aa r X i v : . [ m a t h . N T ] F e b Higher order group cohomology and theEichler-Shimura map ∗ Anton DeitmarDedicated to Christopher Deninger
Abstract:
Higher order group cohomology is defined and first propertiesare given. Using modular symbols, an Eichler-Shimura homomorphism isconstructed mapping spaces of higher order cusp forms to higher order co-homology groups.
Contents ∗ to appear in: J. Reine Angew. Math. IGHER ORDER GROUP COHOMOLOGY Introduction
In the last few years, higher order modular forms have arisen in variouscontexts, for instance in percolation theory [12], Eisenstein series formedwith modular symbols [2, 8], or converse theorems [3, 7]. L-functions ofsecond order forms have been studied in [4, 10], Poincar´e series attached tohigher order forms have been investigated in [11], dimensions of spaces ofsecond order forms have been determined in [5].In this paper we present an approach which focuses on functorial propertiesof higher order cohomology groups . These are the derived functors of higherorder invariant functors, which is a natural generalization of classical groupcohomology. It turns out that these cohomology groups can be representedas Ext-groups over the group ring. Representing the cohomology groupsas Ext-groups has the advantage that the Ext-functors produce long exactsequences out of short exact sequences plugged into either argument. Thistechnique is used extensively throughout the paper.We first introduce higher order group cohomology in general. It turns outthat for finite groups or perfect groups nothing new is gained. For Fuchsiangroups we give exact sequences which allow to compute the dimensions ofcohomology groups inductively. Finally, we define the Eichler-Shimura mapthrough modular symbols. It is a map S qn +2 (Γ) → H q, par (Γ , V n ) , where S qn +2 (Γ) is the space of cusp forms of even weight n + 2 and order q .The space H q, par (Γ , R ) is the higher order group cohomology. For n ≥ n = 0 it has kernel S q − (Γ) andone always has dim S qn +2 (Γ) = dim H q, par (Γ , V n ).I thank Nikolaos Diamantis and Robin Chapman for helpful comments onthe contents of this paper. Throughout, R will be a commutative ring with unit. Let Γ be a group andΣ a normal subgroup. We define a sequence of functors (H ) q =1 , ,... from IGHER ORDER GROUP COHOMOLOGY R [Γ]- modules to the category of R -modules. We start with q = 1. For an R [Γ]-module V letH (Γ , Σ , V ) = V Γ be the usual fix-module, i.e., the set of all v ∈ V for which ( γ − v = 0.Next suppose H already defined, where H (Γ , Σ , V ) is an R - submodule of V . Let H q +1 (Γ , Σ , V ) be the set of all v ∈ V such that ( γ − v ∈ H q (Γ , Σ , V )for every γ ∈ Γ and ( σ − v = 0 for every σ ∈ Σ.If the group Σ is clear from the context, we also write H q (Γ , V ). This will bethe case later for a Fuchsian group Γ, in which case we choose Σ to be thesubgroup generated by all parabolic elements of Γ.The functor H q is a left-exact functor from the category of R [Γ]-modules tothe category of R -modules. Below we will prove this fact by representing H q as a Hom-functor. We denote its right-derived functors by H pq (Γ , Σ , · ).Consider the functor H (Σ , · ) from the category of R [Γ]-modules to the cat-egory of R [Γ / Σ]-modules, mapping V to the Σ-fixed vectors. Then one hasH q (Γ , Σ , · ) = H q (Γ / Σ , , H (Σ , · )). The functor H (Σ , · ) maps injectives toinjectives, so for a Γ-module V there is a Grothendieck spectral sequence E r,s = H rq (Γ / Σ , , H s (Σ , V )) , abutting to H r + sq (Γ , Σ , V ). This has interesting consequences. For instance,if R is a field of characteristic zero and the virtual cohomological dimensionof Γ / Σ is one, then there is an isomorphismH q (Γ , Σ , V ) ∼ = H q (Γ / Σ , , H (Σ , V )) . Let aug : R [Γ] → R be the augmentation map given by aug( P γ a γ γ ) = P γ a γ . Let I = ker(aug) be the augmentation ideal. The following simplelemma will be useful. Lemma 1.1
The ideal I q is the R -span of all elements of the form ( γ − · · · ( γ q − , γ , . . . , γ q ∈ Γ . IGHER ORDER GROUP COHOMOLOGY Proof:
We use induction on q . To start with q = 1 let m ∈ I , then m = X γ m γ γ = X γ m γ γ − X γ m γ = X γ m γ ( γ − . This proves the claim for q = 1. The induction step is clear as I q +1 = I q I . (cid:3) Let I Σ be the augmentation ideal of Σ. As Σ is normal, R [Γ] I Σ is a two-sidedideal of R [Γ]. Let J q be the ideal of R [Γ] generated by the q -th power I q ofthe augmentation ideal together with I Σ , i.e., J q = I q + R [Γ] I Σ . For short we write A for the group algebra R [Γ]. Let V be an A -module. Forany ideal J of A , we write V J for the set of all v ∈ V with J v = 0. There isa natural identification Hom A ( A/J, V ) ∼ = V J . One hasH q (Γ , Σ , V ) = V I q ∩ V I Σ = Hom A ( A/J q , V ) , and hence H pq (Γ , Σ , V ) = Ext pA ( A/J q , V ) . Proposition 1.2
Let Σ ⊂ Γ be an arbitrary normal subgroup. If Γ is finiteand the order | Γ | is invertible in R , then the natural injection H q (Γ , V ) ֒ → H q +1 (Γ , V ) is an isomorphism. Therefore, in this case higher order groupcohomology coincides with classical group cohomology.More generally, the same conclusion holds if there is no non-trivial | Γ | -torsionin V Γ .For arbitrary Γ , the same conclusion holds if Γ coincides with its commutatorsubgroup [Γ , Γ] . Proof:
By induction on q . Suppose first that q = 1. For v ∈ V set P ( v ) = P γ ∈ Γ γv . Then P is a linear map with P = | Γ | P and P = | Γ | on V Γ . Further, for every γ ∈ Γ one has ( γ − v ∈ ker P . Let v ∈ H (Γ , V ).Then ( γ − v ∈ ker P ∩ V Γ = ( | Γ |− tors) ∩ V Γ = 0. This implies a ∈ H (Γ , V ). IGHER ORDER GROUP COHOMOLOGY q > q −
1. Let v ∈ H q (Γ , V ). For γ ∈ Γ one has by induction hypothesis, ( γ − v ∈ H q − (Γ , V ) = H (Γ , V ),and therefore v ∈ H (Γ , V ) = H (Γ , V ).The last assertion is seen as follows. For h, g ∈ Γ, the element ghg − h − − gh − hg ) g − h − = (( g − h − − ( h − g − g − h − belongs to I .Therefore, if Γ = [Γ , Γ], then I = I and hence I q = I for every q . (cid:3) Lemma 1.3 (Cocycle representation)
The module H q (Γ , Σ , V ) is natu-rally isomorphic to Hom A ( J q , V ) /α ( V ) , where α : V → Hom A ( J q , V ) is given by α ( v )( m ) = mv . Proof:
Write J = J q . The exact sequence0 → J → A → A/J → A ( A, V ) → Hom A ( J, V ) → Ext A ( A/J, V ) → Ext A ( A, V )The last term is zero, the first can be identified with V and the first map is α . (cid:3) In the case of classical group cohomology, which is the case q = 1, peopleoften use the following cocycle representation for H (Γ , V ). It is the quotient Z /B , where Z is the space of all maps f : Γ → V such that f ( γτ ) = γf ( τ ) + f ( γ ) and Z is the subspace of all f of the form f ( γ ) = γv − v forsome v ∈ V . Note that for the trivial Γ-module R this identifies H (Γ , R )with the set Hom(Γ , R ) of group homomorphisms into the additive groupof R . A natural isomorphism between these two cocycle representations isgiven by the map Ψ : Z → Hom A ( I, V ),Ψ( f )( γ −
1) = f ( γ ) . Lemma 1.4 (Restriction)
There is a natural restriction map res : H pq (Γ , Σ , V ) → H p (Σ , V ) . The restriction respects the cocycle representations.
IGHER ORDER GROUP COHOMOLOGY Proof:
The restriction is induced by the natural map R [Σ] /I Σ ֒ → A/J k . Thefact that it maps the cocycle representation to the cocycle representationof the group cohomology, follows from the commutativity of the followingdiagram and the exactness of its rows.0 J R [Γ] R [Γ] /J I Σ R [Σ] R [Σ] /I Σ . / / / / / / / / / / ?(cid:31) O O (cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31) / / ?(cid:31) O O (cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31) / / ?(cid:31) O O (cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31) / / (cid:3) The following lemma will be useful later.
Lemma 1.5
The quotients J q − /J q and J q /IJ q are free R = A/I -modules.The natural map I Σ /I → J q /IJ q is injective. Proof:
First note that R [Γ] can be identified with R ⊗ Z [Γ] and then J Rq = R ⊗ J Z q , where we indicate the dependence of the ideal of the ring by anupper index. Next, J Z q is a submodule of the free Z -module Z [Γ], hence free.Finally, as subsets of Q [Γ] one has Q J Z q ∩ J Z q − = J Z q . This implies that J Z q − /J Z q is a free Z -module and as J Rq − /J Rq ∼ = R ⊗ J Z q − /J Z q , the same followsfor R . The proof for J q /IJ q is analogous.For the second assertion consider the map I Σ → J q /IJ q . Its kernel is I Σ ∩ IJ q ,which contains I . To show equality, let f ∈ I Σ ∩ IJ q . Note IJ q = I ( I q + R [Γ] I Σ ) = I q +1 + II Σ . Accordingly, we can write f as X j a j ( γ j,i − · · · ( γ j,q +1 −
1) + X k b k ( γ k − σ k − , where a j , b k ∈ R ; γ j,i ∈ Γ r Σ; γ k ∈ Γ and σ k ∈ Σ. The ideal R [Γ] I Σ isthe kernel of the map R [Γ] → R [Γ / Σ]. It follows that the first summand isannihilated by this map and so there are σ j,k ∈ Σ and σ j ∈ Σ such that X j a j ( γ j, σ j, − · · · ( γ j,q +1 σ j,q +1 − σ j = 0 . IGHER ORDER GROUP COHOMOLOGY X j a j ( γ j, σ j, − · · · ( γ j,q +1 σ j,q +1 − ∈ R [Γ] I Σ . Now ( γ j, σ j, − ≡ ( γ j, −
1) + ( σ j, −
1) mod II Σ , which means that f liesin II Σ . It follows that f can be written as an element of I plus a sum ofthe form X γ ∈ Γ r Σmod Σ ( γ − f γ , where the sum runs over a fixed set of representatives and the f γ are uniquelydetermined. But f ∈ R [Σ] then implies that f γ = 0, so f ∈ I . (cid:3) Let H be the upper half plane in C , which is acted upon via linear fractionalsby the group SL ( R ). As the element − G = PSL ( R ) = SL ( R ) / ±
1. Let Γ be a discrete subgroup of G of finitecovolume which is not cocompact. Then Γ has finitely many equivalenceclasses of cusps c ∈ R ∪ {∞} . For each cusp c fix an element σ c ∈ G suchthat σ c ( ∞ ) = c and σ − c Γ c σ c = (cid:26) ± (cid:18) n (cid:19) : n ∈ Z (cid:27) , where Γ c is thestabilizer subgroup of c in Γ. Let Σ = Γ par be the subgroup generated byall parabolic elements in Γ. Then Σ is the group generated by all stabilizergroups Γ c , where c varies over the cusps of Γ. Since γ Γ c γ − = Γ γc , the groupΣ is normal in Γ.Assume Γ is torsion-free. Then there are hyberbolic generators γ , . . . , γ g and parabolic generators p , . . . , p s such that Γ is the group generated bythese with the only relation[ γ , γ ] · · · [ γ g − , γ g ] p · · · p s = 1 . The number g ≥ genus of Γ. The number s is the number ofinequivalent cusps of Γ. Note that the cohomological dimension of Γ is 1 if s = 0. Otherwise it is 2 (see [13]). IGHER ORDER GROUP COHOMOLOGY g = 0 and q ∈ N , let N g ( q ) = 0. For g ≥
1, let N g ( q ) be the number of alltuples ( i , . . . , i q ) ∈ { , . . . , g } q which do not contain (1 ,
2) as a sub-tuple.Then N g (1) = 2 g and N g (2) = (2 g ) −
1. More generally, one has N g ( q + 1) = 2 gN g ( q ) − N g ( q − , which implies that N g ( q ) = α q + α q − + · · · + α − q , where α = g + p g −
1. Finally, we set N g (0) = 1 for all g . Lemma 2.1
We have dim J q /J q +1 = N g ( q ) . Proof:
As the case g = 0 is trivial, we assume g ≥
1. The exactness of thesequence 0 → J q /J q +1 → A/J q +1 → A/J q → A/J q +1 − dim A/J q . In A/J q one has p j = 1, so we can as well assume s = 0. Then J q = I q .Let ˆ A = lim ←− q A/I q be the I -completion of A . In ˆ A one has the identity γ − = ∞ X n =0 (1 − γ ) n , for every γ ∈ Γ. So we conclude that ˆ A ∼ = ˆ A , where A ⊂ A is the R -subalgebra generated by γ , . . . , γ g , further I = I ∩ A , and ˆ A is the I -completion of ˆ A . Note that A/I q ∼ = ˆ A/ ˆ I q ∼ = ˆ A / ˆ I . It follows that I q /I q +1 is spanned by all elements of the form ( γ i − · · · ( γ i q − ≤ i j ≤ g for each j . The relation [ γ , γ ] · · · [ γ g − , γ g ] = 1is equivalent to γ γ = bγ γ where the element b = [ γ g , γ g − ] · · · [ γ , γ ] iscontained in the subgroup generated by γ , . . . γ g . This means that b lies inthe closed subalgebra of ˆ A generated by γ , . . . γ g . It follows that( γ − γ −
1) = b ( γ − γ −
1) + ( b − γ + γ − . IGHER ORDER GROUP COHOMOLOGY I , the right hand side of this equation can be written as a linearcombination of elements of the form ( γ i − γ i − i , i ) = (1 , γ i − · · · ( γ i q − i ν , i ν +1 ) = (1 ,
2) for every ν = 1 , . . . , q −
1. The latter therefore form a basisof I q /I q +1 . The lemma follows. (cid:3) Let n ≥ P n ( R ) be the R -vector space of homo-geneous polynomials p ( X, Y ) of degree n . So P n ( R ) has dimension n + 1.There is a representation p n of SL ( R ) on the space P n ( R ) given by p n ( γ ) f (cid:18) XY (cid:19) = f (cid:18) γ − (cid:18) XY (cid:19)(cid:19) . Note that the element − ( R ) acts by the scalar ( − n , so that foreven n , one gets a representation of G = SL ( R ) / ±
1. From now on we set R = R , and n ≥ V = V n = P n ( R )is an A = R [Γ]-module, as Γ is a subgroup of G . Lemma 2.2
Assume Γ torsion-free. Let R = R . If n ≥ , then V I q = 0 ,where V = P n ( R ) . Proof:
By Theorem 10.3.5 in [1], the group Γ contains a hyperbolic element γ . Then γ is an element of an R -split torus, which implies that p n ( γ ) isdiagonalizable. As V is a highest weight module, the eigenvalue 1 has multi-plicity one and all other eigenvalues are of absolute value = 1. Let v ∈ V I q ,then for any k , . . . , k q ∈ N we have( γ k − · · · ( γ k q − v = 0 , which implies that v lies in the eigenspace to the eigenvalue 1. As this istrue for every hyperbolic element, v is an element of the intersection U of all1-eigenspaces of hyperbolic elements. This intersection has dimension ≤ P n ( R ). On the other hand, U is invariant under Γ,and as Γ is Zariski-dense in G , the space U is invariant under G . Since therepresentation p n is irreducible, U = 0. (cid:3) Lemma 2.3
Assume Γ torsion-free, let V = P n ( R ) and q ≥ . IGHER ORDER GROUP COHOMOLOGY If s ≥ or n ≥ , then Ext A ( A/J q , V ) = 0 . (b) If n = s = 0 , then the dimension of Ext A ( A/J q , R ) is equal to N g ( q − ,and Ext ( A/J q , R ) = 0 . Proof: (a) In this case, H (Γ , V ) = Ext ( A/I, V ) is zero either because Γhas cohomological dimension 1 or because of Poincar´e duality. The exactsequence 0 → J q → A → A/J q → A ( J q , V ) ∼ = Ext A ( A/J q , V ). For q = 1 the righthand side is zero. This implies Ext A ( I, V ) = 0. By Lemma 1.5 there is anexact sequence 0 → J q → J q − → R N → N . As A/I = R , this gives an exact sequence Ext A ( J q − , V ) → Ext A ( J q , V ) → Ext A ( A/I, V ) N = H (Γ , V ) N = 0. So the first term mapsonto the second and after iteration we find a surjective map Ext A ( I, V ) → Ext A ( J q , V ), hence both are zero.For (b) assume s = 0. Then J q = I q . The cocycle representation readsExt A ( A/I q , R ) ∼ = Hom A ( I q , R ) ∼ = Hom R ( I q /I q +1 , R ), so the diemnsion ofExt A ( A/J q , R ) equals N g ( q ). Consider the exact sequence0 → J q − /J q → A/J q → A/J q − → . It yields the long exact sequence0 → Hom(
A/J q − , R ) ∼ = −→ Hom(
A/J q , R ) −→ Hom( J q − /J q , R ) ∼ = −→ Ext ( A/J q − , R ) −→ Ext ( A/J q , R ) → Ext ( J q − /J q , R ) → Ext ( A/J q − , R ) → Ext ( A/J q , R ) → Ext ( J q − /J q , R ) → ( A/J q , R ) = N g ( q −
1) and Ext ( J q − /J q , R ) ∼ =Ext ( R , R ) N g ( q − has dimension 2 gN g ( q − (Γ , R ) = 2 g by[14] or [9]. Since N g ( q ) = 2 gN g ( q − − N g ( q −
2) and Ext ( J q − /J q , R ) ∼ =Ext ( R , R ) N g ( q − ∼ = R N g ( q − , the latter by Poincar´e duality, we get an exactsequence0 → R N g ( q − → Ext ( A/J q − , R ) → Ext ( A/J q , R ) → R N g ( q − → . IGHER ORDER GROUP COHOMOLOGY ( A/J q , R ) = N g ( q − (cid:3) Theorem 2.4
Assume Γ torsion-free and s > . Let R = R and V = P n ( R ) . (a) If n ≥ , then there is a natural exact sequence → H q (Γ , V ) → H q +1 (Γ , V ) → H (Γ , V ) N g ( q ) → . (b) If n = 0 and s > , then there is an exact sequence → R N g ( q ) → H q (Γ , R ) → H q +1 (Γ , R ) → H (Γ , R ) N g ( q ) → . (c) If n = 0 = s , then there is an exact sequence → R N g ( q ) → H q (Γ , R ) → H q +1 (Γ , R ) → H (Γ , R ) N g ( q ) → R N g ( q − → . Remark.
From [14] or [9] we takedim H (Γ , V n ) = (2 g + s − n + 1) n ≥ , g + s − n = 0 , s > , g n = 0 , s = 0 . Let ¯ N g ( q ) = 1 + N g (1) + · · · + N g ( q ) = dim A/J q +1 . The theorem impliesthat dim H q (Γ , V n ) = ¯ N g ( q − g + s − n + 1) n ≥ , ¯ N g ( q − g + s −
2) + 1 n = 0 , s > ,N g ( q ) n = 0 , s = 0 . Proof:
Consider the exact sequence0 → J q /J q +1 → A/J q +1 → A/J q → , which gives0 → Hom(
A/J q , V ) → Hom(
A/J q +1 , V ) → Hom( J q /J q +1 , V ) → IGHER ORDER GROUP COHOMOLOGY → Ext ( A/J q , V ) → Ext ( A/J q +1 , V ) → Ext ( J q /J q +1 , V ) → . Start with the case when n ≥
1. Then the first row is zero by Lemma 2.2,and the remaining sequence reads0 → Ext ( A/J q , V ) → Ext ( A/J q +1 , V ) → Ext ( J q /J q +1 , V ) → . The first assertion follows. For the second, assume n = 0. Then the firstmap in the long exact sequence is an isomorphism and so the second is zero.We get an exact sequence0 → Hom( J q /J q +1 , R ) → Ext ( A/J q , R ) →→ Ext ( A/J q +1 , R ) → Ext ( J q /J q +1 , R ) → . The last assertion is clear from this. To prove the last item one uses the longexact sequence from the proof of Lemma 2.3 (b). (cid:3)
Let V be a R [Γ]-module. Let C be the set of all cusps of Γ. For c ∈ C , letΓ c be its stabilizer in Γ. We define the parabolic cohomology , H pq, par (Γ , V ) tobe the kernel of the restriction mapH pq (Γ , Γ par , V ) → Y c ∈ C H p (Γ c , V ) . Any γ ∈ Γ induces an isomorphism H p (Γ c , V ) → H p (Γ γc , V ). So it sufficesto extend the product over a set of representatives of Γ \ C . Theorem 2.5
Assume Γ torsion-free. Let R = R and V = P n ( R ) . (a) If n = 0 , then there is a natural isomorphism H q, par (Γ , R ) ∼ = Hom R ( J q /J q +1 , R ) . Consequently, dim H q, par = N g ( q ) . IGHER ORDER GROUP COHOMOLOGY If n ≥ , one has an exact sequence → H q, par (Γ , V ) → H q +1 , par (Γ , V ) → H (Γ , V ) N g ( q ) → . Consequently, dim H q, par (Γ , V ) = ¯ N g ( q − g − n + 1) + sn ) . Proof: (a) Let n = 0. The cocycle representation identifies H q (Γ , R ) withHom A ( J q , R ) ∼ = Hom R ( J q /IJ q , R ). ThenH q, par (Γ , R ) = Hom R ( J q /IJ q + I Σ , R ) ∼ = Hom R ( J q /J q +1 , R ) . (b) Let C be the set of all cusps. We identify Γ \ C with a set of representa-tives. Let c be a cusp and A c = R [Γ c ]. Write I c for its augmentation ideal.Consider the exact sequence of A -modules,0 → J q /J q +1 → A/J q +1 → A/J q → . As A c -modules, these are all direct products of copies of A c /I c ∼ = R , henceby restriction one obtains the sequence0 → R N g ( q ) → R ¯ N g ( q ) → R ¯ N g ( q − → . We get a commutative diagram with exact rows and columns, q, par (Γ , V ) H q +1 , par (Γ , V ) H (Γ , V ) N g ( q ) q (Γ , V ) H q +1 (Γ , V ) H (Γ , V ) N g ( q ) Y c ∈ Γ \ C H (Γ c , V ) ¯ N g ( q − Y c ∈ Γ \ C H (Γ c , V ) ¯ N g ( q ) Y c ∈ Γ \ C H (Γ c , V ) N g ( q ) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / (cid:15) (cid:15) / / (cid:15) (cid:15) (cid:15) (cid:15) / / / / (cid:15) (cid:15) η q / / / / (cid:15) (cid:15) η q +1 (cid:15) (cid:15) ϕ / / / / / / α Here the columns are exact by the very definition of parabolic cohomology.By the snake lemma, the triviality of the cokernel of η will give the desiredexactness. We prove this for q = 1 first. The Γ-module V induces a locally IGHER ORDER GROUP COHOMOLOGY V , on Γ \ H and H (Γ , V ) equals the sheaf co-homology. The restriction H (Γ , V ) → H (Γ c , V ) is the restriction of sheafcohomology to a cusp-section in Γ \ H . As Γ \ H is compact up to cusp sections,one gets an exact sequence,H c (Γ \ H , V ) → H (Γ , V ) → Y c ∈ Γ \ C H (Γ s , V ) → H c (Γ \ H , V ) → . Here H c means cohomology with compact supports and the last zero isH (Γ , V ) = 0. The space H c (Γ \ H , V ) is dual to H (Γ \ H , V ) = H (Γ , V )by Poincar´e duality. The latter space is zero, as we assume n ≥ η and ϕ , so both cokernels are zero. By thesnake lemma, we have an exact sequence,coker( η q ) → coker( η q +1 ) → coker( ϕ ) . For q = 1 the left and right are zero, so the middle vanishes as well. Nextfor q = 2 the same holds true. Inductively, we get coker( η q ) = 0 for every q .Finally, the dimension formula follows fromdim H (Γ , V n ) = ( (2 g − n + 1) + sn n ≥ , g n = 0 . (cid:3) For k ∈ Z and f : H → C define( f | k γ )( z ) = ( cz + d ) − k f ( γz ) , where γ = ± (cid:18) ∗ ∗ c d (cid:19) ∈ G . By linearity, we extend the definition f | k σ toelements σ of the group ring R [Γ]. Let k ≥ S k (Γ) be thespace of cusp forms of weight k , i.e., the complex vector space of all • f : H → C holomorphic, • f | k ( γ −
1) = 0 for every γ ∈ Γ, IGHER ORDER GROUP COHOMOLOGY • for every cusp c of Γ, the function ( f | k σ c )( z ) is O ( e − dy ) as y → + ∞ for some d > γf for f | k ( γ − ), so we can write it as a left action.We now define cusp forms of higher order. First let S k (Γ) = S k (Γ), soclassical cusp forms are of order 1. Next suppose S qk (Γ) is already definedand let S q +1 k (Γ) be the space of all functions f with • f : H → C holomorphic, • f | k ( γ − ∈ S qk (Γ) for every γ ∈ Γ, • for every cusp c , ( f | k σ c )( z ) = O ( e − dy ) as y → ∞ for some d > • f | k ( γ −
1) = 0 for every parabolic element γ ∈ Γ.Note that S qk (Γ) is annihilated by J q with Σ = Γ par . Proposition 2.6
For a, b, k, l ≥ we have S a +1 k (Γ) S b +1 l (Γ) ⊂ S a + b +1 k + l (Γ) . So the space S = M a,k ≥ S a +1 k (Γ) is a bigraded algebra.For f ∈ S qk (Γ) one has f | k γ ∈ S qk (Γ) . Proof:
The first assertion is done by induction on a + b , and the second isan easy calculation. (cid:3) Note [9] that for n ≥ S n +2 (Γ) = (2 g − n + 1) + ns. IGHER ORDER GROUP COHOMOLOGY In this section we will mostly be dealing with the ring R specialized to thefield of real numbers R . Some of the cohomological arguments will be validin greater generality. In those cases we will write R , whereas the use of theletter R indicates that we assume the ground ring to be R here.We define a P n ( C )-valued differential form δ n on H by δ n ( z ) = ( X − zY ) n dz. For a smooth function f on H we set ω ( f ) = 2 πif ( z ) δ n ( z ) . For any P n -valued form ω and γ ∈ G let γ ! ω = p n ( γ ) γ −∗ ω, where γ −∗ = ( γ − ) ∗ . We extend γ γ ! ω linearly to the group ring. Lemma 3.1
Let R = R and q ≥ . For f ∈ S qn +2 (Γ) one has m ! ω ( f ) = 0 for every m ∈ J q . Proof:
A calculation shows that for every γ = (cid:18) ∗ ∗ c d (cid:19) ∈ SL ( R ) one has γ ∗ δ n ( z ) = ( cz + d ) − n − p n ( γ ) δ n ( z ) . This implies p n ( γ − ) γ ∗ ω ( f ) = ω ( f | γ ) for every γ ∈ Γ. Replacing γ with γ − this means γ ! ω ( f ) = ω ( γf ). This extends linearly to m ∈ R [Γ] in place of γ .For m ∈ J q we have mf = 0. (cid:3) Let now f ∈ S qn +2 . For z ∈ H and γ ∈ Γ let ϕ z ( f )( γ ) = p n ( γ ) Z γ − zz Re ω ( f ) ∈ P n ( R ) . IGHER ORDER GROUP COHOMOLOGY f is holomorphic, the integral does not depend on the path from z to γz .One extends the map ϕ z ( f ) linearly to the group ring R [Γ]. This is equivalentto the classical construction of the Eichler-Shimura isomorphism as can befound in Hida’s book [9], where one rather uses the cocycle γ γϕ ( γ − ),however, for a group cocycle the latter agrees with − ϕ ( γ ). Theorem 3.2
Suppose Γ is torsion-free and q ≥ . Let Σ = Γ par . The map m ϕ z ( f )( m ) is in Hom A ( J q , P n ( R )) . The induced element of the space H q (Γ , Γ par , P n ( R )) lies in H q, par (Γ , P n ( R )) and does not depend on the choiceof z ∈ H .If n ≥ , then the map ϕ q thus defined, is an isomorphism of real vectorspaces, ϕ q : S qn +2 (Γ) ∼ = −→ H q, par (Γ , P n ( R )) . If n = 0 , the kernel of ϕ q is S q − (Γ) ⊂ S q (Γ) so ϕ q induces an injection S q (Γ) /S q − (Γ) ֒ → H q, par (Γ , R ) . One has dim S q (Γ) = dim H q, par (Γ , R ) for every q . Proof:
Let γ, τ ∈ Γ and compute ϕ z ( f )( γτ ) = p n ( γ ) p n ( τ ) Z τ − γ − zz Re ω ( f )= p n ( γ ) p n ( τ ) Z τ − zz Re ω ( f ) + p n ( γ ) p n ( τ ) Z τ − γ − zτ − z Re ω ( f )= p n ( γ ) ϕ z ( f )( τ ) + p n ( γ ) Z γ − zz Re τ ! ω ( f ) . By linearity, we can replace τ with m ∈ R [Γ]. In particular, for m ∈ J q weget ϕ z ( f )( γm ) = p n ( γ ) ϕ z ( f )( m ) . This is the desired A -linearity. IGHER ORDER GROUP COHOMOLOGY z . So let z ′ ∈ H and for γ ∈ Γ compute ϕ z ( f )( γ ) − ϕ z ′ ( f )( γ ) = p n ( γ ) Z γ − zz Re ω ( f ) − p n ( γ ) Z γ − z ′ z ′ Re ω ( f )= p n ( γ ) Z z ′ z Re ω ( f ) − p n ( γ ) Z γ − z ′ γ − z Re ω ( f )= p n ( γ ) Z z ′ z Re ω ( f ) − Z z ′ z Re γ ! ω ( f ) . Replacing γ with m ∈ J q one gets, ϕ z ( f )( m ) − ϕ z ′ ( f )( m ) = p n ( m ) Z z ′ z Re ω ( f ) . This means that the left hand side is of the form m mv for some v asclaimed.We next show that ϕ maps to the parabolic cohomology. Note that theexponential decay at the cusps allows to extend the definition of ϕ z ( f )( γ ) = p n ( γ ) R γ − zz Re ω ( f ) to the case when z is replaced by a cusp c , at least if weinsist that the integral path should be the geodesic in H from c to γc . So, if γ ∈ Γ c , i.e., γc = c , then ϕ c ( f )( γ ) = 0, which implies that ϕ c ( f ) zero on Γ c ,so ϕ ( f ) is indeed parabolic.We now show the injectivity of the map ϕ . Note that for γ ∈ Γ we havethat ( γ − J q − ⊂ J q , so multiplication by ( γ −
1) induces a map fromHom R [Γ] ( R [Γ] /J q , V ) to Hom R [Γ] ( R [Γ] /J q − , V ), which is functorial in V . Ascohomology is a universal δ -functor, we get a map( γ −
1) : H q (Γ , Σ , V ) → H q − (Γ , Σ , V ) . We get a commutative diagram S qn +2 (Γ) S q − n +2 (Γ)H q, par (Γ , V ) H q − , par (Γ , V ) . / / ( γ − (cid:15) (cid:15) ϕ q (cid:15) (cid:15) ϕ q − / / ( γ − IGHER ORDER GROUP COHOMOLOGY f be in the kernel of ϕ . By induction, ϕ q − is injective, so then it followsthat ( γ − f is zero for every γ ∈ Γ, hence f ∈ S n +2 (Γ) already. By the case q = 1, the map ϕ ( f ) ∈ Hom A ( J q , V ) then extends to I , hence gives a map I/J q → V . The image of this map ϕ ( f ) lies in V J q , which is zero for n ≥ ϕ ( f ) = 0, so f = 0 by the injectivity of the classical Eichler-Shimura map.This implies the injectivity in case n ≥ n = 0 we use induction on q . For q = 1 the claim follows fromthe classical Eichler-Shimura Isomorphism if we formally set S (Γ) = 0. Nowsuppose q ≥ q −
1. Let f ∈ S q (Γ) be in the kernelof ϕ q . Then ( γ − f ∈ ker ϕ q − = S q − (Γ), hence f ∈ S q − (Γ). For theother inclusion we consider the following diagram which is commutative bythe construction of the Eichler-Shimura map, S q − (Γ) S q (Γ)H q − , par (Γ , R ) H q, par (Γ , R )Hom A ( J q − /I Σ , R ) Hom A ( J q /I Σ , R ) . (cid:31) (cid:127) / / (cid:15) (cid:15) (cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31) ϕ q − (cid:15) (cid:15) (cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31) ϕ q (cid:15) (cid:15) (cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31) ∼ = (cid:15) (cid:15) (cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31) ∼ = / / res The restriction below is equal to zero. Hence S q − n +2 (Γ) maps into the kernelof ϕ q . The surjectivity of ϕ q for n ≥ n = 0 follow from our dimension formulae together with Corollary 3.13 andTheorem 4.1 of [6]. (cid:3) References [1]
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