Invariants, cohomology, and automorphic forms of higher order

0.31 MB·English

by  Anton Deitmar

Higher order invariants, cohomology, and
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automorphic forms
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o Anton Deitmar
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Abstract: Ageneralstructuretheoremonhigherorderinvariantsisproven.

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T For an arithmetic group, the structure of the corresponding Hecke module
N is determined. It is shown that the module does not contain any irreducible
. submodule. This explains the fact that L-functions of higher order forms
h
have no Euler-product. Higher order cohomology is introduced, classical re-
t
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sultsofBorelaregeneralized andahigherorderversionofBorel’sconjecture
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is stated.
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Contents
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1 1 Higher order invariants 3
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1
1.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1
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1.2 Hecke pairs and smooth modules . . . . . . . . . . . . . . . . 5
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:
v 1.3 The case of Z: distributions . . . . . . . . . . . . . . . . . . . 8
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1.4 Arithmetic groups . . . . . . . . . . . . . . . . . . . . . . . . 13
r
a
1.5 Cusp forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 Hodge structure. . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 Higher order cohomology 24
2.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 Sheaf cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Arithmetic groups . . . . . . . . . . . . . . . . . . . . . . . . 27
1
HIGHER ORDER INVARIANTS 2
2.4 The higher order Borel conjecture. . . . . . . . . . . . . . . . 37
Introduction
Higher order modular forms show up in various contexts. For instance, the
limit crossing probabilities in percolation theory, considered as functions of
the aspect ratio, turn out to be higher order forms [17]. As an approach to
the abc-conjecture, Dorian Goldfeld introduced Eisenstein-series twisted by
modular symbols [3, 12, 13], which are higher order forms. Finally, spaces
of higher order forms are natural receptacles of converse theorems [10, 16].
L-functions of second order forms have been studied in [7], Poincar´e series
attached to higher order forms have been investigated in [15], dimensions of
spaces of second order forms have been determined in [9, 8]. Higher order
cohomology has been introduced and an Eichler-Shimura type theorem has
been proven in [5]. In [6] a program has been started which aims at an
understanding of the theory of higher order forms from an representation-
theoretical point of view. The present paper contains a major step in that
direction. It is shown that the order-lowering-homomorphism introduced in
[6] injects the graded module of higher order forms into a canonical ten-
sor product. In the case of cusp forms of weight two and order one it is
shown that the order-lowering-homomorphism is indeed an isomorphism.
So in that case the representation theoretic nature of higher order forms is
completely understood. In the general situation it is not clear under which
circumstances the order-lowering-map is an isomorphism.
The paper is structured as follows. The first part is on higher order invari-
ants, the second on higher order cohomology, which is defined by the right
derived functors of the higher order invariant functor.
The first part starts by setting up the basic concepts of Hecke pairs and
smooth modules. It is shown that the limit order q module, L (V) injects
q
into a canonical tensor product H⊗q ⊗ V∞, where V∞ is the associated
smooth module. Already for the group Γ = Z one gets nontrivial exam-
ples: It turns out, that higher order invariant distributions are precisely
the Fourier transforms of Z-supported tempered distributions or tempered
combs. We then turn to arithmetic groups. In the case of rank zero, the
canonical module H is shown to be a direct sum of irreducible automor-
phic representations. For cusp forms, a similar statement holds by Eichler-
Shimura theory. Also, for cusp forms the completed limit module is indeed
HIGHER ORDER INVARIANTS 3
isomorphic to the completed tensor product. We show that this tensor
product, and thus the completed limit, does not contain any irreducible
subrepresentation. This explains why L-functions of higher order forms do
not possess Euler-products. Finally, it is shown how the order-lowering ho-
momorphism helps to establish Hodge structures on spaces of higher order
forms.
Inthesecondpartthetheoryofhigherordersheafcohomologyisintroduced.
Themostimportantresulthereisthefactthatclassically acyclicsheaves are
higher-order acyclic as well. This facilitates the computation of higher order
cohomology. Finally, classical results of Borel on cohomology of arithmetic
groups are generalized to the higher order context and a generalized version
of Borel’s conjecture is stated.
1 Higher order invariants
1.1 Generalities
Let R be a commutative ring with unit. In the main applications, R will
be the field of complex numbers. Let Γ be a group and let P ⊂ Γ be a
conjugation-invariant subset. Let I denote the augmentation ideal in the
group algebra A= R[Γ]. For an A-module V we define the R-module
H0 (Γ,V)
q,P
of invariants of order q ≥ 0 to be the set of all v ∈ V with Iq+1v = 0 and
pv = v for every p ∈P. Note that for q = 0 one gets the usual invariants
H0 (Γ,V) = H0(Γ,V) = VΓ.
0,P
Another way to describe the space H0 (Γ,V) is as follows. Let hPi ⊂ Γ be
q,P
the subgroup generated by P and let I be the augmentation ideal of the
P
group hPi. As P is conjugation-invariant, the group hPi is normal in Γ and
so AI is a two-sided ideal in A. Let
P
J d=ef Iq+1+AI .
q+1 P
Then J is a two-sided ideal of A and H0 (Γ,V) is the space of all v ∈ V
q+1 q,P
with J v = 0. There is a natural identification
q+1
H0 (Γ,V) ∼= Hom (A/J ,V).
q,P A q+1
HIGHER ORDER INVARIANTS 4
If P is the empty set or {1}, we write H0(Γ,V) for H0 (Γ,V).
q q,P
The sets H0 (Γ,V) ⊂ H0 (Γ,V) form a filtration on V which is not
q,P q+1,P
necessarily exhaustive. Let
H¯ (Γ,V) = GrH0 (Γ,V) d=ef H0 (Γ,V)/H0 (Γ,V)
q,P q,P q,P q−1,P
be the q-th graded piece, where we allow q = 0,1,2,... by formally setting
H0 (Γ,V) = 0.
−1,P
For any group G, let Hom (Γ,G) be the set of homomorphisms from the
P
group Γ to G which send P to the neutral element. So Hom (Γ,G) ∼=
P
Hom(Γ/hPi,G). We will in particular use this notation for G being the
additive group of a vector space W.
Lemma 1.1.1 Suppose that R is a field.
If Γ is finitely generated or dimW <∞, then
Hom (Γ,W) ∼= Hom (Γ,C)⊗W,
P P
where the tensor product is over R.
Proof: There is an injective map φ : Hom (Γ,C) ⊗ W → Hom (Γ,W)
P P
mapping α⊗w to γ 7→ α(γ)w. We have to show that it is surjective under
the conditions given. Pick a basis (w ) of W. If Γ is finitely generated
i i∈I
or dimW < ∞, then, for a given homomorphism η : Γ → W = Cv
i i
there exists finite set F ⊂ I such that η(Γ) ⊂ U = Cw = Cw .
i∈F i i∈LF i
The formula η(γ) = η (γ)w defines η ∈ Hom (Γ,C) and one has
i∈F i i i LP Q
η = φ( η ⊗v ). (cid:3)
i∈F i i P
P
We introduce the order-lowering-homomorphism
ψ :H¯ (Γ,V)→ Hom (Γ,H¯ (Γ,V))
q,P P q−1,P
given by
ψ(v)(γ) = (γ −1)v.
To see that ψ(v) is indeed a group homomorphism, note that in the group
algebra C[Γ] one has (τγ −1) ≡ (τ −1)+(γ −1) mod I2.
Lemma 1.1.2 Let V be an R[Γ]-module which is torsion-free as Z-module.
Let Σ ⊂ Γ be a subgroup of finite index. Then the natural restriction map
res : H¯ (Γ,V) → H¯ (Σ,V)
q,P q,P
is injective. Here on the right-hand-side read P ∩Σ instead of P.
HIGHER ORDER INVARIANTS 5
Note that the torsion-condition is automatic if R contains the field Q.
Proof: Weprovethisbyinductiononq. Forq = 0therestrictionmapisthe
inclusion VΓ ֒→ VΣ and thus injective. For q ≥ 1 we have the commutative
diagram
H¯ (Γ,V) res // H¯ (Σ,V)
q,P (cid:127)_ q,P (cid:127)_
φ φ
(cid:15)(cid:15) (cid:15)(cid:15)
Hom (Γ,H¯ (Γ,V)) res// Hom (Σ,H¯ (Σ,V)).
P q−1,P P q−1,P
To see that the top row is injective, we have to show that the bottom row
is injective. The bottom row is the composition of two maps
Hom (Γ,H¯ (Γ,V)) → Hom (Σ,H¯ (Γ,V))
P q−1,P P q−1,P
→ Hom (Σ,H¯ (Σ,V)),
P q−1,P
the firstof which is injective as Σ is of finite index and V is torsion-free, and
the second is injective by induction hypothesis. (cid:3)
1.2 Hecke pairs and smooth modules
A Hecke pair is a pair (G,Γ) of a group G and a subgroup Γ such that for
every g ∈ G the set ΓgΓ/Γ is finite. We also say that Γ is a Hecke subgroup
of G.
Two subgroups Γ,Λ of a group H are called commensurable, written Γ ∼ Λ,
if the intersection Γ∩Λ has finite index in both. Commensurability is an
equivalence relation which is preserved by automorphisms of H.
The commensurator of a group Γ ⊂ H is
comm(Γ) d=ef {h ∈ H :Γ and hΓh−1 are commensurable}.
Lemma 1.2.1 The commensurator G = comm(Γ) is a subgroup of H. It is
the largest subgroup such that (G,Γ) is a Hecke pair. More precisely,
comm(Γ) = {h ∈ H :|ΓhΓ/Γ|,|Γ\ΓhΓ| < ∞}.
Proof: Let G= comm(Γ) and g ∈ G. Then Γ ∼ gΓg−1. By conjugating we
get g−1Γg ∼ Γ, hence g−1 ∈ G. Next let h ∈ G as well. Then Γ ∼ hΓh−1
HIGHER ORDER INVARIANTS 6
and hence gΓg−1 ∼ ghΓ(gh)−1 so that γ ∼ gΓg−1 implies Γ ∼ ghΓ(gh)−1,
which means that gh ∈ G, so G is indeed a subgroup.
To see the identity claimed in the lemma, one simply observes that for every
h ∈ H the natural map Γ/Γ∩hΓh−1 → ΓhΓ/Γ, mapping the class of γ to
the class of γh is a bijection. (cid:3)
Let G be a group. By a G-module we shall henceforth mean a R[G]-module.
If G is a totally disconnected topological group, an element v of a G-module
V iscalledsmooth ifitisstabilizedbysomeopensubgroupofthetopological
group G. Theset V∞ of all smoothelements is asubmoduleand themodule
V is called smooth if V = V∞.
Drop the condition that G be a topological group and let (G,Γ) be a Hecke-
pair. A congruence subgroup of Γ is any subgroup which contains a group
of the form
Γ∩g Γg−1∩···∩g Γg−1
1 1 n n
forsomeg ,...,g ∈ G. As(G,Γ)isaHeckepair,everycongruencesubgroup
1 n
has finite index in Γ. Note that this definition of a congruence subgroup
coincides with the one given in [6]. For every congruence subgroup Σ equip
the set G/Σ with the discrete topology and consider the topological space
G¯ d=ef limG/Σ,
←
Σ
where the limit is taken over all congruence subgroups Σ.
Lemma 1.2.2 (a) The intersection of all congruence subgroups N = Σ
Σ
is a normal subgroup of G.
T
(b) The natural map p : G → G¯ factors through the injection G/N ֒→ G¯ and
has dense image.
(c) The group multiplication on G/N extends by continuity to G¯ and makes
G¯ to a totally disconnected locally compact group.
We call G¯ the congruence completion of G. Although the notation doesn’t
reflect this, the completion G¯ depends on the choice of the Hecke subgroup
Γ. A Hecke subgroup Γ is called effective, if the normal subgroup N above
is trivial.
HIGHER ORDER INVARIANTS 7
Proof: (a) Let n ∈ N and let g ∈ G. For a given congruence subgroup
Σ we have that n ∈ Σ∩g−1Σg, and so gng−1 ∈ Σ. As Σ varies, we find
gng−1 ∈ N.
(b) Let g,g′ ∈ G with p(g) = p(g′). This means that gΣ = g′Σ for every
congruence subgroup and so gN = g′N. For given (g ) ∈ G¯ the sets
Σ Σ
U = {h ∈ G¯: h Σ = g Σ} form a neighborhood base. Clearly the element
Σ Σ Σ
g ∈ G is mapped into U , so the image of p is dense.
Σ Σ
(c) Let g¯ = (g ) ∈ G¯. Then the net (p(g )) converges to g¯. For h¯ =
Σ Σ Σ Σ
(h ) ∈ G¯ it is easy to see that the net (p(g h )) converges in G¯. We call
Σ Σ Σ Σ
the limit g¯h¯. This multiplication has the desired properties. (cid:3)
The initial topology defined by p on G makes G a topological group with the
congruence groups forming a unit neighborhood base. Clearly every smooth
G¯-moduleisasmoothG-modulebyrestriction. Butalsotheconverseistrue:
Every smooth G-module extends uniquely to a smooth G¯-module and these
two operations of restriction and extension are inverse to each other.
Let P ⊂ G bea conjugation-invariant set. For any subgroupΣ of Γ we write
H0 (Σ,V) for H (Σ,V). Let
q,P q,O∩Σ
L (Γ,V) d=ef limH¯ (Σ,V),
q,P q,P
→
Σ
where the limit is taken over all congruence subgroups of Γ. Note that
L (Γ,V) =V∞. For g ∈ G, the map induced by g:
0,P
H¯ (Σ,V) → H¯0 (gΣg−1,V) −r→es H¯0 (Σ∩gΣg−1,V)
q,P q,P q,P
defines an action of G on L (Γ,V), which makes the latter a smooth mod-
q,P
ule.
Assume from now on that Γ is finitely generated and R is a field. Then
every finite-index subgroup Σ ⊂ Γ is finitely generated as well, so Lemma
1.1.1 applies to all modules V.
Consider the order-lowering map
H¯ (Σ,V) ֒→ Hom (Σ,C)⊗H¯ (Σ,V),
q,P P q−1,P
where Hom (Σ,C) is the set of all homomorphisms Σ → C that vanish on
P
Σ∩P. Iteration gives
H¯ (Σ,V) ֒→ Hom (Σ,C)⊗q ⊗VΣ.
q,P P
HIGHER ORDER INVARIANTS 8
Taking limits we get an injection
⊗q
L (Γ,V)֒→ limHom (Σ,C) ⊗V∞.
q,P P
→
!
Σ
Let H be the space limHom (Σ,C). For g ∈ G one gets a map
Γ,P P
→
Σ
Hom (Σ,C) → Hom (gΣg−1,C) −r→es Hom (Σ∩gΣg−1,C),
P P P
which makes H a smooth module. We have shown:
Γ,P
Proposition 1.2.3 If Γ is finitely generated and R is a field, then there is
a natural injection of smooth modules
L (Γ,V) ֒→ H⊗q ⊗V∞.
q,P Γ,P
Example. Let (G,Γ) = (R,Z), then H = Hom(Γ,C) is one-dimensional.
Γ
For the trivial module V = C one has L (Γ,C) = 0 for q ≥ 1, whereas
q
V∞ = V = C, so, in this case, the map L (Γ,V) → H⊗q ⊗ V∞ is not
q Γ
surjective.
1.3 The case of Z: distributions
In this section we show (Theorem 1.3.3) that the higher order Z-invariant
distributions on R are precisely the Fourier transforms of Z-supported tem-
pered distributions. In this section the ring R will be the field C of complex
numbers.
Let D = C∞(R) and let D′ = C∞(R)′ be the space of distributions. Let
c c
S denote the space of Schwartz functions on R and let S′ be the space of
tempered distributions.
For f ∈ D and T ∈D′ or f ∈S and T ∈ S′ we write hT | fi for T(f).
Let F :S → S denote the Fourier transform normalized by
F(f)(x) d=ef f(y)e−2πixydy.
R
Z
By the same letter we denote the Fourier transform F :S′ → S′ defined by
hF(T) |fi = T | F−1(f) for T ∈ S′ and f ∈ S.
(cid:10) (cid:11)
HIGHER ORDER INVARIANTS 9
The space of Z-periodic distributions is denoted D′Z and for any closed set
S ⊂ R the set of distributions supported in S is denoted by D′ .
S
Lemma 1.3.1 Every element T of D′ can uniquely be written in the form
Z
Nk
T = a ∂nδ ,
k,n k
k∈Zn=0
XX
where δ is the Dirac distribution at k and a ∈ C is arbitrary. The
k k,n
distribution T is tempered if and only if there exist m,N ∈ N and C > 0
such that
• N ≤ N for every k ∈ Z, and
k
• |a |≤ C|k|m for every k ∈ Z and every n = 0,...,N .
k,n k
Proof: A distribution supported at a point x is a linear combination of
derivatives of δ . This proves the first assertion. If the coefficients satisfy
x
the conditions, then it is clear that T is tempered. For the converse assume
thatT istempered. Thetopology of S is definedbythefamily of seminorms
(ρm,n)m,n∈N0, where ρm,n(f) = supx∈R(1+x2m)f(n)(x). As T is continuous,
there exist M,N ∈ N and C > 0 such that |T(f)| ≤ Cρ (f) holds for
m,N
every f ∈ S. Then m,N satisfy the claim. (cid:3)
For a distribution T ∈ S′ = D′ ∩S′ as in the last lemma, we define the
Z Z
order of T to be
ord(T) d=ef max{n : ∃k :a 6= 0}
k,n
if T 6= 0 and ord(0) = 0. The order defines a natural filtration on S′, the
Z
order filtration.
O0S′ ⊂ O1S′ ⊂ ··· ⊂ S′.
Z Z Z
Note that Lemma 1.3.1 implies that this filtration is exhaustive, i.e., that
∞
S′ = OqS′.
Z Z
q=1
[
EveryZ-invariantdistributionisalreadytempered,soD′Z ⊂ S′,sowemight
as well write this space as S′Z. The Fourier transform maps S′Z into S′.
Z
HIGHER ORDER INVARIANTS 10
Lemma 1.3.2 The image F(S′Z) is the set of distributions T ∈ S′ of order
Z
zero.
Proof: Viewing a distribution as a section of the sheaf of distributions,
it becomes clear that the set of Z-invariant distributions can be identified
with the set of distributions on the manifold R/Z, i.e., there is a canonical
isomorphismψ : C∞(R/Z)′ → S′Z.Thisisomorphismisdefinedbytheequa-
c
Z Z
tion hψ(T) | fi = T | f , where f (x) = f(x+k). Taking Fourier
k∈Z
coefficients is a linear isomorphism
(cid:10) (cid:11) P
Cc∞(R) → {(ck)k∈Z : the sequence is rapidly decreasing}.
So a distribution on R/Z is given by a moderately growing sequence (dk)k∈Z
via the map (ck)k∈Z 7→ kckdk. (cid:3)
P
Let
∞
H0 (Z,S′) d=ef H0(Z,S′)
∞ q
q=0
[
be the space of higher order Z-invariants in S′.
Theorem 1.3.3 For every q ≥ 0 we have H0(Z,D′) = H0(Z,S′). The
q q
Fourier transform is a linear bijection F : H0 (Z,S′)−∼=→ S′, which respects
∞ Z
order filtrations, i.e.,
F(Hq0(Z,S′)) = OqSZ′
holds for every q ∈ N.
Proof: For a polynomial p(x) = a +a x+···+a xn in C[x] we write
0 1 n
|p| = max|a |.
j
j
A family (pk)k∈Z of polynomials is said to be of moderate growth, if there
exist d ∈ N and C > 0 with |p |≤ C(1+|k|d) holds for every k ∈ Z.
k
Lemma 1.3.4 Let q ∈N and let (pk)k∈Z be a family of polynomials in C[x]
of degree ≤ q. Then there exists a unique family (qk)k∈Z of polynomials with
q (x+1)−q (x) = p (x)
k k k
and q (0) = 0. It follows that the degree of each q is ≤ q.
k k
If the family (p ) is of moderate growth, then so is the family (q ).
k k