsatisfying v λ | t = λ ( t ) v λ ∀ t ∈ T ( O L ) , v λ | n = v λ ∀ n ∈ N opp ( O L ) . In particular, we can define a map f λ : G ( O L ) −→ V λ ( O L ) g v λ | g. Since we have invariance under N opp , this function descends to N opp \ G . We can thenrestrict this function to (the O L -points of) X . Lemma 4.9.
Let λ = ( k , k , ∈ Z . Then V λ ( O L ) can be realised as a subrepresenta-tion of Sym k ( O L ) ⊗ Sym k ( O L ) , and the highest weight vector is v λ = k X i =0 ( − i (cid:18) k i (cid:19) X k − i Y i ⊗ U i V k − i , where a general element has form P P ( X, Y, Z ) ⊗ Q ( U, V, W ) . Proof.
See [PP09], Remark 2.4.3.
Proposition 4.10.
We have f λ (cid:12)(cid:12)(cid:12)(cid:12) X ∈ A Pλ ( O L ) ⊗ V λ ( O L ) .Proof. We explicitly compute v λ | g , where g = x y z . We see that this is equal to v λ | g = k X i =1 ( − i (cid:18) k i (cid:19) ( X + xY + yZ ) k − i ( Y + zZ ) i ⊗ ( U + xV + yW ) i ( V + zW ) k − i . (4)It’s easy to see from this that the coefficient of each monomial is an element of A Pλ ( O L )(and in particular that the maximal degree of z in this expression is k ), and we concludethe result.For a distribution µ ∈ D Pλ ( O L ), define an ‘evaluation at A Pλ ( O L ) ⊗ V λ ( O L )’ map bysetting µ ( f ⊗ v ) = µ ( f ) ⊗ v ∈ V λ ( O L ) . In particular, we can evaluate at f λ . Definition 4.11.
Define the specialisation map at weight λ to be the mappr λ : D Pλ ( O L ) −→ V λ ( O L )given by evaluation at f λ ∈ A Pλ ( O L ) ⊗ V λ ( O L ) . ifting non-ordinary cohomology classes for SL Chris Williams
This map is I -equivariant, but not π i -equivariant. As in [PP09], we introduce a twistedaction of π i to get around this. Definition 4.12.
Define a (right) action of Σ on V λ ( L ) by v ? γ = v | γ, γ ∈ I,v ? π i = λ ( π i ) − v | π i . Let V ?λ ( L ) denote the module V λ ( L ) with this twisted action.Then we see that: Lemma 4.13.
The map pr λ : D Pλ ( L ) −→ V ?λ ( L ) is Σ -equivariant. Definition 4.14.
Let L λ ( O L ) .. = pr λ ( D Pλ ( O L )) ⊂ V ?λ ( L ) . Note that this is stable underthe ? -action of Σ since pr λ is Σ-equivariant.We have an action of Γ ⊂ I on these coefficient spaces. In particular, we can define thegroup cohomology of these coefficient spaces, and then note that, for each integer r , themap pr λ induces a map ρ λ .. = ρ λ ( r ) : H r (Γ , D Pλ ( O L )) −→ H r (Γ , L λ ( O L )) . These spaces come equipped with the natural Hecke action on cohomology, and theaction of the U p operator is given by the matrix π = π π .
5. Filtrations and control theorems for SL λ = ( k , k , ∈ Z , we defined a space L λ ( O L ) of classical coefficients, a space D Pλ ( O L ) of partially overconvergent coefficients,and a space D λ ( O L ) of fully overconvergent coefficients (where D λ ( O L ) is as definedin [PP09]). We also defined maps pr iλ between these coefficient modules, and theseinduce mapsH r (Γ , D λ ( O L )) ρ λ −−−−−→ H r (Γ , D Pλ ( O L )) ρ λ −−−−−→ H r (Γ , L λ ( O L ))on the cohomology.In this section, we prove that if we restrict to the simultaneous small-slope eigenspacesof the operators on the cohomology given by π and π , the composition ρ λ of thesemaps is an isomorphism. For posterity, we give the definition of small slope now. Definition 5.1.
Let U p,i be the operator on the cohomology induced by the element π i of Remark 4.5, for i = 1 ,
2. We call these operators the
Hecke operators at p . Definition 5.2.
Let φ be an eigensymbol at p (with classical or overconvergent coeffi-cients) of weight λ = ( k , k , U p,i φ = α i φ for i = 1 ,
2. We say said to be small slope at p if v p ( α ) < k − k + 1 and v p ( α ) < k + 1 . In particular, we will show that the restriction of ρ λ to the small slope subspaces is anisomorphism. We use two applications of Theorem 1.2 to prove this.17 ifting non-ordinary cohomology classes for SL Chris Williams
We now define a filtration on the modules D Pλ ( O L ) that allows us to apply Theorem 1.2. D Pλ ( O L ) Definition 5.3.
Define F N D Pλ ( O L ) .. = n µ ∈ D Pλ ( O L ) : µ ( x r y s z t ) ∈ π N − ( r + s ) L O L o . Proposition 5.4.
The filtration F N D Pλ ( O L ) is stable under the action of Σ .Proof. Let µ ∈ F N D Pλ ( O L ) . We know that, for γ = ( a ij ) ∈ I , we have γ · x r y s z t =( a + a x + a y ) r ( a + a x + a y ) s × ( − m + m z − ( m z ) x + m y ) t ( a + a x + a y ) k − k − r − s × ( m − m z − ( m z ) x − m y ) k − t , where m ij is the ( i, j )th minor of γ , using Lemma 4.6. Write this as µ | γ ( x r y s z t ) = X a,b ≥ β ab ( z ) x a y b , where β ab ( z ) is a polynomial in z of degree at most t . Then note that p divides the terms a , a , a , m , m , and m , whilst the terms a , a , a , m and m are all p -adicunits. In particular, we examine the p -divisibility conditions on the coefficients β ab ( z ).Any monomial x a y b coming from the first bracket in this expression has coefficientdivisible by p a + b − r , since p | a . Similarly, any such monomial in the second bracket hascoefficient divisible by p a + b − s . Moreover, since in the remaining three brackets p dividesthe coefficient of both x and y before expanding, we see that any monomial including x a y b in the expanded expression is divisible by p a + b . Accordingly, by combining this,we see that p a + b − ( r + s ) | β ab ( z ). Since we already know that µ ( x a y b z c ) ∈ π N − ( a + b ) L O L forany c ≤ t , we now see that µ ( β ab ( z ) x a y b ) ∈ p a + b − ( r + s ) π N − ( a + b ) L O L ⊂ π N − ( r + s ) L O L , as required.Since π and π act on such monomials by multiplying by a non-negative power of p ,they also preserve the filtration. Thus the filtration is stable under the action of Σ.We actually need a slightly finer filtration. Definition 5.5.
Define F N D Pλ ( O L ) .. = F N D Pλ ( O L ) ∩ ker(pr λ ) . Since pr λ is Σ-equivariant, this filtration is also Σ-stable. The crux of our argument isthen: Proposition 5.6.
Suppose µ ∈ ker(pr λ ) . Then µ ( x r y s z t ) = 0 for all r + s ≤ k − k , ≤ t ≤ k . ifting non-ordinary cohomology classes for SL Chris WilliamsProof.
We explicitly examine the map pr λ . Earlier, in equation (4), we gave a formulafor the expression f λ ( x, y, z ). If µ ∈ ker(pr λ ), then in particular µ ( f λ ( x, y, z )) = 0.We consider the monomials including the term U k , keeping the notation of previously.Such a term can occur only for i = k , so that these terms all appear in( − k ( X + xY + zZ ) k − k ( Y + zZ ) k ⊗ U k . By expanding out this bracket, and considering the coefficients of each monomial, wesee that we have µ ( x r y s z t ) = 0 for at least the range of r, s and t specified by theproposition. Remark:
Note that, for general λ , this condition on r + s is optimal. In particular,consider λ = ( k, , k ≥
1. Then if µ ∈ ker( ρ λ ), then we do notnecessarily have µ ( x k ) = 0, so in particular we can’t say anything general about thevalues µ ( x r y s ) where r + s > k − Lemma 5.7.
Let µ ∈ F N D Pλ ( O L ) , and let α ∈ O L with v p ( α ) < k − k + 1 . Then µ | π ∈ α F N +1 D Pλ ( O L ) . Proof.
We have µ | π ( x r y s z t ) = p r + s µ ( x r y s z t ). From Proposition 5.6, we see that if r + s ≤ k − k , we have µ ( x r y s z t ) = 0. In particular, from this, we have µ | π ( x r y s z t ) ∈ p k − k +1 π N − ( r + s ) L O L . As v p ( α ) < k − k + 1, and it must be divisible by an integral power of π L , we have p k − k +1 ∈ απ L O L , so that µ | π ( x r y s z t ) ∈ απ N − ( r + s ) L O L . Thus µ ∈ α F N +1 D Pλ ( O L ), as required. D Pλ ( O L )We require one further definition before we can apply Theorem 1.2; namely, a submoduleof D Pλ ( O L ) that will play the role of D α in condition (v) in Notation 1.1. Definition 5.8.
Let α ∈ O L . Define D P,αλ ( O L ) .. = n µ ∈ D Pλ ( O L ) : µ ( x r y s z t ) ∈ αp − ( r + s ) O L o . Proposition 5.9.
The subspace D P,αλ ( O L ) is stable under the action of Σ .Proof. Let µ ∈ D Pλ ( O L ) and γ ∈ I , and recall the proof of Proposition 5.4, and inparticular, the computation µ | γ ( x r y s z t ) = X a,b ≥ µ ( β ab ( z ) x a y b ) , where p a + b − ( r + s ) | β ab ( z ). Now take µ to be in the smaller space D P,αλ ( O L ). Then µ ( β ab ( z ) x a y b ∈ p a + b − ( r + s ) αp − ( a + b ) O L = αp − ( r + s ) . Thus µ | γ ∈ D P,αλ ( O L ), as required.As π and π act on monomials by multiplying by non-negative powers of p , stability inthis case is clear. 19 ifting non-ordinary cohomology classes for SL Chris Williams
Lemma 5.10.
Suppose µ ∈ D P,αλ ( O L ) . Then µ | π ∈ α D Pλ ( O L ) . Proof.
Consider µ | π ( x r y s z t ) = p r + s µ ( x r y s z t ). Since µ ∈ D P,αλ ( O L ), we see that µ | π ( x r y s z t ) ∈ α O L , and the result immediately follows. We can now apply Theorem 1.2 to the small slope subspace in this situation. In par-ticular, in the set-up of this theorem, let D = D Pλ ( O L ) and D α = D P,αλ ( O L ). Thenwe have written down a filtration of this space that satisfies the conditions of Theorem1.2. In particular, we have all the objects of Notation 1.1 (i)-(v), and then we’ve showncondition (a) of the theorem in Lemma 5.10 and condition (b) in Lemma 5.7. So we’veproved: Proposition 5.11.
Let α ∈ O L with v p ( α ) < k − k + 1 . Let D Pλ ( O L ) be the mod-ule of partially overconvergent coefficients defined in Section 4.3, and let L λ ( O L ) =pr λ ( D Pλ ( O L )) . Then the restriction ρ λ : H r (Γ , D Pλ ( O L )) U p, = α ∼ −→ H r (Γ , L λ ( O L )) U p, = α of ρ λ to the α -eigenspaces of the U p, operator is an isomorphism. We now change direction and focus on the action of the U p, operator induced from π .In particular, by applying the theorem again with the U p, operator, we can lift frompartial to fully overconvergent coefficients. As the results are very similar to, and inmany cases simpler than, those above, we present the material here in less detail.Define a filtration on D λ ( O L ) by F N D λ ( O L ) .. = (cid:8) µ ∈ D λ ( O L ) : µ ( x r y s z t ) ∈ π N − tL O L (cid:9) ∩ ker(pr λ ) . This is Σ-stable by a very similar argument to previously. We also define, for α ∈ O L , D αλ ( O L ) .. = (cid:8) µ ∈ D λ ( O L ) : µ ( x r y s z t ) ∈ αp − t O L (cid:9) , which is also easily seen to be Σ-stable and satisfies the conditions required of D α inTheorem 1.2. When v p ( α ) < k + 1 , we see that if µ ∈ F N D λ ( O L ), then µ | π ∈ α F N +1 D λ ( O L ), again by a similar argument before after studying the kernel of pr λ .Putting this together and using Theorem 1.2, we get: Proposition 5.12.
Let α ∈ O L with v p ( α ) < k + 1 . Let D λ ( O L ) and D Pλ ( O L ) bethe modules of fully and partially overconvergent coefficients respectively, as defined inSection 4.3. Then the restriction ρ λ : H r (Γ , D λ ( O L )) U p, = α ∼ −→ H r (Γ , D Pλ ( O L )) U p, = α of ρ λ to the α -eigenspaces of the U p, operator is an isomorphism. ifting non-ordinary cohomology classes for SL Chris Williams
We can combine the results of Propositions 5.11 and 5.12 to obtain the following con-structive non-critical slope control theorem for SL . Theorem 5.13.
Consider the set-up of Notation 0.1 in the Introduction. In particular,let λ = ( k , k , be a dominant algebraic weight, and let α , α ∈ O L with v p ( α ) Israel J. Math. , 153:319 – 354, 2006. (Cited onpage 1.)[GM14] Xavier Guitart and Marc Masdeu. Overconvergent cohomology and quaternionic Dar-mon points. J. Lond. Math. Soc. , 90 (2):495 – 524, 2014. (Cited on page 2.)[Gre07] Matthew Greenberg. Lifting modular symbols of non-critical slope. Israel J. Math. ,161:141–155, 2007. (Cited on pages 2, 4, and 10.)[PP09] David Pollack and Robert Pollack. A construction of rigid analytic cohomology classesfor congruence subgroups of SL ( Z ). Canad. J. Math. , 61:674–690, 2009. (Cited onpages 1, 2, 5, 6, 11, 12, 14, 15, 16, and 17.)[PS11] Robert Pollack and Glenn Stevens. Overconvergent modular symbols and p -adic L -functions. Annales Scientifique de l’Ecole Normale Superieure , 2011. (Cited on pages 1and 11.)[PS12] Robert Pollack and Glenn Stevens. Critical slope p -adic L -functions. J. Lond. Math.Soc. , 2012. (Cited on page 1.)[Ste94] Glenn Stevens. Rigid analytic modular symbols. Preprint, 1994. (Cited on page 1.)[Urb11] Eric Urban. Eigenvarieties for reductive groups. Ann. of Math. , 174:1695 – 1784, 2011.(Cited on page 2.)[Wil17] Chris Williams. P -adic L -functions of Bianchi modular forms. Proc. Lond. Math. Soc. ,114 (4):614 – 656, 2017. (Cited on pages 2, 4, 9, 10, and 11.),114 (4):614 – 656, 2017. (Cited on pages 2, 4, 9, 10, and 11.)