Version of 12.02.97
On dynamical systems and their possible
significance for arithmetic geometry
by Christopher Deninger
0
Introduction
In the papers [D1], [D5], [D3] a cohomological formalism for algebraic schemes X0
over spec ZZ or spec Q was conjectured which would explain many of the expected
properties of motivic L-series. All consequences of this very rigid formalism that I
could imagine turned out to be either provable [D2], [D4], [DN], [Sa] or to amount
to some well known conjectures on L-series of motives as for example the Riemann
hypotheses and the Artin conjecture – both generalized to the context of motives –
and the Bloch Beilinson conjectures on vanishing orders.
The cohomologies in question are equipped with an endomorphism Θ whose eigenvalues on the H 1 of spec ZZ for example should be the zeroes of ζ(1−s). For varieties
over finite fields the cohomology theory can be constructed from l-adic cohomology
but this definition is quite artificial and should rather become a comparison isomorphism at a later stage of the theory. Still it shows that Θ behaves as a derivation
with respect to cup product. It is therefore natural to consider etΘ for real t which
acts by endomorphisms of the cohomology ring. Hence we are suggested the following possibility: To every X0 as above there should be functorially attached a
dynamical system (X, φt ) consisting of some space X with an action φt of lR or
lR≥0 . The proposed cohomologies should arise by applying a suitable “dynamical”
cohomology theory to (X, φt ) and we should have φt∗ = etΘ where φt∗ denotes the
(semi)-flow induced on cohomology by the (semi)-flow φt on X.
For this approach two questions have to be answered:
a. What is the relevant dynamical system (X, φt ) attached to a scheme X0 ?
b. What is the “suitable dynamical cohomology theory” above?
The system in a should have the following basic property: The non-constant closed
orbits or at least certain distinguished ones should be in bijection with the closed
points x0 of X0 . The minimal period of the closed orbit corresponding to x0 should
equal log N (x0 ). The only natural candidates (X, φt ) at least for X0 the spectrum
of the ring of integers in a number field k that are known to me were constructed
by Connes in a somewhat different setting in the very interesting note [C2]. There
the space X is the quotient of IAk /k ∗ viewed as a non-commutative space and the
flow under consideration is given by the natural action of the idèle class group.
On the other hand the above mentioned conjectural cohomological and sheaf theoretic formalism of [D1], [D5] which was abstracted from properties of L-series runs
1
parallel to e.g. the formalism of l-adic cohomology for varieties over finite fields.
This suggests that a natural solution to a can even be found with X a “commutative” topological space but no natural candidate has yet been constructed. For
definiteness we assume that X can be taken to be a Banach manifold although other
possibilities must be kept in mind. Our strategy in this paper is to first adress question b to some extent and then to investigate what properties dynamical systems
must have in order that their dynamical cohomologies be isomorphic to the conjectured cohomology groups described in [D1], [D5]. In this program we are partly
successful: In section two we single out a class of dynamical systems that should
contain all systems attached to schemes but is still much too vast. On the other
hand from a dynamical systems point of view this class is very special. In particular
the dynamical systems given by the geodesic flow on locally symmetric spaces do
not belong to it which may account for the differences in the analytic behaviour of
arithmetic and geometric zeta functions. After a study of this class in section three
we pursue our strategy in section four and obtain topological information on the
systems that should appear in a and some possibly useful hints as to their construction. Also our study shows that the present approach of realizing the conjectured
cohomologies of [D1], [D5] as cohomologies of dynamical systems requires X0 /ZZ to
be in some sense ordinary if X0 ⊗ Q is empty e.g. if X0 is defined over a finite field.
Let us call a dynamical cohomology theory “topological” resp. “coherent” if there is
a “dynamical” Lefschetz trace formula which accounts for the closed orbits in terms
of cohomology in much the same way as the usual topological resp. holomorphic
Lefschetz trace formulas account for fixed points. Compare 3.20 resp. 2.2 for the
kind of trace formulas we have in mind. In b a topological dynamical cohomology
theory is required. Unfortunately in spite of much effort no satisfactory dynamical
cohomology neither topological nor coherent has yet been developed. However Patterson has suggested in [P] that for Anosov flows on finite dimensional manifolds a
version of the leafwise cohomology with respect to the unstable foliation might be
a good candidate for a coherent dynamical cohomology theory. Our approach to
question b in section two is based on his inspiring ideas.
Motives appear for two reasons in this paper: On the one hand the dynamical
formalism should ultimately shed light on the properties of general motivic L-series.
On the other hand even if one is interested only in the Riemann zeta function
a study of the dynamical system (X, φt ) that should be attached to spec ZZ will
benefit from the theory of motives for the following reason: In our picture motives
over Q with good reduction everywhere will give rise to local systems on X and hence
give information on the fundamental group of X, one of its most basic topological
invariants.
Section one of the present paper which contains background material on flows and
the exterior calculus adapted to foliations on Banach manifolds and also section
three do not depend on any conjecture. On the other hand sections two and four
do where we compare dynamical and arithmetic cohomologies.
I would like to express my hearty thanks to Y. Ihara for inviting me to Kyoto where
much of this work was done and for his interest. I would also like to thank A.
2
Beilinson, N. Kurokawa, S.J. Patterson, M. Puschnigg, A.J. Scholl and W. Singhof
very much for helpful remarks and discussions. I am very grateful to the RIMS for
providing ideal working conditions and for supporting my stay.
3
1
Background on flows and the exterior calculus
adapted to foliations
In this section we first indicate a couple of notions and facts concerning connections
and differential forms on foliated manifolds. We will work in the category of real
smooth paracompact Hausdorff Banach manifolds and smooth maps between them.
The manifolds are not assumed to be connected. Vector bundles are modelled on
Banach spaces and are also supposed to be smooth. In particular subbundles are
always closed. For such bundles E, T, . . . the sheaves of smooth local sections are
denoted by E, T , . . . Some background for the following in the infinite dimensional
context which requires some care can be found in [L], [V]. The sheaf of germs of
smooth real resp. complex valued functions on a Banach manifold X is denoted by
A = AX resp. Ac = AcX . We write Λp E1∗ ⊗ Λq E2∗ ⊗ E3 for the sheaf of smooth sections
of the bundle of continuous multilinear maps E1p × E2q → E3 which are alternating
in the first p and the last q coordinates c.f. [L] III § 4. Note that this is an abuse of
notation if E1 or E2 has infinite dimensional fibres.
For a Banach manifold X with tangent bundle T = T X the algebraic direct sum
ΛT ∗ :=
M
Λp T ∗
p≥0
still carries an exterior product ∧ satisfying the usual rules [V] IV, 3.
A subbundle T0 ⊂ T is said to be integrable if one of the following equivalent
conditions holds:
a) [T0 , T0 ] ⊂ T0 i.e. T0 is a sub Lie algebra of T .
b) dJ ⊂ J where J ⊂ ΛT ∗ is the ideal of forms which vanish on T0 i.e. if all
coordinates are in T0 .
The equivalence of these conditions follows from the formula:
hdλ, Y0 × . . . × Yp i =
p
X
(−1)i Yi hλ, Y0 × . . . × Ŷi × . . . × Yp i
i=0
(1)
+
X
(−1)i+j hλ, [Yi , Yj ] × Y0 × . . . × Ŷi × . . . × Ŷj × . . . × Yp i
i<j
for local sections λ of Λp T ∗ and Y0 , . . . , Yp of T .
The Frobenius theorem holds in our context [L] VI and hence T0 induces a smooth
foliation on X. If T0 is integrable we define the leafwise exterior derivative
dT0 : Λp T0∗ −→ Λp+1 T0∗
by setting hdT0 λ, Y0 × . . . × Yp i equal to the right hand side of (1) for λ ∈ Λp T0∗ and
Y0 , . . . , Yp ∈ T0 . One checks that d2T0 = 0 and that we obtain a map of complexes
(Λ• T ∗ , d) −→→ (Λ• T0∗ , dT0 ) .
More generally for any two integrable subbundles T0 ⊂ T 0 ⊂ T we get a natural
map of complexes:
0
(Λ• T ∗ , dT 0 ) −→→ (Λ• T0∗ , dT0 ) .
4
Any subbundle T1 ⊂ T 0 with T0 ⊕ T1 = T 0 gives a splitting of this map i.e. the
leafwise derivative dT0 can also be described as the composition:
(2)
dT0 : Λp T0∗ ,→ Λp T
0∗
d
0
T
−→
Λp+1 T
0∗
−→→ Λp+1 T0∗
where the first map depends on T1 .
Let E be a vector bundle on X with associated sheaf of sections E. Given an
integrable subbundle T0 of T a T0 -connection on E or E is a map
δ = δT0 : E −→ T0∗ ⊗ E
which satisfies the Leibnitz rule:
δ(f e) = dT0 f ⊗ e + f δ(e)
for local sections f of A and e of E. The associated connection 5 : T0 ⊗lR E → E
given by the formula:
5Y e = 5(Y ⊗ e) := hδe, Y i for e ∈ E, Y ∈ T0
then satisfies the following condition: In any local chart the Christoffel symbol
Γ5 (Y, e) = 5Y (e) − hDe, Y i
is given by a smooth map from the chart to the space of continuous bilinear maps
from IB × IE to IE where IB and IE are the Banach spaces on which X and E are
modelled [V] IV.4.
Given δ we can extend it to a map
δ : Λp T0∗ ⊗ E −→ Λp+1 T0∗ ⊗ E
for all p ≥ 0
by setting:
(3)hδλ, Y0 × . . . × Yp i =
p
X
i=0
+
(−1)i 5Yi hλ, Y0 × . . . × Ŷi × . . . × Yp i
X
(−1)i+j hλ, [Yi , Yj ] × . . . × Ŷi × . . . × Ŷj × . . . × Yp i
i<j
for λ ∈ Λp T0∗ ⊗ E and Y0 , . . . , Yp ∈ T0 .
The curvature of δ is the A-linear map
R = δ ◦ δ : E −→ Λ2 T0∗ ⊗ E .
The T0 -connection δ is called flat or integrable if R = 0. In this case the diagram
(4)
δ
δ
δ
E −→ T0∗ ⊗ E −→ Λ2 T0∗ ⊗ E −→ . . .
is a complex and by the Frobenius theorem E becomes a T0 - foliated vector bundle
i.e. a bundle whose transition functions are foliated in some and hence every T0 foliated atlas for X. Recall that functions or more generally sections are said to be
5
T0 -foliated if they are locally constant on the leaves of the T0 -foliation. We write E T0
or E δ for E with this reduction of the structural group. The sheaf E T0 of smooth
foliated sections of E is given by
δ
E T0 = E δ := Ker (E −→ T0∗ ⊗ E) .
On the other hand given a T0 -foliated vector bundle F the leafwise exterior derivative
dT0 can be extended to a flat T0 -connection on the sheaf of all smooth sections of
F . One obtains an equivalence between the category of vector bundles with a flat
T0 -connection on X and the category of T0 -foliated bundles on X. Using this remark
or working directly with (E, δ) the usual proof of the Poincaré lemma e.g. [V] Ch.
5 extends to give:
1.1 Foliated Poincaré Lemma: If δ is a flat T0 -connection on E the complex (4)
is a resolution of E δ . If X admits smooth partitions of unity the sheaves in (4) are
fine.
We next investigate a notion of compatibility for connections on E with respect to
two different subbundles of T . First we note the following facts:
1.2 Proposition: Consider integrable subbundles T0 , T1 , T 0 of T such that
T 0 = T0 ⊕ T1 . Then we have the following assertions:
0
1) The exterior algebra Λ• T ∗ inherits a natural bigraduation by the sheaves
Λp T0∗ ⊗ Λq T1∗ and dT 0 decomposes as dT 0 = d0 + d1 where d0 has bidegree (1, 0)
and d1 has bidegree (0, 1).
2) The restriction di : Λ• Ti∗ → Λ•+1 Ti∗ agrees with dTi for i = 0, 1.
3) The restriction d0 : Λp T0∗ ⊗ Λq T1∗ → Λp+1 T0∗ ⊗ Λq T1∗ equals the extension of the
flat T0 -connection on Λq T1∗ induced by the flat T0 -connection d0 |T1∗ : T1∗ → T0∗ ⊗ T1∗ .
4) Consider the family of isomorphisms
∼
∧ : Λq T1∗ ⊗ Λp T0∗ −→ Λp T0∗ ⊗ Λq T1∗
given by ∧ (β ⊗ α) = (−1)pq α ⊗ β .
in finite dimensions and by
h∧η, Y1 × Y0 i = (−1)pq hη, Y0 × Y1 i for Y0 ∈ T0p , Y1 ∈ T1q in general .
Then ∧−1 ◦ d1 ◦ ∧ : Λq T1∗ ⊗ Λp T0∗ → Λq+1 T1∗ ⊗ Λp T0∗ equals the extension of the flat
T1 -connection on Λp T0∗ induced by the flat T1 -connection ∧−1 ◦ d1 |T0∗ : T0∗ → T1∗ ⊗T0∗ .
Proof: 1) Set Apq = Λp T0∗ ⊗ Λq T1∗ . We have to show that
dT 0 Apq ⊂ Ap+1,q ⊕ Ap,q+1 .
The assertion for dT 0 A10 for example follows from the formula:
hdT 0 ω, Y0 × Y1 i = Y0 (hω, Y1 i) − Y1 (hω, Y0 i) − hω, [Y0, Y1 ]i
6
and the integrability of T1 which implies that dT 0 ω has no component in A02 if
ω ∈ A10 . The general case follows similarly using equation (1). The remaining
assertions of the proposition are clear.
We now assume that in the situation of the proposition we are given flat Ti -connections
δi on a vector bundle E over X for i = 0, 1. Using the flat T0 -connection
d0 : Λq T1∗ −→ T0∗ ⊗ Λq T1∗ ,
we get a flat T0 -connection
δ0 : Λq T1∗ ⊗ E −→ T0∗ ⊗ Λq T1∗ ⊗ E
which in the finite dimensional case is given by:
δ0 (β ⊗ e) = d0 (β) ⊗ e + (−1)q β ∧ δ0 (e)
where
∼
∧ : Λq T1∗ ⊗ Λp T0∗ ⊗ E −→ Λp T0∗ ⊗ Λq T1∗ ⊗ E
is defined by ∧(β ⊗ α ⊗ e) = (−1)pq α ⊗ β ⊗ e. In general δ0 is given by the formula
hδ0 (η), Y0 ×. . .×Yq i = hhδ0 (η), Y1 ×. . .×Yq i, Y0 i+
q
X
(−1)j hη, [Y0 , Yj ]×Y1 ×. . .×Ŷj ×. . .×Yq i
j=1
for η ∈ Λq T1∗ ⊗ E, Y0 ∈ T0 and Y1 , . . . , Yq ∈ T1 .
We thus get a complex for every q:
(5)
δ
δ
0
0
Λ2 T0∗ ⊗ Λq T1∗ ⊗ E −→ . . .
T0∗ ⊗ Λq T1∗ ⊗ E −→
Λq T1∗ ⊗ E −→
On the other hand the above flat T1 -connection:
∧−1 ◦ d1 : Λp T0∗ −→ T1∗ ⊗ Λp T0∗
together with the flat T1 -connection δ1 on E induces similarly a flat T1 -connection
δ1 : Λp T0∗ ⊗ E −→ T1∗ ⊗ Λp T0∗ ⊗ E ,
and hence a complex:
(6)
δ
δ
1
1
Λp T0∗ ⊗ E −→
T1∗ ⊗ Λp T0∗ ⊗ E −→
Λ2 T1∗ ⊗ Λp T0∗ ⊗ E −→ . . .
which in turn gives the complex:
(7)
δ
δ
1
1
Λp T0∗ ⊗ E −→
Λp T0∗ ⊗ T1∗ ⊗ E −→
Λp T0∗ ⊗ Λ2 T1∗ ⊗ E −→ . . .
where
δ1 : Λp T0∗ ⊗ Λq T1∗ ⊗ E −→ Λp T0∗ ⊗ Λq+1 T1∗ ⊗ E
7
is defined by δ1 = ∧ ◦ δ1 ◦ ∧−1 .
Concerning the ensuing diagram
E
(8)
δ1 y
T1∗ ⊗
E
δ1 y
δ
0
−→
δ
0
−→
δ
δ
T0∗⊗ E
Λ2 T0∗ ⊗ E
0
−→
yδ1
T0∗ ⊗T1∗ ⊗ E
yδ1
δ
0
−→
δ
yδ1
∗
Λ2 T0∗ ⊗
T1 ⊗ E
yδ1
−→ . . .
−→ . . .
0
0
2 ∗
Λ2 T1∗ ⊗ E −→
T0∗ ⊗ Λ2 T1∗ ⊗ E −→
Λ2 T0∗ ⊗ Λ
T1 ⊗ E −→ . . .
y
y
y
..
.
..
.
..
.
we have the following information:
1.3 Proposition: Assume that we are in the situation of proposition 1.2.
0
1) Let δ 0 : E → T ∗ ⊗ E be a flat T 0 -connection on E and let δ0 , δ1 be the T0 -resp.
T1 -connection on E defined by δ 0 = δ0 + δ1 . Then δ0 , δ1 are flat and the diagram (8)
is a double complex i.e. (δ0 + δ1 )2 = 0 in all degrees.
2) Conversely assume that we are given flat Ti -connections δi on E such that the
diagram
δ0
−→
T0∗⊗ E
E
δ1 y
yδ1
δ
0
T0∗ ⊗ T1∗ ⊗ E
T1∗ ⊗ E −→
is anticommutative. Then δ 0 = δ0 + δ1 is a flat T 0 -connection on E i.e. (8) is a double
complex.
Proof: 1) follows using the decomposition according to bidegree dT 0 = d0 + d1 of
proposition 1.2. For 2) note the easily established formula
(δ0 δ1 + δ1 δ0 )(ω ⊗ e) = (d0 d1 + d1 d0 )(ω) ⊗ e + ω ⊗ (δ0 δ1 + δ1 δ0 )(e)
= ω ⊗ (δ0 δ1 + δ1 δ0 )(e)
for ω ∈ Λp T0∗ ⊗ Λq T1∗ and e ∈ E. This implies the assertion in the case required
later that E has finite rank. The proof in general is only slightly more involved.
In the situation of proposition 1.2 we shall call flat T0 -resp. T1 -connections δ0 resp.
δ1 on E compatible if δ 0 = δ0 + δ1 is a flat T 0 - connection. Note that dT0 , dT1 are
compatible on E = A.
1.4 Proposition: Consider integrable subbundles T0 , T1 , T 0 of T such that
T 0 = T0 ⊕ T1 and let E be a vector bundle on X with compatible flat T0 -resp.
T1 -connections δ0 and δ1 . Set δ 0 = δ0 + δ1 . Then the complexes
(Λ• T1∗ ⊗ E)δ0
and (Λ• T0∗ ⊗ E)δ1
0
are resolutions of E δ and we have canonical isomorphisms:
0
0
H n (X, E δ ) ∼
= H n (X, (Λ• T ∗ ⊗ E)δ1 )
= H n (X, (Λ• T ∗ ⊗ E)δ0 ) ∼
= H n (X, Λ• T ∗ ⊗ E) ∼
1
for all n. The same is true for cohomology with supports.
8
0
0
Proof: By construction Λ• T ∗ ⊗ E is the simple complex associated to the double
complex (8). On the other hand the rows resp. columns of (8) give resolutions of
(Λq T1∗ ⊗ E)δ0 resp. (Λp T0∗ ⊗ E)δ1 for all p, q ≥ 0 by the foliated Poincaré lemma 1.1.
Thus the spectral sequence:
E2pq =
II H
p
( I Hq (Λ• T0∗ ⊗ Λ• T1∗ ⊗ E)) =⇒ Hp+q (Λ• T
0∗
⊗ E)
degenerates into isomorphisms:
0
Hp ((Λ• T1∗ ⊗ E)δ0 ) ∼
= Hp (Λ• T ∗ ⊗ E) .
0
Since the right hand sheaves are zero for p ≥ 1 and isomorphic to E δ for p = 0 by
1.1 we see that (Λ• T1∗ ⊗ E)δ0 and by the analogous argument also (Λ• T0∗ ⊗ E)δ1 are
0
resolutions of E δ .
We now consider the following situation which arises in the context of Anosov systems for example:
1.5 Assume we are given integrable subbundles T u , T s , T 0 , T 0 of T with
T 0 = T u ⊕ T s ⊕ T 0 and such that T u ⊕ T 0 and T s ⊕ T 0 are integrable but not
necessarily T u ⊕ T s . Applying 1.2 to the decomposition T 0 = (T u ⊕ T 0 ) ⊕ T s we get
a flat T u ⊕ T 0 -connection on T s∗ and hence a flat T u -connection du on T s∗ which is
dT 0
0
0
also described as the composition T s∗ → T ∗ →
Λ2 T ∗ −→→ T u∗ ⊗ T s∗ . Similarly
we get a flat T s -connection ds on T u∗ and the composition
∧
ds
ds : T u∗ −→ T s∗ ⊗ T u∗ ∼
= T u∗ ⊗ T s∗
0
d
0
0
T
equals the composition T u∗ → T ∗ →
Λ2 T ∗ −→→ T u∗ ⊗ T s∗ . We thus get flat
T u -resp. T s -connections on Λq T s∗ resp. Λp T u∗ for all p, q ≥ 0. Given a vector
bundle E on X with flat T u -resp. T s -connections δ u resp. δ s we obtain flat T u -resp.
T s -connections δ u resp. δ s on all the sheaves Λq T s∗ ⊗ E and Λp T u∗ ⊗ E. These
connections and their extensions to the sheaves Λp T u∗ ⊗ Λq T s∗ ⊗ E give rise to a
diagram as in (8) with rows and columns complexes. However this will not be a
double complex in general i.e. the squares will not be anticommutative if T u ⊕ T s is
not integrable. In fact consider the trivial bundle E = X × lR with E = A together
with its canonical flat T u -resp. T s -connections du , ds . Then we already have the
following assertion:
Proposition: The square
du
T u∗
A −→
ds y
s
yd
du
T s∗ −→ T u∗ ⊗ T s∗
is anticommutative if and only if T u ⊕ T s is integrable.
Proof: Assume that ds du + du ds = 0 and fix f ∈ A, Y0 ∈ T u∗ , Y1 ∈ T s∗ . For any
Z ∈ T 0 write Z = Z u ⊕ Z s ⊕ Z 0 according to the decomposition T 0 = T u ⊕ T s ⊕ T 0 .
9
We have:
hds du f, Y0 × Y1 i = hdT 0 du f, Y0 × Y1 i
= Y0 hdu f, Y1 i − Y1 hdu f, Y0 i − hdu f, [Y0 , Y1 ]i
= −Y1 (Y0 (f )) − hdT 0 f, [Y0 , Y1 ]u i
and similarly
hdu ds f, Y0 × Y1 i = Y0 (Y1 (f )) − hdT 0 f, [Y0 , Y1 ]s i .
Hence
0 = hds du f + du ds f, Y0 × Y1 i = [Y0 , Y1 ]0 (f )
and thus [T u , T s ] ⊂ T u ⊕ T s so that T u ⊕ T s is integrable. The converse is clear.
1.6 We end this section by setting up elementary notions related to flows. Consider
a Banach manifold X and a C ∞ -function φ : D ⊂ lR × X → X where D is open in
lR × X and such that for all x ∈ X there is some −∞ ≤ t(x) < 0 with
D ∩ lR × {x} = (t(x), ∞) × {x}. Setting φt (x) = φ(t, x) we assume φ0 = id and
that φt (φs (x)) = φt+s (x) in the following sense: If φs (x) is defined then one side is
defined iff the other is and in this case they are equal. Thus φ is the maximal flow
associated to the corresponding vector field Y = Yφ defined by
Yφ,x = (T φx )0
d
|t=0
dt
!
for x ∈ X
where φx (t) = φt (x). A partially defined flow as above will be called a Frobenius
flow. Example: Translation on (0, ∞).
Heuristics: The reasons why we think Frobenius flows may appear in the expected
dynamical systems associated to arithmetic schemes are these:
1 In sections 3, 4 we will see that these dynamical systems should be generated
by smooth vector fields. Hence the domain of definition D of the corresponding
maximal flow will have the above properties that D ∩ (lR × {x}) = Ix × {x} where Ix
is an open interval with 0 ∈ Ix and that φt (φs (x)) = φt+s (x) if either side is defined.
2 In characteristic p the semigroup (ZZ≥0 , +) acts on schemes by the powers of
the Frobenius endomorphism. By analogy this suggests that lR≥0 × X should be
contained in D and also that φ does not necessarily extend to a complete flow
φ : lR × X → X. A further reason why the flow should be defined for all t ≥ 0 (or
t ≤ 0) is that otherwise we do not get an induced action on global cohomologies c.f.
the end of 1.6. Finally there are the numerical investigations by Berry [B]. They
suggest that there should be a quantum physical system whose Hamiltonian H has
the imaginary parts of the nontrivial zeroes of ζ(s) as its eigenvalues. Moreover the
semiclassical limit of this system should not allow time reversal. Thse ideas fit with
our’s as follows: Given the correct dynamical system Y for spec ZZ = spec ZZ ∪ {∞}
we have shown in [D3] § 2 , [D5] § 1 that the intersection cohomology
H0 = IH 1 (Y, C) = H 1 (Y, j∗ C)
10
in the notation of [D5] should be equipped with a scalar product (, ) with respect to
which we have for cohomological reasons:
(Θh1 , h2 ) + (h1 , Θh2 ) = (h1 , h2 )
for the infinitesimal generator Θ of the induced flow on H0 . Moreover the eigenvalues
of Θ would be the nontrivial zeroes of ζ(s). The completion H of H0 to a Hilbert
space together with the unbounded self-adjoint “Hamiltonian” H = i(1/2 − Θ)
should be the quantum physical system postulated by Hilbert and Berry. We interpret Berry’s assertion on the semiclassical limit as saying that the flow on Y should
exist in positive time but should not be complete.
We should point out that this discussion on the nature of the flows to expect in
arithmetic is more speculative than the other main conclusions in section 4 which
to a large extent are deduced comparing the approaches of Patterson and myself in
[P] and [D1].
For a Frobenius flow φ on X the open subsets X t = {x ∈ X | t > t(x)} of X satisfy
∼
X t = X for all t ≥ 0 and φt : X t −→ X −t ⊂ X is an open embedding for every t.
The orbit or trajectory of a point x ∈ X under a Frobenius flow is the set γx =
{φt (x) | t > t(x)}. If γx is compact then t(x) = −∞ and either γx = {x} or there
exists a unique positive number l(γx ) > 0 the length of γx such that we have a
diffeomorphism:
∼
lR/l(γx )ZZ −→ γx , t 7−→ φt (x) .
In this case the orbit is said to be closed or periodic of period l(γx ) and the length
depends only on the orbit γ = γx and not on the choice of representative x ∈ γ.
Given a Frobenius flow φ on X an action ψ opposite to φ on a vector bundle E over
X is given by homomorphisms of vector bundles ψ t : φt∗ E → E |X t for t ∈ lR where
φt : X t ,→ X such that ψ 0 = id and ψ t+s = ψ t ◦ φt∗ (ψ s ) over X s+t for all x, t ∈ lR. We
will assume that ψ is smooth in the sense that the induced map η : (lR × E) |D → E
defined by η(t, ex ) = ψφ−tt (x) (ex ) is smooth. The maps
ψ t := T ∗ φt : φt∗ T ∗ X −→ T ∗ X t = T ∗ X |X t
on the cotangent bundle give such an action opposite to φ for example. More generally for any subbundle T 0 ⊂ T X which is φ-invariant in the sense that T φt (T 0 |X t ) ⊂
T 0 for all t the maps
0
0
ψ t = T ∗ φt : φt∗ T ∗ −→ T ∗ |X t
0
give an action on T ∗ opposite to φ. Similarly for any sheaf G of vector spaces on X
an action ψ opposite to φ is given by linear maps
ψ t : (φt )−1 G −→ G
such that ψ t+s = ψ t ◦ (φt )−1 (ψ s ) over X s+t .
Since X t = X for all t ≥ 0 it induces an action on cohomology by the monoid lR≥0
via:
φt∗
ψt
ψ∗t : H n (X, G) −→ H n (X, (φt )−1 G) −→ H n (X, G) for t ≥ 0
and similarly on cohomology with any family of supports stable under inverse images
by all φt : X → X for t ≥ 0.
11
Let T0 ⊂ T = T X be an integrable φ-invariant subbundle. Then φt maps leaves
of the T0 -foliation on X t into leaves of the T0 -foliation on X. Let ATX0 resp. ATX0t
denote the sheaf of germs of smooth real valued functions on X resp. X t which are
locally constant along the leaves of the T0 -foliation. Then pullback by φt induces an
isomorphism:
∼
ψ t = φt∗ : (φt )−1 ATX0 −→ ATX0t , f 7−→ f ◦ φt .
For a sheaf G of ATX0 -modules we also write φt∗ G for (φt )−1 G with the induced ATX0t module structure i.e.
φt∗ G = (φt )−1 G ⊗(φt )−1 AT0 ATX0t .
X
For example if E/X is a T0 -foliated vector bundle then φt∗ E T0 is the sheaf over X t
of foliated sections of the T0 -foliated bundle φt∗ E T0 .
We say that an action ψ on a bundle E/X is compatible with a T0 -foliated structure
on E if the natural AX t -linear map ψ t : φt∗ E → E |X t induced by ψEt maps φt∗ E T0
into (E |X t )T0 = E T0 |X t for all t. For E of finite rank the compatibility is equivalent
to the commutativity of the diagram:
φt∗E
ψt
y
E |X t
φt∗ (δ)
→ φt∗ (T0∗ ⊗ E) =
δ
φt∗ T0∗ ⊗ φt∗ E
t
yψ ⊗ ψ t
(T0 |X t )∗ ⊗ E |X t .
−→
1.7 For a vector bundle E with action ψ opposite to a Frobenius flow φ the restriction of ψ to the fibre at x ∈ X t is a map
ψxt : Eφt (x) −→ Ex .
If γ is a closed orbit and x ∈ γ then x ∈ X t for all t ∈ lR and ψxνl(γ) is an endomorphism of Eφνl(γ) (x) = Ex for every integer ν and if E has finite rank:
det(id − ψγνl(γ) | Eγ ) := det(id − ψxνl(γ) | Ex )
and
Tr(ψγνl(γ) | Eγ ) := Tr(ψxνl(γ) | Ex )
are independent of the choice of x ∈ γ.
1.8 We now relate actions opposite to a flow to connections. This will be used in
§ 3. Assume the Frobenius flow φ on the Banach manifold X has no constant orbits
and let T 0 = T 0 X be the rank one subbundle of T X generated by the tangents to
the flowlines i.e. by the vector field Yφ . Consider a vector bundle E on X with
action ψ opposite to φ. We define a homomorphism of sheaves ψ̇ = ψ̇E : E → E by
the following formula for local sections e ∈ E:
1 t
ψx (eφt (x) ) − ex
t→0 t
ψ̇(e)x = lim
12
where the limit is taken in the topology of Ex . The natural action opposite to φ on
the trivial bundle E = X × lR induces in this way the map
φ̇ : A → A given by φ̇(f ) = Yφ (f ) .
On a φ-invariant subbundle T 0 ⊂ T X and in particular for T 0 = T 0 we get an action
opposite to φ:
ψ t = φt∗ (T φ−t ) : φt∗ T 0 −→ T 0 |X t
noting that φt : X t → X −t is a diffeomorphism with inverse φ−t . On fibres we have:
ψxt = Tφt (x) (φ−t ) : Tφ0 t (x) −→ Tx0
for x ∈ X t .
According to [KoN] Prop. 1.9 which extends to the setting of Banach manifolds the
induced map
(9)
ψ̇ : T 0 −→ T 0 is given by ψ̇(Z) = [Yφ , Z] .
Returning to general E, ψ the formula ψ t (f e) = φt∗ (f )ψ t (e) for local sections e of
(φt )−1 E and f of (φt )−1 A implies that
ψ̇(f e) = φ̇(f )e + f ψ̇(e) = hdT 0 f, Yφ ie + f ψ̇(e) for e ∈ E and f ∈ A .
Hence the map
δ ψ,0 : E −→ T 0∗ ⊗ E ,
δ ψ,0 (e) = Yφ∗ ⊗ ψ̇(e)
defines a T 0 -connection (trivially flat) on E which for E = X × lR agrees with
dT 0 : A → T 0∗ . Note that the vector field Yφ defines a trivialization
∼
Y φ : A −→ T 0 , f 7−→ f · Yφ
which is equivariant in the sense that the following diagram commutes:
t −1
(φ ) A
φt∗ y
(φt )−1 (Y φ )
∼
A |X t
−→
Yφ
∼
−→
(φt)−1 T 0
t
yψ
T 0 |X t .
For this observe that the equation
(10)
ψxt (Yφ,φt (x) ) = Yφ,x
where ψxt = Tφt (x) (φ−t )
implies that:
Y φ (φt∗ f ) = φt∗ f · Yφ = ψ t (f · Yφ ) = ψ t (Y φ (f ))
for local sections f of (φt )−1 A.
We will often denote the maps φ̇ : A → A and ψ̇ : E → E by Θ and also the induced
maps on cohomologies.
Given the action ψ opposite to φ on E we have constructed a T 0 -connection δ ψ,0 on
E. Conversely if we are given a T 0 -connection δ 0 : E → T 0∗ ⊗ E on E we obtain by
integration a unique smooth action ψ opposite to φ on E with ψ̇(e) = hδ 0 (e), Yφ i.
The two constructions are inverse to each other.
13
1.9 We close this section by defining a weak Anosov property for complete flows
on a Banach manifold. Given a continuous norm k k on T X define T u X ⊂ T X to
be the set of vectors v ∈ T X such that
kT φt (v)k −→ 0 for t −→ −∞ .
Similarly T s X ⊂ T X is to be the set of v ∈ T X such that
kT φt (v)k −→ 0 for t −→ +∞ .
We will call (X, φ) weakly Anosov for k k if T u X and T s X are continuous subbundles
of X and T X decomposes as the continuous direct sum
T X = T uX ⊕ T sX ⊕ T 0X .
If X is compact this property and the bundles T u X and T s X are independent of
the choice of k k and the usual Anosov systems are weakly Anosov in our sense.
2
Heuristics: Comparing dynamical and motivic
formalisms
In this section we give two different reasons why a certain (still very wide) class
of dynamical systems may contain systems of arithmetic interest. One reason is
cohomological and based on comparing the largely speculative works [P] of Patterson
and [D1] of myself. The other is based on a proposition by Katsuda and Sunada
[KS] and on the assumption motivated by [D1] that continuous Tate twists should
exist for “arithmetic” dynamical systems.
2.1 We begin by recalling parts of [P] as we understand them. Consider a finite
dimensional real analytic manifold X with a real analytic lR-operation
φ : lR × X → X without constant orbits. We assume that for some choice of a
continuous norm we have
T X = T u X ⊕ T sX ⊕ T 0 X
in the analytic category where T s X is the stable and T u X is the unstable bundle of
the flow and we write T ? for T ? X. Let Ω0 be the set of closed orbits of φ and let Ω
be the closure of the union of these orbits in X. The following dynamical conditions
are assumed:
1) Ω is compact.
T
2) For some neighbourhood U of Ω we have t∈lR φt (U ) = Ω.
3) For any two open sets V1 , V2 intersecting Ω nontrivially and any t0 > 0 there
exists some t > t0 such that φt (V1 ) ∩ (V2 ∩ Ω) 6= ∅ .
4) If x ∈ X is such that for any neighbourhood V of x and any t0 > 0 there exists
some t > t0 such that φt (V ) ∩ V 6= ∅ then x ∈ Ω.
14
Next consider a real analytic vector bundle E over X equipped with a flat T u connection δ u and an lR-action ψ on E opposite to φ which is compatible with the
T u -foliated structure of E. According to 1.5 and 1.6 the dual stable bundle T s∗ and
hence also the bundles Λi T s∗ ⊗E are equipped with such connections and lR-actions.
Note that the T u -covariant derivative on T s used in [P] which is essentially given by
the Lie bracket:
[,]
T u ⊗lR T s −→ T −→→ T s
induces the same T u -connection on T s∗ as above. Patterson suggests that a formula
of the following kind should hold:
2.2 Coherent dynamical Lefschetz trace formula: As distributions in D 0 (lR+ )
we have:
X
(−1)
ν
u
Tr(ψ∗ | Hcν (X, E T ))dis
ν
=
X
l(γ)
γ∈Ω0
X
k≥1
Tr(ψγkl(γ) | Eγ )
kl(γ)
det(id − ψγ
| Tγs∗ )
δkl(γ)
where δy is the Dirac distribution in y > 0 and Tr(ψ | H)dis is to denote a suitable
distribution valued trace. Note that by our definition of the action on T s∗ we have
det(id − ψγkl(γ) | Tγs∗ ) = det(id − Tx φkl(γ) | Txs ) > 0 for any x ∈ γ
the latter expression appearing in [F] section 2.
If I understood Patterson correctly the basic idea was this: Quite generally in order
to have a Lefschetz trace formula the operator in question must be contractive near
the fixed points i.e. near the closed orbits in our case. For the leafwise cohomology
with respect to the unstable foliation the dynamical system appears as though there
were no unstable orbits and in particular the flow appears contractive near the closed
orbits.
For a complex valued L1loc -function f on lR+ let hf i be the associated distribution
in D 0 (lR+ ). In the most favourable case (“discrete spectrum”) we set:
Tr(ψ | H)dis =
(11)
X
mα hetα i in D 0 (lR+ )
α∈C
if the algebraic multiplicities:
n
mα = dim h ∈ HC | ex. L ≥ 1 s.t. (ψ t − etα id)L (h) = 0 for all t ∈ lR
o
are finite and the series converges in D 0 (lR+ ) i.e. by Banach- Steinhaus if the series
hTr(ψ | H)dis, ϕi =
X
α∈C
mα
Z
∞
0
etα ϕ(t) dt
P
converges for all ϕ ∈ D(lR+ ). As an example note that the series ρ hetρ i where ρ
runs over the non-trivial zeroes of the Riemann zeta function converges in D 0 (lR+ )
P
e.g. [DSchr] (1.3) but that ρ etρ does not converge for any real t.
In general the definition of a good distribution valued trace on the usually infinite
15
u
dimensional spaces Hcν (X, E T ) is an unsolved problem. It seems probable that
u
only a certain subquotient of Hcν (X, E T ) will contribute to the trace. For finite
dimensional X a natural candidate may be the maximal Hausdorff quotient with
respect to the topology induced by the usual Fréchet topologies on the global sections
u
of the C ∞ -resolution of E T .
For technical reasons Patterson considers only the sheaves of those smooth sections
that in addition are real analytic on Ω, but we omit this point from the discussion.
Formula 2.2 would give a cohomological expression for the Selberg zeta function as
defined in [F] section 2 of the dynamical system (X, φt ) with coefficients in the dual
bundle E ∗ . Note that the action ψ opposite to φ on E induces a flow on E ∗ covering
φ. To deal with Ruelle zeta functions Patterson states a second formula involving
u
the cohomology groups Hcj (X, (Λi T s ⊗ E)T ) which is supposed to follow from 2.2.
This I do not see since to get actions φt∗ T s → T s one has to use the flow at time
−t and then the determinant in 2.2 does not cancel. For our purposes it is essential
though to work with the exterior powers of the dual bundle T s∗ anyhow and then
this little problem disappears: The following formula – speculative as the former –
is an immediate consequence of 2.2:
2.3 Consequence of the coherent dynamical Lefschetz trace formula: As
distributions in D 0 (lR+ ) we have:
X
u
(−1)i+j Tr(ψ∗ | Hcj (X, (Λi T s∗ ⊗ E)T ))dis =
i,j
X
l(γ)
γ∈Ω0
X
Tr(ψγkl(γ) | Eγ )δkl(γ) .
k≥1
2.4 We now compare the preceeding dynamical formalism with the conjectural
arithmetic cohomology theory discussed in [D1] § 7.
Consider a finite extension k/Q with ring of integers o = ok and a finite set S of
prime ideals in o and set spec oS = (spec o) \ S. Let us write (“spec oS ”, φt ) for the
hypothetical dynamical system of [D1] § 7 associated to spec oS . Granting the usual
conjectures on the analytic behaviour of motivic L-series and the existence of the
cohomology theory postulated in loc. cit. and denoted ? H ν (“spec oS ”, F (M )) etc.
in the following one gets e.g. by [DSchr] § 3 the following formula in D 0 (lR+ ) which
has some similarities with 2.3:
X
(−1)ν Tr(ψ∗ | ? Hcν (“spec oS ”, F (M )))dis =
ν
X
p∈S
/
log N p
X
I
Tr(Frkp | Ml,ιp )δk log N p .
k≥1
Here M is any mixed motive over k and the remaining notations are as in [DSchr]
§§ 2, 3. Comparing with 2.3 we see that the closed orbits of (“spec oS ”, φt ) should
correspond to the prime ideals p ∈
/ S the length of the orbit γp =
b p being l(γp ) =
I
log N p. We also see that at least the semisimplifications of the pairs (Ml,ιp , Frp ) and
(Ex , ψxlog N p ) for x ∈ γp should be isomorphic. Since E was supposed to be a vector
I
bundle the fibre dimensions dim Ex are constant whereas the dimension dim Ml,ιp
drops for the bad reduction primes. Hence the above formula can be a special case
16
of 2.3 only in case M has good reduction at all primes p ∈
/ S. This is quite natural:
the sheaf F (M ) over “spec o” has to be thought of as constructible. It becomes
“locally free” only after restriction to “spec oS ” for S containing the bad reduction
primes. Moreover formula 2.3 depends on the flat T u -connection on E to define the
u
sheaves (Λi T s∗ ⊗ E)T . But “vector bundles with a connection” will only describe
“locally constant” sheaves, the general case requiring a theory of D-modules.
Attempts to define E starting from the proposed fibre Ml,ι with its Gkp -action, p /| l
are probably hopeless. I will explain what I consider to be the right strategy in 2.8
after the situation has become clearer. That strategy also avoids the difficulties due
to the fact that the good category of mixed motives has yet to be constructed.
Comparing the formulas in 2.3 and 2.4 suggests that for the dynamical systems of
arithmetic interest there should exist a natural equivariant spectral sequence:
u
E1ij = Hcj (X, (Λi T s∗ ⊗ E)T ) =⇒ Hci+j (X, F )
(12)
where F is some sheaf or complex of sheaves attached to E.
The simplest reason for which such a spectral sequence could exist is this: There
actually exists a complex
u
u
u
E T −→ (T s∗ ⊗ E)T −→ (Λ2 T s∗ ⊗ E)T −→ . . .
(13)
and the group Hcν (X, F ) is the ν-th cohomology with supports of this complex.
Then (12) would result from the E1 -hypercohomology sequence of (13). The most
natural way in which a complex (13) could come about and one which would place
the stable and the instable leaves on an equal footing is as follows:
We assume that in addition to δ u the bundle E carries a flat T s -connection δ s
compatible with ψ t such that the diagram constructed in 1.5 for T 0 = T X etc. is
actually a double complex:
δ
(14)
E
s
y
T s∗ ⊗
E
δ
s
y
δu
−→
δu
−→
δu
T u∗ ⊗ E
s
yδ
T u∗ ⊗T s∗ ⊗ E
s
yδ
δu
−→
δu
−→
δu
Λ2 Tu∗ ⊗ E
s
yδ
s∗
Λ2 T u∗ ⊗
T ⊗E
s
yδ
−→ . . .
−→ . . .
u∗
Λ2 T s∗
⊗ Λ2 T s∗ ⊗ E −→ Λ2 T u∗ ⊗
Λ2 T s∗ ⊗ E −→ . . .
⊗ E −→ T
y
..
.
y
y
..
.
..
.
For the trivial one dimensional bundle in particular which should correspond to the
motive Q(0) this condition forces the bundle T u ⊕ T s to be integrable as we have
seen in the proposition of 1.5. This property drastically limits the class of Anosov
systems under consideration and is the most important insight of this section. For
the geodesic flow on the sphere bundle of a quotient of the upper halfplane by a
discrete cocompact subgroup of PSL2 (lR) the bundle T u ⊕ T s is not integrable. In
my opinion this is the deeper reason for the differences in the analytic behaviour and
17
functional equation of motivic L-functions as compared to Ruelle’s zeta functions
attached to the geodesic flow.
Given that T0 = T u ⊕ T s is integrable the anticommutativity of the diagram:
E
(15)
δs y
δu
T u∗ ⊗ E
−→
s
yδ
δu
T s∗ ⊗ E −→ T u∗ ⊗ T s∗ ⊗ E
will already ensure by proposition 1.3 that δ0 = δ u + δ s is a flat T0 -connection on E
i.e. that (14) is a double complex. The latter induces a complex
u
δs
u
δs
u
E T −→ (T s∗ ⊗ E)T −→ (Λ2 T s∗ ⊗ E)T −→ . . .
which is a resolution of E T0 by proposition 1.4 so that we get canonical isomorphisms
u
Hcν (X, E T0 ) = Hcν (X, (Λ• T s∗ ⊗ E)T )
and thus the desired spectral sequence (12) with F = E T0 .
Patterson’s finite dimensional setup will probably not suffice to treat the expected dynamical systems of arithmetic origin. It seems more realistic to take infinite
dimensional manifolds into account. The preceeding considerations suggest the following picture:
2.5 To spec oS there should be associated:
– a Banach manifold X = XS with a Frobenius flow φt without constant orbits whose
closed orbits correspond to the prime ideals p ∈
/ S and such that l(γp ) = log N p if
γp =
b p.
– a decomposition T X = T0 X ⊕ T 0 X where T0 X is an integrable T φt -invariant
complement to the one-dimensional bundle T 0 X of tangents to the flowlines. A
further structure on X is suggested by later considerations in 3.24.
To every motive M over k with good reduction outside S there should correspond
a vector bundle E with a flat T0 -connection δ0 on X or equivalently a T0 -foliated
bundle F on X c.f. the remarks before 1.1. Moreover (E, δ0 ) resp. the foliated
bundle F should be equipped with a compatible action ψ opposite to φ such that
the semisimplifications of the pairs (Ml,ι , Frp ) and (Ex , ψxlog N p ) for x ∈ γp are isomorphic for all p ∈
/ S.
If FR (M ) = E T0 denotes the sheaf of foliated sections of F there should be a subquoν
tient H c (X, FR (M )) of Hcν (X, FR (M )) which the distributional trace of ψ∗ “sees”
such that
?
Hcν (“spec oS ”, F (M ))
? ν
H (“spec oS ”, F (M ))
ν
∼
= H c (X, FR (M ))C
ν
∼
= H (X, FR (M ))C
and
as modules under the monoid action by lR≥0 . There is good reason for the cautious
formulation in terms of subquotients here, c.f. (4.1). A topological dynamical
Lefschetz formula having the form (24) below would then imply that:
X
(−1)ν Tr(ψ∗ | Hcν (X, FR (M )))dis =
ν
X
p∈S
/
18
log N p
X
k≥1
Tr(Frkp | Ml,ι )δk log N p .
In section 4 we will investigate the consequences for (X, φt ) that these connections to
arithmetic geometry would imply. That study will be based on section 3 where the
elementary theory of general dynamical systems (X, φt ) with an integrable invariant
complement to T 0 X is described from our point of view.
Let us now briefly consider an extension of the preceeding ideas to flat regular
quasi-projective schemes X0 /ZZ. Let us write (X, φt ) for the hypothetical dynamical
system of [D1] (7.28) corresponding to X0 and write ? Hcν (X, C) for the cohomology
groups with action by lR≥0 postulated in loc. cit. As in [DSchr] § 3 one sees that
the conjectural Lefschetz trace formula of [D1] (7.29) would imply an equality of
distributions in D 0 (lR+ ):
X
ν
(−1)ν Tr(φ∗ | ? Hcν (X, C))dis =
X
x0 ∈|X0 |
log N x0
X
δk log N x0 .
k≥1
Comparing with 2.3 the same arguments as before therefore suggest the following
picture:
2.6 To X0 /ZZ as above there should be functorially associated
– a Banach manifold X with a Frobenius flow φ without constant orbits whose
closed orbits correspond to the closed points of X0 and such that l(γx0 ) = log N x0
if γx0 =
b x0 .
– a decomposition T X = T0 X ⊕ T 0 X where T0 X is integrable and φ-invariant. For
a further structure see 3.24.
Let R resp. C denote the sheaf of smooth real resp. complex valued functions on
X that are locally constant on the leaves of the T0 X-foliation. Then the looked
for cohomologies ? H ν (X, C) of [D1] § 7 should be suitable subquotients of the sheaf
cohomologies H ν (X, C) that are “seen” by the distributional trace of the flow induced
by φ on H ν (X, C) and correspondingly for cohomology with supports.
2.7 The flatness condition in 2.6 is due to later considerations c.f. 4.7. According
to them the assertions of 2.6 cannot be expected generally if X0 has characteristic
p. If the variety X0 /IFp is ordinary in a suitable sense however they should apply.
For finite fields in particular there is a natural finite dimensional dynamical system
with the appropriate cohomologies. For X0 = spec IFq with q = pr consider X =
lR/ log qZZ with lR acting by translation. In this case T0 X = 0 corresponds to the
foliation of X by points so that C is the sheaf of smooth C-valued functions on X.
Hence Hcν (X, C) = H ν (X, C) = 0 for ν ≥ 1. On the other hand by Fourier theory
the direct sum of the algebraic eigenspaces of Θ on H 0 (X, C):
0
H (X, C) :=
M
H 0 (X, C)Θ∼α ⊂ H 0 (X, C)
α∈C
is the ring of Laurent polynomials:
"
!
2πi
2πi
x , exp −
x
lLq = C exp
log q
log q
19
!#
0
with lR-action by translation. Hence H ν (X, C) for ν ≥ 1 resp. H (X, C) agree with
the cohomologies H ν (spec IFq /lLq ) of [D1] § 4. Note that the assignment spec IFq 7→
(X = lR/ log qZZ, φt ) is functorial! I think that the expected functor from schemes
to dynamical systems will associate to spec IFq some system which is T0 -diffeotopic
in an obvious sense to the one above i.e. to its only closed orbit.
2.8 We now explain a strategy to obtain the sheaves FR (M ) given that 2.6 has
been extended to a functor from smooth quasiprojective varieties X0 /k to dynamical
systems (X, φt ) with a decomposition T X = T0 X ⊕ T 0 X as above. We also write
“X0 ” = X. For such a variety π0 : X0 → spec k let
π = “π0 ” : “X0 ” → “spec k”
be the associated morphism of dynamical systems. The functor
X0 ; Ri π∗ (RX )
defines a cohomology theory with values in the abelian category of R“spec k” -modules
with action opposite to the Frobenius flow on “spec k”. Using simplicial techniques in
particular cohomological descent this functor is extended to eventually non- smooth
simplicial varieties over k. By the universality of the good category of mixed motives
MMk we will get an induced functor from MMk to such modules which we denote
by M ; G(M ). The morphism j0 : spec k → spec o will induce a morphism of
dynamical systems j = “j0 ” and we should have FR (M ) = j∗ G(M ). If M has good
reduction outside of S the restriction of FR (M ) to “spec oS ” should be a locally free
R“spec k” -module of rank the rank of M . This restriction is also denoted FR (M ) and
should equal jS∗ G(M ) where jS = “jS0 ” and jS0 : spec k → spec oS is the natural
morphism. Thus we see that a good theory of motives will be useful to produce many
sheaves F = FR (M ) but also that at least the problems one faces with respect to
motives are no obstacle to building up the dynamical theory we are hoping for.
2.9 We close this section by giving another heuristic argument why the condition
T u ⊕ T s integrable should be essential in our context. Consider a smooth Anosov
flow (X, φt ) on a compact manifold X and let h be the topological entropy of the
flow. For any unitary character χ : H1 (X, ZZ) → U (1) set
L(χ, s) =
Y
(1 − χ([γ])e−sl(γ) )−1
γ
where γ runs over the closed orbits of the flow. Then L(χ, s) has the line Re s = h
as its line of absolute convergence and defines a holomorphic function in Re s > h
with a meromorphic continuation to an open neighbourhood of Re s = h. The next
result is proved in [KS] 2.1:
Proposition: The following conditions are equivalent:
1) There exists a non-trivial character χ such that L(χ, s) has a pole on the line
Re s = h.
2) T u X ⊕ T s X is integrable.
20
The implication 2) ⇒ 1) depends on the fact that under condition 2) there is a
character χ as above with χ([γ]) = exp 2πil(γ) for all closed orbits γ. See 3.11
below for a more general statement. The other direction lies deeper and is based
on a result of Guillemin and D. Kazhdan [GK] on triviality of certain dynamical
1-cocycles. We assume that the proposition extends to a context which includes the
searched for dynamical systems X attached to spec o for number fields k/Q. The
unramified quasi-characters χλ = k kλ : IA∗k /k ∗ → C∗ are unitary and non-trivial
for λ = it, t ∈ lR∗ and should correspond to unitary characters of H1 (X, ZZ). Since
L(χit , s) = ζ(s + it) has Re s = 1 as its line of convergence with a pole at s = 1 − it
the proposition implies that T u X ⊕ T s X is integrable. The same argument applies
to schemes X0 /ZZ as in 2.6. Conversely we will see in the next section that an
integrable invariant complement to T 0 X implies the existence of continuous “Tate
twist” local systems on X.
3
Basic properties of flows with an integrable
invariant complement
In this section we develop the elementary theory of leafwise cohomologies attached
to the flows in the title from our point of view. In particular we stress analogies
with l-adic cohomology of varieties over the algebraic closure of a finite field.
Definition: A Frobenius system is a triple (X, φt , T0 ) where X is a Banach manifold,
φ is a Frobenius flow on X without constant orbits and T0 ⊂ T X is a φ-invariant
integrable subbundle such that T X = T0 ⊕ T 0 c.f. 1.6, 1.8.
A morphism of Frobenius systems π : (X, φtX , T0 X) → (X 0 , φtX 0 , T0 X 0 ) is a smooth
map π : X → X 0 with T π(T0 X) ⊂ T0 X 0 and such that π ◦ φtX = φtX 0 ◦ π whenever
both sides are defined.
Example: If (X, φt , T0 ) is a Frobenius system and π : X̃ → X is a covering then
there is a unique lift of the Frobenius structure to one on X̃ such that π becomes a
morphism of Frobenius systems.
3.1 The Frobenius system is said to be complete if the flow is complete. Examples
of such systems are obtained a follows: Let M be a Banach manifold with an action
of a subgroup Λ ⊂ lR by diffeomorphisms and define an action of Λ on M × lR by
setting (x, t) · λ = ((−λ) · x, t + λ) for λ ∈ Λ. Assume that this action is properly
discontinuous so that X = M ×Λ lR := (M × lR)/Λ is a manifold. Then
(X, φ, T0 ) = (M ×Λ lR, τ, T M ×Λ lR)
where τ is translation in the second factor is a complete Frobenius systems. We
will see later 3.12 that every such system is obtained in this manner. Note that
the (closed) orbits of the flow on X correspond to the Λ-orbits on M (which are
pointwise fixed by some 0 6= λ ∈ Λ). For m ∈ M we have for the isotropy group Λm
that either Λm = 0 or that Λm = lZZ for some 0 < l ∈ Λ. In the second case Λ · m
corresponds to a closed lR-orbit on X of length l.
21
The decomposition T = T0 ⊕ T 0 induces a surjection T −→→ T0 and hence an
extension by zero map T0∗ ,→ T ∗ . For example we will consider the extension by
zero ωφ ∈ Γ(X, T ∗ ) of the section Yφ∗ ∈ Γ(X, T 0∗ ) defined by hYφ∗ , Yφ i = 1.
The decomposition T X = T0 ⊕ T 0 leads to a bigraduation of Λ• T ∗ by the sheaves
Λp T0∗ ⊗ Λq T 0∗ and the exterior derivative decomposes as a sum d = d0 + d0 where
d0 has bidegree (1, 0) and d0 has bidegree (0, 1). The restriction of d0 to T 0∗ :
d0 : T 0∗ −→ T0∗ ⊗ T 0∗
is a flat T0 -connection on T 0∗ c.f. proposition 1.2.
3.2 Proposition: For a Frobenius system we have d0 Yφ∗ = 0 and dωφ = 0.
Proof: Since T0 is integrable and ωφ vanishes on T0 the form dωφ vanishes on T0 as
well. To show that dωφ = 0 it suffices therefore and since T 0 has rank one to show
that
hdωφ , Y0 × Y 0 i = 0 for Y0 ∈ T0 and Y 0 ∈ T 0 .
Clearly we may assume that Y 0 = Yφ and then:
hdωφ , Y0 × Yφ i = Y0 hωφ , Yφ i − Yφ hωφ , Y0 i − hωφ , [Y0 , Yφ ]i
= Y0 (1) + hωφ , [Yφ , Y0 ]i = 0
since [Yφ , Y0 ] = ψ̇(Y0 ) ∈ T0 by (9) and since ωφ |T0 = 0. From dωφ = 0 it follows that
d0 ωφ = 0 and hence that d0 Yφ∗ = 0.
Note that the form ωφ is φ-invariant in the sense that φt∗ ωφ = ωφ |X t for all t in lR:
For any Z ∈ T |X t we have
hφt∗ ωφ , Zi = hωφ , T φt ◦ Zi ◦ φt
and hence
hφt∗ ωφ , Zi = 0 = hωφ |X t , Zi
for Z ∈ T0 |X t since T0 is φinvariant. On the other hand for x ∈ X t setting y = φt (x) ∈ X −t we find:
(10)
hφt∗ ωφ , Yφ ix = hωφ,y , ψy−t (Yφ,φ−t (y) )i = hωφ,y , Yφ,y i = 1 = hωφ , Yφ ix .
Thus we have φt∗ ωφ = ωφ |X t as claimed.
3.3 Remark: For any morphism π : (X, φtX , T0 X) → (X 0 , φtX 0 , T0 X 0 ) of Frobenius
systems we have:
T π ◦ Y φX = Y φX 0 ◦ π
and π ∗ ωφX 0 = ωφX .
The second equation follows because π ∗ ωφX 0 is zero on T0 X and satisfies:
hπ ∗ ωφX 0 , YφX ix = hωφX 0 , T π ◦ YφX iπ(x) = hωφX 0 , YφX 0 iπ(x) = 1 .
The data defining a Frobenius system can be described differently c.f. [Pl] § 2:
22
3.4 Proposition: Let φ be a Frobenius flow without constant orbits on a Banach
manifold X. To specify a subbundle T0 ⊂ T X such that (X, φt , T0 ) is a Frobenius
system is equivalent to specifying a closed φ-invariant one-form ω on X up to multiplication by a locally constant nowhere vanishing function, such that hω, Yφ ix 6= 0
for all x ∈ X. It then follows that hω, Yφi is locally constant.
Proof: We have seen that given a Frobenius system the form ω = ωφ has the
required properties. On the other hand a form ω as in the proposition has an
integrable kernel T0 = Ker ω as follows from the formula
0 = hdω, Y0 × Y1 i = Y0 hω, Y1 i − Y1 hω, Y0 i − hω, [Y0 , Y1 ]i .
As hω, Yφi does not vanish on X we get a splitting T X = T0 ⊕ T 0 . Finally we have
for Z ∈ T0 |X t that
hω, T φt (Z)i ◦ φt = hφt∗ ω, Zi = hω |X t , Zi = 0
and hence T φt (T0 |X t ) ⊂ T0 i.e. T0 is φ-invariant. If ω 0 is another closed φ-invariant
one-form such that Ker ω 0 = T0 then ω 0 = f ω for a smooth nowhere vanishing
function f . As 0 = dω 0 = df ∧ ω we get for Y ∈ T0 that
1
0 = hdf ∧ ω, Y × Yφ i = Y (f )hω, Yφi
2
i.e. that Y (f ) = 0. On the other hand by φ-invariance of ω, ω 0 we find φt∗ f = f |X t
and hence Yφ (f ) = 0. Thus f is locally constant. The last assertion follows since
hωφ , Yφ i = 1.
Finally we point out that in certain cases the complement T0 to T 0 is uniquely
determined by the flow, c.f. [KS] p. 15:
3.5 Proposition: Consider a complete flow φ on a Banach manifold X and assume
that for some continuous norm k k on T X the flow is weakly Anosov in the sense
of 1.9 and kYφ k is bounded along every flow line. Then a continuous φ-invariant
integrable complement T0 to T 0 exists if and only if T u ⊕ T s is integrable in which
case T0 = T u ⊕ T s is uniquely determined.
Proof: If T u ⊕ T s is integrable it provides the desired complement. On the other
hand assume that T0 is an integrable invariant complement to T 0 with associated
one-form ωφ . By assumption kωφ k is bounded along every flow line. For v ∈ Txs and
t ≥ 0 we have:
|hωφ , vix | = |hφt∗ ωφ , vix | = |hωφ , Tx φt (v)iφt (x) |
≤ kωφ kφt (x) kTx φt (v)kφt (x) .
For t → ∞ we find hωφ , vix = 0 i.e. v ∈ T0,x and hence T s ⊂ T0 . Similarly we get
T u ⊂ T0 and hence T u ⊕ T s ⊂ T0 . By assumption T X = T u ⊕ T s ⊕ T 0 = T0 ⊕ T 0 .
Hence T u ⊕ T s = T0 and the proposition is proved.
23
Remark: If the flow is not complete but only Frobenius we still get that T s ⊂ T0
where T s is defined as in 1.9.
We now establish certain equivalences of categories of sheaves with some extra structure on Frobenius systems (X, φt , T0 ). Let E be a locally free sheaf of A-modules
of finite rank equipped with a flat T0 -connection δ0 : E → T0∗ ⊗ E. Recall from
1.6 that an action ψ opposite to φ is compatible with δ0 iff the following diagram
commutes:
φt∗ (δ0 ) t∗
φt∗E
→ φ (T0∗ ⊗ E) =
φt∗ T0∗ ⊗ φt∗ E
t
ψty
yψ ⊗ ψ t
δ
(T0 |X t )∗ ⊗ E |X t .
0
−→
E |X t
On the other hand by proposition 1.3 a T 0 -connection δ 0 : E → T 0∗ ⊗ E on E is
compatible with δ0 iff the diagram:
E
δ0 y
δ
0
−→
δ
T0∗⊗ E
0
yδ
0
T 0∗ ⊗ E −→
T0∗ ⊗ T 0∗ ⊗ E
is anticommutative. In 1.8 we have seen that actions ψ opposite to φ on E correspond to T 0 -connections δ 0 on E. Using the two diagrams above it follows that
compatibility with δ0 is preserved under this bijection.
By definition to give compatible flat T0 - and T 0 -connections δ0 and δ 0 is equivalent
to give a flat connection δ : E → T ∗ ⊗ E on E in the ordinary sense: δ = δ0 + δ 0 .
Let R resp. C be the sheaf of smooth real resp. complex valued functions that are
locally constant along the leaves of the T0 -foliation. Note that the pullback action
t −1
φt∗ : (φt )−1 A → A |X t opposite to φ restricts to an action φt∗
R : (φ ) R → R |X t since
t∗
T0 is φ-invariant. Similarly we get an action φC opposite to φ on C. Incidentially
note that every morphism π : (X, φX , T0 X) → (X 0 , φX 0 , T0 X 0 ) of Frobenius systems
induces maps of ringed spaces
π : (X, RX ) −→ (X 0 , RX 0 ) and π : (X, CX ) −→ (X 0 , CX 0 ) .
It can be shown that the pullback map
π ∗ : π −1 RX 0 −→ RX , f 7−→ f ◦ π
is actually an isomorphism. For a sheaf F of RX 0 -modules we write π ∗ F for π −1 F
with the induced RX -module structure i.e.
π ∗ F = π −1 F ⊗π−1 RX 0 RX 0 .
Clearly π ∗ is an exact functor. Similar remarks apply to C instead of R.
3.6 Proposition: On a Frobenius system we have the following equivalences of
24
categories:
locally free A-modules E of rank r with a flat T0 -connection δ0 and with an
action ψ opposite to φ compatible with δ0 (equivalently: and with a T 0 - connec
tion δ 0 compatible
with δ0 )
x
x
(2)y
(3)y
(
)
locally free R-modules F of rank r with
(1)
local systems F of lRan action ψ opposite to φ (equivalently:
.
←→
vector spaces of rank r
0
0
with a T -connection δ )
Functors (1) are given by:
F 7−→ F = F ⊗lR R + ψ t = idF ⊗ φt∗
R
F = Fδ
0
(resp. + δ 0 = id ⊗ d0 |R )
←− F .
Functors (2) are given by:
F 7−→ E = F ⊗lR A + ψ t = idF ⊗ φt∗ (resp. + δ 0 = idF ⊗ d0 )
+ δ0 = idF ⊗ d0
δ
δ = δ0 + δ 0 , δ 0 = δ 0,ψ .
F = E ←− E where
Finally (3) are given by:
F 7−→ E = F ⊗R A + ψ t = ψFt ⊗ φt∗
+ δ0 = idF ⊗ d0
F = E δ0 ←− E .
(resp. + δ 0 = δF0 ⊗ id + id ⊗ d0 )
If we complexify everything we get analogous equivalences of categories for sheaves
of C-vector spaces which will still be denoted by F, F , E usually.
Remark: The preceeding proposition can also be formulated in terms of foliated
vector bundles and it then extends to bundles of infinite rank.
We now introduce “continuous Tate twists”: For α ∈ lR resp. α ∈ C and a locally
free R-resp. C-module F of finite rank with an action ψF opposite to φ we define
F (α) to be F as an R-resp. C-module but with the new action:
ψFt (α) = e−tα ψFt .
In terms of Θ = ψ̇ c.f. 1.8 the twist is given by:
ΘF (α) = ΘF − α id
and hence
δF0 (α) (e) = δF0 (e) − αYφ∗ ⊗ e .
0
If F = F δ is the local system corresponding to (F , ψ) we set
0
0
0
F (α) = F (α)δ and in particular lR(α) = R(α)δ and C(α) = C(α)δ .
25
Note that
F (α) = F ⊗R R(α) and F (α) = F ⊗lR lR(α)
and hence:
R(α + β) = R(α) ⊗R R(β) and lR(α + β) = lR(α) ⊗lR lR(β) .
The analogous assertions hold with R, lR replaced by C, C. If we define the twists
of vector spaces with an lR≥0 -action similarly as the twist of sheaves we have:
H i (X, F (α)) = H i (X, F )(α) as lR≥0 -modules c.f. 1.6.
There is the following proposition comparing the cohomologies of F and F :
3.7 Proposition: Let F and (F , ψ) be related as in 3.6 and let ΘF = ψ̇F : F → F
be the derivative of the action. Then there is a short exact sequence of sheaves
Θ
F
0 −→ F −→ F −→
F −→ 0
and hence for all n ≥ 0 a short exact sequence:
(16) 0 −→ H n−1 (X, F )/ΘH n−1 (X, F ) −→ H n (X, F ) −→ H n (X, F )Θ=0 −→ 0
where Θ = (ΘF )∗ and similarly for cohomology with supports.
Proof: Let (E, ψ, δ0 ) correspond to F and (F , ψ) as in 3.6. By Prop. 1.4 we have
a short exact sequence:
δ0
0
0 −→ E δ0 +δ −→ E δ0 −→ (T 0∗ ⊗ E)δ0 −→ 0
where δ 0 = δE0,ψ i.e. a short exact sequence:
(17)
δ0
0 −→ F −→ F −→ (T 0∗ ⊗ E)δ0 −→ 0 .
The A-linear isomorphism:
∼
E −→ T 0∗ ⊗ E , e 7−→ Yφ∗ ⊗ e
induces because of d0 Yφ∗ = 0 c.f. Prop. 3.2 an R-resp. C-linear isomorphism
∼
E δ0 −→ (T 0∗ ⊗ E)δ0 ,
which fits into a commutative diagram:
δ0
F −→ (T 0∗ x⊗ E)δ0
ΘF y
o
F =
E δ0
26
Combining this with (17) we get the exactness of the sequence:
Θ
F
0 −→ F −→ F −→
F −→ 0 .
Remark: Given a suitable topology on H i (X, F ) we can consider the infinitesimal
generator Θψ∗ of the lR≥0 -action on its domain of definition D = D(Θψ∗ ) in the
maximal Hausdorff quotient of H i (X, F ). The map Θψ∗ should equal the map
induced by Θ = (ΘF )∗ on D c.f. proposition 3.19 for a simple example.
The exact sequence (16) is analogous to the short exact sequence for a variety X 0 /IFq
that relates the l-adic cohomologies over X0 ⊗IFq IFq and over X0 . The mechanism is
different though: In l-adic cohomology the space is changed from X0 to X0 ⊗IFq IFq
whereas in our case it remains the same and the sheaf is changed from F to F . In
the terminology of arithmetic geometry the groups H i (X, F ) should be viewed as
“arithmetic” cohomologies and the groups H i (X, F ) as “geometric” cohomologies.
In support of this note for example that the groups H i (X, F ) commute with twists
whereas H i (X, F (α)) is not in general isomorphic to H i (X, F (β)) if α 6= β.
Consider the following compositions arising from (16) for n = 1 and F = R resp.
F = C where we suppose that X is connected and x0 ∈ X is a point:
exp∗
∼
H 0 (X, R) −→ H 1 (X, lR) ∼
= Hom(π1 (X, x0 ), lR) −→ Hom(π1 (X, x0 ), lR>0 )
exp∗
H 0 (X, C) −→ H 1 (X, C) ∼
= Hom(π1 (X, x0 ), C) −→ Hom(π1 (X, x0 ), C∗ ) .
For α ∈ H 0 (X, R) let ρα : π1 (X, x0 ) → lR>0 ⊂ GL 1 (lR) be the corresponding
representation and similarly in the complex case. We then have the following nice
fact:
3.8 Proposition: For α ∈ lR ⊂ H 0 (X, R) the representation ρα is the one belonging to the local system lR(α) on X and similarly for α ∈ C ⊂ H 0 (X, C).
Proof: Going through the construction of (16) one sees that for general (F , ψ) the
homomorphism:
δ
H 0 (X, F )/ΘH 0 (X, F ) ,→ H 1 (X, F ) ⊃
Ker (H 0 (X, T ∗ ⊗ E) → H 0 (X, Λ2 T ∗ ⊗ E))
δ
Im (H 0 (X, E) → H 0 (X, T ∗ ⊗ E))
maps the class of f ∈ H 0 (X, F ) to the class of the closed form
ωφ ⊗ f ∈ H 0 (X, T ∗ ⊗ E) .
Regarding the above inclusion observe that we did not assume the existence of
partitions of unity on X c.f. 1.1. Note that dωφ = 0 by 3.2 and that ωφ ∧ δf =
ωφ ∧ δ 0 f = 0 since δ0 f = 0. It follows that for any c ∈ π1 (X, x0 ) represented by a
smooth path γ : [0, 1] → X we have:
(18)
ρα (c) = exp
Z
27
γ
αωφ
.
For any g ∈ A we have d0 g = Yφ (g)Yφ∗ . A local section g ∈ lR(α) satisfies d0 g = 0
and hence:
dg = Yφ (g)ωφ = φ̇(g)ωφ = αgωφ .
Thus:
d(γ ∗ g) = γ ∗ g(γ ∗ (αωφ )) where defined.
For all h ∈ Γ([0, 1], γ −1 lR(α)) we therefore find:
dh = hγ ∗ (αωφ )
since h has the form γ ∗ g locally. As γ −1 lR(α) is constant there is some h which does
not vanish in any point of [0, 1] and we get:
ρα (c) = exp
Z
1
0
∗
Z
γ (αωφ ) = exp
1
0
dh
h
!
=
h(1)
.
h(0)
Since the representation ρ : π1 (X, x0 ) → GL 1 (lR) corresponding to lR(α) is determined on c by h(1) = ρ(c)h(0) we find ρα = ρ. The same argument works for α ∈ C
and C(α) of course.
3.9 Remark: More generally one can define F (α) for any α ∈ H 0 (X, R) resp.
H 0 (X, C) by the same formula as before and this gives sheaves R(α), lR(α) etc.
Then the proposition 3.8 holds more generally for these α the proof being identical.
Note that if we are given a closed orbit γ : lR/l(γ)ZZ → X and if x0 = γ(0) we can
consider its monodromy action on lR(α) for α ∈ lR which is given by:
ρα ([γ]) = exp
Z
γ
Z
αωφ = exp
l(γ)
0
φ∗x0 (αωφ )
!
.
As φ∗x0 ωφ = dt if t is the coordinate on lR/l(γ)ZZ it equals:
(19)
ρα ([γ]) = exp
Z
l(γ)
0
!
α dt = exp(αl(γ)) .
On the hoped for dynamical system “spec ZZ” we would thus get ρα ([γp ]) = pα for
example. Similar remarks apply to C(α) for α ∈ C. If we allow non-constant α as
in 3.9 we get more generally:
ρα ([γ]) = exp
Z
l(γ)
0
α ◦ φtx0 dt
!
for the monodromy action of γ on lR(α) or C(α).
3.10 Corollary: Consider a connected Frobenius system (X, φ, T0 ) and let dco be
the dimension of the Q-sub vector space of lR generated by the lengths l(γ) of the
closed orbits γ. Then we have:
1) dimQ (π1ab (X) ⊗ZZ Q) ≥ dco .
2) If dco ≥ 1 then the map ρ : lR −→ Hom(π1ab (X), lR>0 ) , α 7−→ ρα is injective.
3) For dco ≥ 2 the map ρ : C −→ Hom(π1ab (X), C∗ ) , α 7−→ ρα is injective as
well.
28
Proof: Whatever the base point x0 any closed orbit γ gives a conjugacy class in
π1 (X, x0 ) and hence a well defined element [γ] in π1ab (X). For ρα : π1ab (X) → C∗ we
have by (19) that ρα ([γ]) = exp(αl(γ)). Now 2) and 3) follow. 1) is a consequence
of the next observation:
3.11 Facts: a) In the situation of 3.10 there is a homomorphism
l = lX : π1ab (X) −→ lR
R
with l([γ]) = l(γ) for closed orbits γ. It is given by l = log ρ1 i.e. l[γ] = γ ωφ .
Its image Λ = ΛX ⊂ lR is called the group of periods of (X, φ, T0 ) e.g. [Pl] p. 740. If
there is a local system Q(1) of Q-vector spaces on X such that Q(1)⊗Q lR = lR(1) then
Λ ⊂ log Q∗+ since in that case ρ1 = exp ◦ l must take values in GL1 (Q(1)x0 ) = Q∗ .
b) For any morphism ρ : (X, φtX , T0 X) → (X 0 , φtX 0 , T0 X 0 ) of Frobenius systems we
have a commutative diagram:
π1ab (X)
ρ∗ y
lX
&
lX 0
π1ab (X 0 )
lR
%
so that in particular ΛX ⊂ ΛX 0 . This follows from the equation ρ∗ ωφX 0 = ωφX of
remark 3.3.
Remark: For an algebraic IFq -scheme X0 every closed point x ∈ |X0 | determines a
Frobenius conjugacy class in the fundamental group of X0 with respect to any base
point and hence a well defined Frobenius element Fx in π̂1ab (X0 ). For the canonical
projection:
deg : π̂1ab (X0 ) −→→ π̂1ab (spec IFq ) = ẐZ
induced by X0 → spec IFq we have:
deg(Fx ) = deg x = (κ(x) : IFq ) for all x ∈ |X0 | .
This projection is also obtained as the image of 1 ∈ ẐZ under the map ẐZ →
Homcont (π̂1ab (X0 ), ẐZ) obtained by taking the projective limit of the edge morphisms:
ZZ/N ⊂ H 0 (X0 ⊗ IFq , ZZ/N ) −→ H 1 (X0 , ZZ/N ) = Hom(π1ab (X0 ), ZZ/N ) .
By functoriality it suffices to check this assertion for X0 = spec IFq where it is easy.
Note that on the abstract (not profinite) subgroup of π̂1ab (X0 ) generated by all Fx
the homomorphism log q · deg has values in (log q)ZZ ⊂ lR. For those X0 /IFq to which
the discussion of 2.6 applies (c.f. 2.7) it should be closely related to the map l of
3.11 for the hoped for Frobenius system X attached to X0 .
Remark: In Selberg’s theory the log of the norm of a conjugacy class does not
come from a homomorphism on the abelianized fundamental group. This is another
manifestation of the fact that T u ⊕ T s is not integrable in that context. Note here
29
that in proposition 2.9 we can add the following equivalent condition:
(3) The length function extends to a homomorphism l : π1ab (X) → lR.
The structure of complete Frobenius systems is clarified by the following result which
essentially already occurs in the work of Reeb, Novikov and Plante:
3.12 Theorem (c.f. [Pl] 2.4, 2.6): Consider a complete Frobenius system
(X, φ, T0 ) with group of periods Λ and assume that X is connected. Then for any
leaf M of the T0 -foliation we have Λ = {t ∈ lR | φt (M ) = M } and hence the connected Banach manifold M inherits a Λ-action by letting λ ∈ Λ act via φλM = φλ |M .
With notations as in 3.1 there is then an isomorphism of Frobenius systems:
∼
(X, φ, T0 ) −→ (M ×Λ lR, τ, T M ×Λ lR)
and in particular an exact sequence:
l
1 −→ π1 (M, x0 ) −→ π1 (X, x0 ) −→ Λ −→ 0
for any x0 ∈ M .
Proof: We first assume that Λ = 0. Then ωφ = df for a smooth function f : X → lR.
As
d(φt∗ f − f ) = φt∗ ωφ − ωφ = 0
we have φt∗ f = f + ct for some smooth function c : lR → lR. Since φt is a flow, c is
additive and c0 = 0 so that c = c1 · t. Because of
1 = hωφ , Yφ i = Yφ (f ) =
d
| (φt∗ f ) = c1
dt t=0
we see that φt∗ f = f + t. Thus f : X → lR is surjective and lR-equivariant if lR acts
via φ on X and via translation τ on lR. Because of ωφ,x 6= 0 for all x ∈ X the map
f is a submersion and for all s ∈ lR the lR-equivariant map
g : f −1 (s) × lR −→ X , g(y, t) = φt (y)
is a diffeomorphism with inverse g −1 (x) = (φs−f (x) (x), f (x) − s). In particular all
the manifolds f −1 (s) are diffeomorphic and connected. Since ωφ vanishes on T0 the
map f maps leaves of the T0 -foliation to points. It follows that the T0 -leaves are just
the manifolds f −1 (s) for s ∈ lR. Now let Λ be arbitrary and let (X̃, φ̃, T̃0 ) be the
induced complete Frobenius system on the covering π : X̃ → X with Galois group
Λ defined by the exact sequence:
l
1 −→ N −→ π1 (X, x0 ) −→ Λ −→ 0 .
The preceeding discussion applies to (X̃, φ̃, T̃0 ) and we find some f : X̃ → lR with
ωφ̃ = df which may be chosen such that f (x̃0 ) = 0 for some x̃0 ∈ X̃ over x0 . Let
M̃ be any T̃0 -leaf contained in π −1 (M ). We claim that the map π |M̃ : M̃ → M is
a diffeomorphism. The only issue is injectivity: Assume x̃1 , x̃2 ∈ M̃ are mapped to
30
the same point in M . Choose a C 1 -curve γ̃ in M̃ connecting x̃1 and x̃2 . Its image
γ in M is a closed curve and we find:
Z
γ
ωφ =
Z
γ̃
ωφ̃ = f (x̃2 ) − f (x̃1 ) = 0
since by the above M̃ is just a fibre of f . It follows that the deck transformation
mapping x̃1 to x̃2 corresponds to an element of N . But by construction N operates
trivially on X̃ so that x̃1 = x̃2 . We next claim that:
π −1 (M ) =
a
φ̃λ (M̃ ) .
λ∈Λ
Since the leaves of the T̃0 -foliation on X̃ over M are just the connected components
of π −1 (M ) they are permuted by the π1 (X, x0 )/N -action on X̃. Thus it suffices to
check that
σc (M̃ ) = φ̃l(c) (M̃ )
for all deck transformations σc corresponding to elements c ∈ π1 (X, x0 ). But this
follows from the equation
f (σc (x̃)) =
Z
σc (x̃)
x̃0
ωφ̃ = f (x̃) +
Z
σc (x̃)
x̃
ωφ̃ = f (x̃) +
= f (x̃) + l(c) = f (φl(c) (x̃)) .
Z
c
ωφ
From what we have seen above the composition:
G : M × lR
(π |M̃ )−1 ×idlR
∼
−→
g
∼
M̃ × lR −→ X̃
where g(y, t) = φ̃t (y)
is an lR-equivariant diffeomorphism. We have
M = πσc (M̃ ) = π φ̃l(c) (M̃ ) = φl(c) (M )
and hence Λ operates on M via the diffeomorphisms φλ |M .
As in 3.1 we let Λ act on M × lR by:
(x, t) · λ = (φ−λ (x), t + λ) for λ ∈ Λ .
On the other hand via the isomorphism Λ ∼
= π1 (X, x0 )/N the group Λ acts on X̃
by deck transformations and it only remains to check that G is Λ-equivariant. This
follows by a short calculation using the commutative diagrams where λ = l(c):
φ̃λ
σ
M̃ −→
φ̃λ (M̃ )
yπ |φ̃λ (M̃ )
π |M̃ y
c
λ
M̃ −→
φ̃
(M̃ )
yπ |φ̃λ (M̃ )
π |M̃ y
M = φλ (M )
φλ
M −→ M = φλ (M )
and the fact that φ̃t commutes with the deck transformations σc .
3.13 In the following proposition and remark the Frobenius systems are not assumed to be complete.
31
Proposition: Consider a connected Frobenius system (X, φ, T0 ) and a point
x0 ∈ X.
1) Let (X̃, φ̃, T̃0 ) be the induced Frobenius system on the covering π : X̃ → X
corresponding to the kernel of l : π1 (X, x0 ) → lR with Galois group ΛX . Then there
is a ΛX -equivariant morphism of Frobenius systems
f : (X̃, φ̃, T̃0 ) −→ (lR, τ, 0)
where ΛX acts on X̃ by deck transformations and on lR by translation. The functions
f : X̃ → lR which occur here are those with ωX̃ = df . In particular they are uniquely
determined up to an additive constant.
2) If in 1) the fibres of f are connected (e.g. if the flow is complete) and if Λ ⊂ lR
is dense then H 0 (X, C) = C.
3) There are morphisms of Frobenius systems:
f : (X, φ, T0 ) −→ (lR, τ, 0)
resp.
f : (X, φ, T0 ) −→ (lR/lZZ, τ, 0) for some l ∈ lR, l > 0
if and only if ΛX = 0 resp. ΛX ⊂ lZZ.
Proof: 1) By construction ΛX̃ = 0. Hence ωX̃ = df for a smooth function
f : X̃ → lR and an argument as in the beginning of the proof of theorem 3.12
shows that f provides a map f : (X̃, φ̃, T̃0 ) → (lR, τ, 0). It is ΛX -equivariant because
for c ∈ π1 (X, x0 ) we have:
Z
f (σc (x̃)) − f (x̃) =
σc (x̃)
x̃
ωφ̃ =
Z
c
ωφ = l(c) for all x̃ ∈ X̃ .
On the other hand by 3.3 we have for any morphism f : (X̃, φ̃, T̃0 ) → (lR, τ, 0) that
ωφ̃ = f ∗ (dt) = df .
2) For h in H 0 (X, C) consider its Λ-invariant lift to a function h̃ = h ◦ π in H 0 (X̃, C).
Let x̃1 , x̃2 be two points in the same fibre of f . They can be connected by a C 1 -curve
γ̃ in this fibre and we have:
h̃(x̃1 ) − h̃(x̃2 ) =
Z
γ̃
dh̃ =
Z
γ̃
Yφ̃ (h̃)ωφ̃ = 0 since ωφ̃ = df .
As Λ is non-zero f is surjective and using 1) it follows that h̃ factors over a continuous
Λ-invariant function on lR i.e. over a constant function. Thus h is constant as well.
3) Because of 3.11 the conditions are necessary. An application of part 1) shows
that they are also sufficient.
Remark: If ΛX has rank higher than one the map f in 1) may be viewed as a map
f : X → “non-commutative lR/ΛX ”. More precisely it induces a map
C0 (lR)× ΛX −→ C0 (X̃)× ΛX
between crossed product algebras, at least if X is locally compact. Since ΛX being
abelian is amenable the crossed product algebras equal their reduced versions. Since
C0 (X̃)× ΛX is Morita equivalent to C0 (X) we get in particular a map
Kn (C0 (lR)× ΛX ) → Kn (C0 (X)) = K n (X) for every n ≥ 0 .
32
The left hand group can be calculated by the following argument which was communicated to me by M. Puschnigg.
Let Γ be a discrete abelian group operating on a C ∗ -algebra A. Then there is a
natural convergent spectral sequence
2
Epq
= Hp (Γ, Kq (A)) =⇒ Kp+q (A× Γ) .
Assume for simplicity that Γ is torsionfree and write Γ = lim Γi a filtering direct limit
−→
of free abelian subgroups Γi ⊂ Γ of finite rank. Then one checks that lim A× Γi =
−→
A× Γ and hence
K∗ (A× Γ) = lim K∗ (A× Γi ) .
−→
The spectral sequence is then obtained as the direct limit of the corresponding
spectral sequences for the Γi -action of A. These in turn are obtained iteratively
from the case Γ = ZZ which follows from the following result of Pimsner–Voiculescu:
Let A be a C ∗ -algebra, α ∈ Aut A an automorphism. Then there is a natural exact
sequence:
1−α∗
K∗ (A)
−→ K∗ (A) −→ K∗ (A×
x
α ZZ)
∂
y∂
1−α
K∗+1 (A× α ZZ) ←− K∗+1 (A) ←−∗
K∗+1 (A) .
Since Kq (C0 (lR)) = ZZ if q ≡ 1 mod 2 and = 0 if q ≡ 0 mod 2 the spectral sequence
for the ΛX operation on C0 (lR) degenerates and if ΛX is free abelian therefore gives
(uncanonical) isomorphisms:
Kn (C0 (lR)× ΛX ) ∼
=
M
Λn+2r+1 (ΛX ) .
r∈ZZ
The situation can be analyzed further by considering the morphism of spectral
sequences:
Hp (ΛX ,K q (lR)) =⇒ Kp+q (C0 (lR)×
ΛX )
y
Hp (ΛX , K q (X̃)) =⇒
y
K p+q (X)
which come from the above. With foliations and ergodic group actions present
there is a lot of noncommutative geometry that can be applied to the study of
Frobenius systems e.g. [C1]. Apart from the above remark on the map from the
K-theory of “lR/ΛX ” to the one of X however we want to draw attention to one
more point only: Assume that ΛX ⊂ lR is free abelian of rank greater than one.
Then the von Neumann algebra L∞ (lR)× ΛX is a factor of type Π∞ . In the case
of number theoretical interest where ΛX = log(Q∗+ ) is a possibility c.f. Remark 4.2
the dimension theory of this factor can perhaps be used to realize the well known
hope of interpreting the volumes of cohomologies in Arakelov theory as real-valued
dimensions. However this is pure fancy at the present stage.
3.14 We now incorporate vector bundles into the picture: Assume that we are in
the situation of theorem 3.12 and let ρ : E → X be a vector bundle on X with an
lR-operation ψ t opposite to φt . Let ρM : EM → M denote the restriction of E to
33
a vector bundle on the leaf M . It is equipped with a Λ-operation opposite to the
Λ-operation φλM on M given by
λ
ψM
:= ψ λ |φλ∗
: φλ∗
M EM −→ EM
M EM
for λ ∈ Λ .
−λ
λ
Thus η t = ψ −t and ηM
= ψM
fit into commutative diagrams:
ηλ
ηt
E −→ E
ρy
yρ
and
φt
M
E
EM −→
M
ρM y
y ρM
φλ
X −→ X
M
M −→
M.
Set EM ×Λ lR = (EM × lR)/Λ where Λ operates on EM × lR via
−λ
λ
(e, t) · λ = (ηM
(e), t + λ) = (ψM
(e), t + λ).
Proposition: There are commutative diagrams:
∼
E
×Λ lR
−→ EM
ρy
yρM × id
∼
X −→ M ×Λ lR
(20)
and
φλ∗ E ∼
= (τ −λ )∗ (EM ×Λ lR) = φλ∗
M EM
×Λ lR
λ
λ
ψ y
yψM × id
∼
E
−→
EM ×Λ lR .
(21)
Proof: Let ρ̃ : Ẽ → X̃, η̃ t be the pullbacks of ρ : E → X, η t to the covering
π : X̃ → X used in the proof of 3.12. Note that Ẽ carries a Λ-operation over the
one on X̃ such that Ẽ/Λ = E. One checks that with notations as in loc. cit.
∼
i) gẼ : ẼM̃ × lR −→ Ẽ , gẼ (e, t) = η̃ t (e) is an isomorphism
ii) the isomorphism
GE : EM × lR
(π |M̃ )−1 ×id
∼
−→
gẼ
∼
ẼM̃ × lR −→ Ẽ
is Λ-equivariant, and
iii) the diagrams
G
G
ρM
E
EM× lR −→
Ẽ
× idy
yρ̃
and
E
EM× lR −→
Ẽ
t
t
yη̃
id × τ y
G
G
E
EM × lR −→
Ẽ
M × lR −→ X̃
are commutative. Thus (20) follows and using the equality:
λ
id × τ λ = ηM
× id : EM ×Λ lR −→ EM ×Λ lR
we get (21) as well.
34
A (flat) T0 -connection δ0 on E induces a (flat) connection δM on EM . If δ0 is
∼
compatible with ψ t then the isomorphism E −→ EM ×Λ lR identifies δ0 with the
canonical T0 (M ×Λ lR) = T M ×Λ lR-connection on EM ×Λ lR induced by δM . In
this case the sheaves F and F on X attached to (E, ψ, δ0 ) by proposition 3.6 can
δM
be described over M ×Λ lR as follows: Let FM = EM
be the locally constant sheaf
×AlR
on M of horizontal sections of (EM , δM ). The exterior tensor products FM 2
and FM 2
×lR on M × lR carry natural Λ-operations covering the one on M × lR,
and descend to sheaves FM 2
×Λ AlR and FM 2
×Λ lR on M ×Λ lR. If E is real then F
corresponds to FM 2
×Λ AlR and F to FM 2
×Λ lR and similarly in the complex case. In
particular:
×Λ AlR and CX =
b CM 2
×Λ AclR .
RX =
b lRM 2
To see that F corresponds to FM 2
×Λ AlR note that the canonical Λ-equivariant inclusion
×AlR ,→ F̃
i : FM 2
of sheaves on M × lR ∼
= X̃ is actually an isomorphism: A section s = s(m, t) of F̃
over U × I where U ⊂ M is open and connected and I ⊂ lR is open is by definition
locally constant and hence constant in the variable m ∈ U . Thus s lies in the image
of i which therefore induces isomorphisms
∼
Γ(U × I, FM 2
×AlR ) −→ Γ(U × I, F̃) .
Hence i is an isomorphism on stalks and the claim follows.
λ
We also point out that the maps ψM
: φλ∗
M EM → EM above induce a Λ-operation
λ
λ −1
ψM : (φM ) FM → FM opposite to the Λ-operation on M and that the composition:
φλ∗
ψλ
M
M
λ
n
n
λ −1
n
ψM
∗ : H (M, FM ) −→ H (M, (φM ) FM ) −→ H (M, FM ) for λ ∈ Λ
defines a Λ-operation on the cohomologies H n (M, FM ).
3.15 We next make some remarks on the calculation of the group H 1 (X, F ) for
a local system F of C- vector spaces of finite rank on a connected not necessarily
complete Frobenius system X. Similar assertions hold for local systems of lR-vector
spaces. Let X̃ be the covering of X with Galois group Λ defined by the exact
sequence:
l
1 −→ N −→ π1 (X, x0 ) −→ Λ −→ 0
where x0 ∈ X is a base point. Let F̃ be the pullback of F to X̃ and consider the
spectral sequence
H i (Λ, H j (X̃, F̃ )) =⇒ H i+j (X, F ) .
It gives rise to a short exact sequence:
0 −→ H 1 (Λ, H 0 (X̃, F̃ )) −→ H 1 (X, F ) −→ H 1 (X̃, F̃ )Λ −→ H 2 (Λ, H 0 (X̃, F̃ )) −→ H 2 (X, F ) .
The space H 0 (X̃, F̃ ) ∼
= FxN0 being finite dimensional the first group H 1 (Λ, H 0 (X̃, F̃ ))
can sometimes be approached by the following lemma. Note by the way that if
X happens to be complete X ∼
×Λ C then by homotopy
= M ×Λ lR and F ∼
= FM 2
invariance:
λ
H j (X̃, F̃ ) = H j (M, FM ) with Λ-operation via ψM
∗ .
35
Lemma: Let A be an abelian group and let V be a C[A]-module which is finite
dimensional as a C-vector space. Write V unip for the C[A]-submodule of V of vectors
which for all a in A are annihilated by (1 − a)L for some L ≥ 1 (equivalently: for
L = dimC V ). Then the inclusion V unip ⊂ V induces an isomorphism:
H 1 (A, V unip ) = H 1 (A, V ) .
In particular we have
H 1 (Λ, H 0 (X̃, F̃ )) ∼
= H 1 (Λ, FxN0 unip ) .
in the above sequence.
Corollary: For any α ∈ C define the Λ-module C(α) to be C with action by e−λα
for λ in Λ. Then for any connected Frobenius system X we have an exact sequence:
0 → H 1 (Λ, C(α)) → H 1 (X, C(α)) → HomΛ (Γ, C(α)) → H 2 (Λ, C(α)) → H 2 (X, C(α))
where Γ = N ab / torsion and where:
Hom(Λ, C) if α = 0 or
if 0 6= α ∈ ilR and Λ ⊂
H (Λ, C(α)) =
0
in all other cases.
1
2πi
ZZ
α
Proof of the lemma: 1) First consider the special case where there is some b in A
such that 1 − b is invertible on V . We then have to show that H 1 (A, V ) = 0. So let
ϕ : A → V be a cocycle i.e. ϕ(a1 + a2 ) = ϕ(a1 ) + a1 · ϕ(a2 ). As A is commutative
we have
(1 − a1 ) · ϕ(a2 ) = (1 − a2 ) · ϕ(a1 ) .
Set v := (1 − b)−1 ϕ(b). Then we have for any a in A
(1 − a)v = (1 − b)−1 (1 − a)ϕ(b) = (1 − b)−1 (1 − b)ϕ(a) = ϕ(a)
and hence ϕ is a coboundary.
2) In the general case set
a Vµ
= {v ∈ V | (µ − a)L v = 0 for some L ≥ 1}
for any a in A, µ in C and let a V 0 be the (direct) sum of the generalized eigenspaces
6 1. Then we have
a Vµ for µ =
V = a V1 ⊕ a V 0
as C[A]-modules
and step 1 shows that
H 1 (A, a V1 ) = H 1 (A, V )
since 1 − a is invertible on a V 0 . For a suitable choice of elements a1 , . . . , an in A we
have
unip
an (. . . a2 ( a1 V1 )1 . . .)1 = V
and the lemma follows.
36
3.16 We now prove the topological dynamical Lefschetz trace formula for compact
Frobenius systems whose group of periods Λ is infinite cyclic. This is by far the
easiest case and certainly not the one relevant for the applications we have in mind.
Still it shows to some extent that the picture we have been developing is not unreasonable. We hope to return to Lefschetz trace formulas for more general Frobenius
systems in the future.
For the present case we need some preparations. Let K = lR or C. For l ∈ lR, l > 0
set Λ = lZZ ⊂ lR and let Rep K,f (Λ) be the abelian category of finite dimensional
linear representations of Λ over K. We denote the operation of λ ∈ Λ on
λ
H ∈ Rep K,f (Λ) by ψ λ = ψH
. For µ ∈ C∗ we set:
Hµ = {h ∈ HC | ex. L ≥ 1 s.t. (ψ λ − µλ/l id)L (h) = 0 for all λ ∈ Λ}
where HC = H ⊗K C.
The group (lR, +) acts on C ∞ (lR) by τ t where (τ t g)(s) = g(s + t). We define a
Λ-operation on H ⊗ C ∞ (lR) by
−λ
λ · (h ⊗ g) = ψH
(h) ⊗ τ λ g .
Let Rep K (lR) be the abelian category of K-vector spaces with a linear (lR, +)-action.
For D in Rep K (lR) and α ∈ C we set:
Dα = {d ∈ DC | ex. L ≥ 1 s.t. (ψ t − etα id)L (d) = 0 for all t ∈ lR}
where ψ t denotes the (lR, +)-operation on D and DC = D ⊗K C. Consider the
additive functor:
ID∞ : Rep K,f (Λ) −→ Rep K (lR)
defined by
ID∞ (H) = (H ⊗lR C ∞ (lR))Λ
with lR-operation via τ t =
b idH ⊗ τ t .
Note that ID∞ (H) is naturally a Fréchet space and the action
lR × ID∞ (H) −→ ID∞ (H) , (t, d) 7−→ τ t (d)
is smooth. In particular the infinitesimal generator of τ exists. By the mean value
theorem it equals:
d
ΘID∞ (H) := idH ⊗
dt
∞
as a continuous linear operator on ID (H). Hence the elements d ∈ ID∞ (H)α satisfy:
(ΘID∞ (H) − α id)L (d) = 0 for some L 0 ,
so that in particular they are in (H ⊗ C e (lR))Λ where C e (lR) is the space of real
analytic lR-valued functions on lR whose Taylor series are everywhere convergent.
We define the smooth part Dsm ⊂ D as the Gal(C/K)-fixed space of
37
L
α
Dα in DC .
3.17 Proposition: For all H in Rep K,f (Λ) we have (H ⊗ C ∞ (lR))Λ = 0 so that in
particular the functor ID∞ is exact. Moreover
dim ID∞ (H)α = dim Hexp(lα)
for all α ∈ C
and
ID∞ (H)sm ⊂ (H ⊗ C e (lR))Λ
is dense in ID∞ (H) .
Restricted to ID∞ (H)sm it holds that τ t = exp(tΘID∞ (H) ).
Remark: The Riemann Hilbert correspondence on G| m,K provides a functor ID from
Rep K,f (Λ) to the category of regular singular differential
equations on G| m,K and in
d
particular to Rep K (lR) if we let lR act by exp tl 5 z dz . We have ID ∼
= ID∞
sm . In
∞
particular IDsm is exact.
Proof of 3.17: We may assume that K = C. Identifying π1 (lR/Λ, 0) with Λ any H
in Rep C,f (Λ) defines a flat vector bundle H on lR/Λ. Let S be the sheaf of smooth
sections of H. As π : lR → lR/Λ is a covering π −1 S is identified with the sheaf of
smooth sections of the constant bundle π ∗ H. In particular π −1 S is a fine sheaf. The
spectral sequence of this covering:
H i (Λ, H j (lR, π −1 S)) =⇒ H i+j (lR/Λ, S)
therefore degenerates into isomorphisms:
H i (Λ, H 0 (lR, π −1 S)) = H i (lR/Λ, S) .
Since S is fine and
H 0 (lR, π −1 S) = C ∞ (lR, H) = H ⊗ C ∞ (lR)
the first assertion follows taking i = 1.
λ
For µ ∈ C∗ let C(µ) be C with ZZ-operation ψC(µ)
= µλ/l for λ ∈ Λ. Choose any
β ∈ C with exp(2πiβ) = µ and set
gβ (t) = exp(2πil−1 βt) .
Then
ID∞ (C(µ)) ∼
= gβ · ID∞ (C) as topological (lR, +)-modules.
By definition
ID∞ (C) = C ∞ (lR, C)Λ = C ∞ (lR/Λ, C) ,
hence using Fourier theory we see that:
ID∞ (C(µ))α = gβ+ν · C if lα = 2πi(β + ν) , ν ∈ ZZ
and
ID∞ (C(µ))α = 0 for all other values of α .
Thus
dim ID∞ (C(µ))α = dim C(µ)exp(lα)
38
for all α ∈ C
and again by Fourier theory we get for H = C(µ) that
ID∞ (H)sm =
M
ID∞ (H)α =
α∈C
M
ID∞ (H) 2πi (β+ν)
l
ν∈ZZ
is dense in ID∞ (H) .
The general case follows from this since the functors ID∞ and ID∞
sm are exact and
since every object of Rep C,f (Λ) is a finite successive extension of C(µ)’s. For the
last assertion note that τ t = exp(tΘID∞ (H) ) after restriction to (H ⊗ C e (lR))Λ .
We can now calculate the leafwise cohomologies in the following situation:
3.18 (X, φt , T0 ) is a compact and hence complete, connected Frobenius system
whose group of periods Λ is infinite cyclic Λ = lZZ ⊂ lR for some l > 0. Thus by
3.12 the leaves of the T0 -foliation are also compact. F is a free R- or C-module of
finite rank on X with an action ψ opposite to φ. Correspondingly set K = lR resp.
K = C.
3.19 Proposition: In the situation 3.18 let M be a T0 -leaf of X with the induced
Λ-action and let FM be the restriction of F to a locally constant sheaf on the compact
λ
manifold M . Recall from 3.14 that H n (M, FM ) is a Λ-module via ψM
∗ for λ ∈ Λ
n
t
and from 1.6 that H (X, F ) is an (lR, +)-module via ψ∗ for t ∈ lR. The Fréchet
topologies on the global sections of the natural C ∞ -resolution of F
δ
δ
δ
0
0
0
0→F →E →
T0∗ ⊗ E →
Λ2 T0∗ ⊗ E →
...
where (E, δ0 , ψ) corresponds to (F , ψ) via proposition 3.6 induce a Fréchet topology on H n (X, F ), the point being Hausdorffness. There is a canonical topological
isomorphism of Fréchet spaces with a smooth (lR, +)- action:
H n (X, F ) = ID∞ (H n (M, FM )) .
In particular the smooth part H n (M, F )sm is dense in H n (X, F ) and only countably
many H n (X, F )α are nonzero. The infinitesimal generator Θψ∗ of ψ∗t exists as a
continuous linear operator on H n (X, F ):
1
Θψ∗ (h) := lim (ψ∗t (h) − h) for h ∈ H n (X, F ) .
t→0 t
It agrees with the map Θ = (ΘF )∗ induced on cohomology by the map of sheaves
ΘF : F → F . On H n (X, F )sm we have:
ψ∗t = exp(tΘψ∗ ) = exp(tΘ) .
Remark: Before giving the proof of the proposition we show by a well known
example that for Frobenius systems with non-compact T0 - leaves the cohomology
can have a more complicated structure.
For l ∈
/ Q set Λ = ZZ + lZZ ⊂ lR and consider the natural projection π : Λ → lZZ. Let
λ ∈ Λ act on M = lR by translation with π(λ) and consider the Frobenius system
39
X = M ×Λ lR = lR ×lZZ (lR/ZZ) whose flow is 1-periodic. Its T0 -foliation is identified
with the foliation by the flowlines of the Kronecker flow with slope l on the 2-Torus.
We have
× AlR )
H q (M × lR, R) = H q (M × lR, lR 2
q
= H (lR, AlR ) by homotopy invariance
= 0 for q ≥ 1 and = C ∞ (lR) for q = 0 .
Hence the spectral sequence
H p (Λ, H q (M × lR, R)) =⇒ H p+q (X, R)
degenerates into isomorphisms:
H n (X, R) = H n (Λ, C ∞ (lR)) .
By proposition 3.17 we know that
H q (ZZ, C ∞ (lR)) = 0 for all q ≥ 1 .
Hence the spectral sequence
H p (lZZ, H q (ZZ, C ∞ (lR))) =⇒ H p+q (Λ, C ∞ (lR))
for the group extension 0 → ZZ → Λ → lZZ → 0 gives isomorphisms
H n (Λ, C ∞ (lR)) = H n (lZZ, C ∞ (lR/ZZ)) ,
so that we get:
H n (X, R) = H n (lZZ, C ∞ (lR/ZZ)) .
The right hand group has been determined for n = 1 by Haefliger in [H] 2.1 giving
the first part of the following proposition. Call a number l ∈
/ Q Liouville if it does
not satisfy a diophantine condition of the following type: There are positive numbers
s and c such that
|ml + n| ≥ c(1 + m2 )−s
for any (m, n) 6= (0, 0) in ZZ2 .
Proposition: 1) H 0 (X, R) = lR , H n (X, R) = 0 for n ≥ 2 and
dimlR H 1 (X, R) = ∞ resp. = 1 according to whether l is a Liouville number or not.
1
The maximal Hausdorff quotient H (X, R) of H 1 (X, R) is one-dimensional.
2) For α ∈ C we have
n
dimC H (X, C)
Θ=α
=
(
1 for n = 0, 1 and α = 0
0 in all other cases
where Θ = (ΘC )∗ .
The second assertion follows from the exact sequence (16):
0 → H 0 (X, C)/(Θ − α)H 0 (X, C) → H 1 (X, C(α)) → H 1 (X, C)Θ=α → 0
40
since H 1 (X, C(α)) is two-dimensional for α = 0 and zero for α 6= 0 by Corollary
3.15.
Proof of proposition 3.19: We may assume that F is a C-module i.e. K = C.
According to 3.12 we have canonically X = M ×Λ lR. Let π : M × lR → X be
the canonical projection and set F̃ = π −1 F , so that F̃ = FM 2
×AclR . Applying the
Künneth formula of [Br] Th. 8.3 + Ex. 19 we get an isomorphism:
M
∼
H p (M, FM ) ⊗C H q (lR, AclR ) −→ H n (M × lR, F̃ )
p+q=n
and hence the canonical map
(22)
∼
α : H n (M, FM ) ⊗lR C ∞ (lR) −→ H n (M × lR, F̃)
is an isomorphism. Its inverse is given as follows: Any cohomology class ξ in
H n (M × lR, F̃ ) is represented by a smooth map
ω : M × lR −→ Λp T ∗ M ⊗ EM
with ω(m, t) ∈ (Λp T ∗ M ⊗ EM )m
for all m ∈ M
such that for every t ∈ lR the form ωt ∈ Γ(M, Λp T ∗ M ⊗ EM ) defined by ωt (m) =
ω(m, t) is closed. Then α−1 (ξ) is given by the map t 7→ [ωt ] as an element of
C ∞ (lR, H n (M, FM )) = H n (M, FM ) ⊗lR C ∞ (lR) .
Note here that H n (M, FM ) is finite dimensional. Because of this the differentials in
the complex Γ(M, Λ• T ∗ M ⊗ EM ) are closed maps in the natural Fréchet topologies
and hence a Fréchet structure is induced on H • (M, FM ) which by finite dimensionality is the natural topology on a finite dimensional C- vector space. Thus α and
α−1 are continuous so that in particular H n (M × lR, F̃ ) is a Fréchet space. Thus the
differentials on the global sections of the C ∞ -resolution of F̃ are closed and hence
also the differentials of the closed subcomplex Γ(X, Λ• T0∗ ⊗ E) of Λ-invariant such
sections. Hence H n (X, F ) is a Fréchet space as well. Because of 3.17 and (22) the
spectral sequence
H p (Λ, H q (M × lR, F̃)) =⇒ H p+q (X, F )
degenerates into (algebraic) isomorphisms
∼
H n (X, F ) −→ H n (M × lR, F̃ )Λ
which are clearly continuous and by the open mapping theorem for Fréchet spaces
therefore topological isomorphisms. Thus we get a natural (lR, +)-equivariant topological isomorphism
∼
ID∞ (H n (M, FM )) −→ H n (X, F ) where ΘID∞ (H n (M,FM )) =
b Θ.
Now the remaining assertions follow from the corresponding facts for ID∞ (H).
We can now prove a topological dynamical Lefschetz trace formula for compact
Frobenius systems:
41
3.20 Theorem: In the situation 3.18 assume that the set Ω0 of closed orbits is
finite and that for any γ ∈ Ω0 the sign:
ε(γ) = sgn det(1 − Tγ φl(γ) | T0γ )
is well defined the determinant being nonzero. Let E be the flat vector bundle on X
corresponding to F as in 3.6. Then the following topological dynamical Lefschetz
trace formula holds in D 0 (lR+ ):
X
(−1)n Tr(ψ∗ | H n (X, F ))dis =
n
X
l(γ)ε(γ)
γ∈Ω0
X
Tr(ψγkl(γ) | Eγ )δkl(γ)
k≥1
where the distributional trace on the left hand side is defined by formula (11). Note
that by definition for any point x ∈ γ we have:
Tr(ψγkl(γ) | Eγ ) = Tr(ψxkl(γ) | Ex ) = Tr(ψxkl(γ) | Fx /mx Fx )
where mx ⊂ Rx resp. Cx is the ideal of functions vanishing at x c.f. [P] § 2.
Proof: We may assume that F is a C-module. As X is compact the system is
complete and hence for any T0 -leaf M we have X = M ×Λ lR by 3.12. With notations
as before f = φlM is a diffeomorphism of the compact manifold M and the map
γ 7→ γM = M ∩ γ gives a bijection between the closed orbits γ of the lR-action
on X and the finite orbits γM of the ZZ-action on M by the powers of f with
l
l(γ) = |γM | · l. The map fE = ψM
is an automorphism opposite to f of the flat
vector bundle EM = E |M .
Because of the relation:
ε(γ) = sgn det(1 − Tx f |γM |k | Tx M ) =: εk (γM ) for any x ∈ γM , k ≥ 1
we find:
X
l(γ)ε(γ)
γ
=
X
X X
νl
X
ν≥1
= l
X
γM
|γM |=ν
X
x∈M
δnl
X
n≥1
kν
εk (γM )Tr(fE,x
| EM,x)δkνl
where x = xγM ∈ γM
k≥1
f ν (x)=x
f ν−1 (x)6=x
n≥1
= l
Tr(ψγkl(γ) | Eγ )δkl(γ)
k≥1
ν≥1
= l
X
X
X
kν
sgn det(1 − Tx f kν | Tx M )Tr(fE,x
| EM,x )δkνl
k≥1
n
sgn det(1 − Tx f n | Tx M )Tr(fE,x
| EM,x )
x∈M
f n (x)=x
δnl
X
(−1)ν Tr((fE , f )∗n | H ν (M, FM ))
ν
by the usual Lefschetz fixed point formula. Here FM is the sheaf of locally constant
sections of EM . Using the isomorphism of proposition 3.19:
H ν (X, F ) = ID∞ (H ν (M, FM )) where ψ∗ =
b τ∗
and the next lemma the result follows.
42
3.21 Lemma: For H ∈ Rep K,f (Λ), Λ = lZZ, l > 0 we have:
Tr(τ∗ | ID∞ (H))dis = l
X
nl
δnl Tr(ψH
| H) in D 0 (lR+ )
n≥1
where Tr(τ∗ | ID∞ (H))dis is defined by formula (11)
Proof: We may assume that K = C. For α ∈ C we have by 3.17 that
mα = dim ID∞ (H)α = dim Hexp(lα) .
Thus for any test function ϕ ∈ D(lR+ ) we find:
∞
hTr(τ∗ | ID (H))dis , ϕi =
X
mα
α∈C
= l
X
Z
lR
etα ϕ(t) dt
dim Hµ
µ∈C∗
XZ
n∈ZZ lR
etLog µ ϕ(lt)e−2πint dt .
According to the Poisson summation formula we have for all ψ ∈ D(lR):
X
ψ(n) =
ψ̂(x) =
Hence we get:
ψ̂(n)
n∈ZZ
n∈ZZ
where
X
Z
lR
ψ(t)e−2πixt dt .
hTr(τ∗ | ID∞ (H))dis , ϕi = l
X
dim Hµ
µ∈C∗
= l
X
X
ϕ(ln)µn
n∈ZZ
nl
ϕ(ln)Tr(ψH
| H)
n≥1
since supp ϕ ⊂ (0, ∞).
3.22 The following discussion leads to a more sheaf theoretic interpretation of the
local terms in topological dynamical Lefschetz trace formulas. It was suggested by
the considerations in [D1] § 7, [D2].
Let X = (X, φ, T0 ) be a Frobenius system and consider an orbit γ in X. Every point
x ∈ γ determines a morphism of Frobenius systems as follows. Set S = (t(x), ∞) if
γ is not closed and let S = lR/lZZ if γ is closed of lenght l = l(γ). Then the map
γ = (γ, x) : S = (S, τ, 0) −→ X
defined by (γ, x)(s) = φs (x)
defines an S-valued point of X in the category of Frobenius systems. Now note that
sheaves of RS - modules are acyclic since RS = AS is fine and that as pointed out
before 3.6 the pullback of RX -modules via γ to RS -modules is an exact functor.
Hence the functors
F 7−→ Fγ = Γ(S, γ ∗ F )
43
from R-modules to C ∞ (S)-modules are exact. Note that Fγ = Γ(S, γ −1 F ) as lRvector spaces. We think of Fγ as the stalk of F in the geometric point γ determined
by x ∈ γ over γ. If F carries an action ψ opposite to φ then Fγ becomes an (lR≥0 , +)module via ψ∗t c.f. 1.6. For different choices of γ over γ the functors F 7→ Fγ into
C ∞ (S)-modules (eventually with a semilinear (lR≥0 , +)-action) are isomorphic. The
same facts hold for C-modules of course.
Corollary 3.23 Let X be as above and let F be a locally free R-module of finite
rank on X with an action ψ opposite to φ. Let E be the flat vector bundle on X
corresponding to F as in 3.6. Then the following formula holds for any closed orbit
γ on X in D 0 (lR+ ):
Tr(ψ∗ | Fγ )dis = l(γ)
X
Tr(ψγkl(γ) | Eγ )δkl(γ)
k≥1
where the distributional trace is defined by (11). On (Fγ )sm we have ψ∗t = exp(tΘ)
where Θ is induced by the map of sheaves ΘF : F → F .
Proof: The result follows by applying 3.19 and 3.20 to S and γ ∗ F noting that γ ∗ E
corresponds to γ ∗ F as in 3.6 and that Θγ ∗ F = γ ∗ ΘF .
Remark: Under the correspondence 2.5 if p =
b γp on X = XS for p ∈
/ S and F (M )
is the sheaf of R-modules corresponding to a motive M over k with good reduction
outside of S we should have:
F (M )γ p
sm
,Θ ∼
= (Fp (M ), Θ)
where the right hand side was constructed in [D1] § 3. That construction depended
on choices which conjecturally lead to isomorphic pairs.
Using the preceeding corollary the trace formula of 3.20 can be rewritten in the form
(23)
X
(−1)n Tr(ψ∗ | H n (X, F ))dis =
n
X
ε(γ)Tr(ψ∗ | Fγ )dis .
γ∈Ω0
3.24 Comparing the trace formulas of 2.5 and 3.20 we are suggested to look for a
natural class of Frobenius systems such that for the finite dimensional systems in
this class ε(γ) = 1 for all closed orbits γ for which ε(γ) is defined. For example
we could consider Frobenius systems with T0 = T s where T s is defined with respect
to some norm k k on T X to be the set of vectors v ∈ T X with kT φt (v)k → 0 for
t → +∞. Note that for any point x on a closed orbit γ the eigenvalues of Tx φl(γ)
acting on Txs have absolute value less than one. We will call such Frobenius systems
stable. Other natural classes of Frobenius systems with the above property are the
following. Assume that the Frobenius system (X, φ, T0 ) carries a diffeomorphism of
bundles J : T0 → T0 such that J 2 = −id and J ◦ T φt = T φt ◦ J. In other words the
leaves of the T0 -foliation carry almost complex structures which vary differentiably
with the leaves and are respected by the flow. For a closed orbit γ and any x ∈ γ
the endomorphism T φl(γ) of T0x is then complex linear with respect to the complex
44
structure Jx so that ε(γ) = 1 if it is defined and X is finite dimensional. We will
call the structure (X, φ, T0 , J) an almost complex Frobenius system. It is called a
complex Frobenius system if J induces complex structures on all the leaves of the
T0 -foliation. These objects are made into categories in the obvious way. Consider a
Banach manifold M with an (almost) complex structure JM and a subgroup Λ ∈ lR
which acts on M by (almost) complex isomorphism such that the Λ-action on M ×lR
is properly discontinuous. Then (M ×Λ lR, τ, T M ×Λ lR, JM ×Λ id) is a complete
(almost) complex Frobenius system and an adaptation of the proof of theorem 3.12
shows that all complete (almost) complex Frobenius systems have this form. I expect
that the systems in 2.5 and 2.6 actually carry a complex structure J as above. This is
in accordance with the ideas at the beginning of section 3 in [D3] on a decomposition
of lH1 (Y, C) into a holomorphic and an antiholomorphic part. It would be interesting
to single out a class of stable or of of almost complex Frobenius systems for which
a topological dynamical trace formula of the following form is valid in D 0 (lR+ ):
(24)
X
(−1)n Tr(ψ∗ | Hcn (X, F ))dis =
n
X
l(γ)
γ∈Ω0
X
Tr(ψγkl(γ) | Eγ )δkl(γ)
k≥1
for locally free finite rank R-modules F with action ψ opposite to φ. Equivalently
by 3.23 it would take the form:
(25)
X
n
(−1)n Tr(ψ∗ | Hcn (X, F ))dis =
X
Tr(ψ∗ | Fγ )dis .
γ∈Ω0
Here “c” is to denote a suitable family of supports on X compatible with the flow.
Also a suitable notion of constructible sheaves of R-modules should be introduced
and (24) and (25) be generalized accordingly.
4
Frobenius systems in arithmetic geometry?
The content of this section is speculative. We take 2.5, 2.6 and 2.8 as our working
hypotheses and use the theory of Frobenius systems of section 3 to investigate how
this would fit with the motivic picture. In this way we get new information about
the topology of e.g. the dynamical system that should be attached to spec ZZ.
For the ring oS of S-integers in a number field k/Q consider the Frobenius system
(X, φt ) = (“spec oS ”, φt ) that should be attached to spec oS as in 2.5. To every
motive M of rank r over k with good reduction outside of S the construction of 2.8
would associate functorially a locally free sheaf FR (M ) of R-modules on X of rank
r equipped with an action ψ opposite to φ. To M = Q(n) there should correspond
the R-module R(n). We set
FC (M ) = FR (M ) ⊗R C = FR (M ) ⊗lR C .
The lR≥0 -equivariant isomorphisms postulated in 2.5:
ν
ν
H c (X, FC (M )) ∼
= ? H ν (“spec oS ”, F (M ))
= ? Hcν (“spec oS ”, F (M )) and H (X, FC (M )) ∼
45
where the right hand groups are the ones of [D1] § 7 imply the following relation to
partial L-series:
(26)
LS (M, s) =
2
Y
det∞
ν=1
and
(27)
1
Y
∗
LS (M (1), s) =
ν=0
det∞
1
ν
(s − Θψ∗ ) | H c (X, FR (M ))
2π
(−1)ν+1
1
ν
(s + Θψ∗ ) | H (X, FR (M ))
2π
(−1)ν+1
if the formalism works as expected in [D1] (7.4) and [D5] (3.1).
In particular infinitesimal generators Θψ∗ of the induced semiflow ψ∗t should exist on
ν
ν
H c (X, FR (M )) and H (X, FR (M )). In order to proceed we make the assumption
motivated by 3.19 that Θψ∗ agrees on these subquotients of cohomology with the
operator Θ = (ΘFR (M ) )∗ induced by the map of sheaves ΘFR (M ) = ψ̇ : FR (M ) →
FR (M ) c.f. 1.8. The notion of the infinitesimal generator Θψ∗ of ψ∗t presupposes the
ν
ν
existence of some Hausdorff topologies on H c and H . As mentioned before it seems
ν
ν
possible that H c and H are the smooth parts of the maximal Hausdorff quotients
of Hcν resp. H ν with respect to some locally convex topologies on the global sections
of the natural C ∞ -resolution
δ
δ
δ
0
0
0
0 −→ FR (M ) −→ E −→
T0∗ ⊗ E −→
Λ2 T0∗ ⊗ E −→
···
where (E, δ0 , ψ) corresponds to (FR (M ), ψ) via proposition 3.6. The differentials
should be continuous with respect to these topologies. For these heuristic reasons
it seems reasonable to assume that we can take:
0
H (X, FR (M )) = H 0 (X, FR (M ))sm .
(28)
We will see in a moment that on the other hand H 1 (X, R)sm must be much bigger
1
than H (X, R)!
ν
ν
Of course in order to be less conjectural on the relation between the H c and H groups to the cohomologies Hcν and H ν a study of dynamical Lefschetz trace formulas
for non-compact Frobenius systems generalizing Theorem 3.20 is necessary!
The considerations in [D1] § 7, [D5] § 3 together with (28) give that
0
H 0 (X, C)sm = H (X, C) = C with Θ = 0 .
(29)
From (27) applied to M = Q(0) we therefore get:
(30)
1
dim H (X, C)Θ∼α = νoS (α) := ords=1−α ((s − 1)ζk (s)
Y
(1 − N p−s )) .
p∈S
Recall that if the locally free C-module F corresponds to the local system F of
C-vector spaces as in proposition 3.6 then proposition 3.7 gives an exact sequence:
(31) 0 → H n−1 (X, F )/(Θ − α)H n−1 (X, F ) → H n (X, F (α)) → H n (X, F )Θ=α → 0
for every α ∈ C and in particular exact sequences:
(32)0 → H n−1 (X, C)/(Θ − α)H n−1 (X, C) → H n (X, C(α)) → H n (X, C)Θ=α → 0 .
46
Hence from (29) we get H 0 (X, C) = C i.e. that X is connected, and H 0 (X, C(α)) = 0
for all α 6= 0 which is compatible with the following consequence of formula (18):
Fact: On any Frobenius system the local systems C(α) are without non-trivial
global sections for all α 6= 0 if and only if the group of periods Λ ⊂ lR is not cyclic.
In our case Λ contains all the numbers log N p for p ∈
/ S since these are supposed to
be the lengths of the closed orbits and hence Λ has infinite rank.
4.1 In this subsection we explain why we expect H 1 (X, C)Θ=0 to be (uncountably)
1
infinite dimensional in contrast to the group H (X, C)Θ=0 which should be zero by
equation (30). Consider the exact sequence:
δ
0 −→ H 0 (X, C)/ΘH 0 (X, C) −→ H 1 (X, C) −→ H 1 (X, C)Θ=0 −→ 0 .
The group of periods Λ of X contains log N (Jk,S ) where Jk,S is the group of fractional
L
ZZ. Hence
ideals in k prime to S. Thus π1ab (X) has a subquotient isomorphic to p∈S
Q /
1
ab
H (X, C) = Hom(π1 (X), C) contains a C-subspace isomorphic to p∈S
/ C. On the
other hand strengthening equation (29) I expect H 0 (X, C) = C i.e. that the space
of T0 -leaves on X is sufficiently complicated so that any smooth complex valued
function on X factoring over it is constant. Since Λ ⊂ lR is dense and X connected
this would for example follow from proposition 3.13 if the fibres of a function f in
the notation of loc. cit. are connected. The latter condition is satisfied if the flow
on X is complete for example.
4.2 We now indicate how Ext-groups of motives should relate to dynamical cohomologies. Let MMoS resp. MMfoS be the full subcategories of the Q-linear
category of mixed motives over k which are integral in the sense of Scholl [Sch]
resp. which have good reduction at all maximal ideals p ∈
/ S of o. Then we have
MMfoS ⊂ MMoS and for all M 0 , M 00 in MMfoS :
(33)
ExtνMMf (M 00 , M 0 ) = ExtνMMo (M 00 , M 0 ) if ν = 0, 1 .
S
oS
We have to check this for ν = 1 only: Let 0 → M 0 → M → M 00 → 0 be an extension
in MMoS and choose an isomorphism
W
0
W
00
∼
.
Gr W
• M = Gr • M ⊕ Gr • M
By integrality we know that for every p ∈
/ S and any prime number l such that p /| l
we have
∼
for N = M 0 , M, M 00
Gr W
• Nl = Nl
as modules under the inertia group Ip . Hence Ml ∼
= Ml0 ⊕ Ml00 as Ip -modules. Since
Ip acts trivially on Ml0 , Ml00 by assumption it does so on Ml as well and (33) follows.
As pointed out in [D1] (7.21) and [D5] (2.2) the Bloch–Beilinson conjectures on
vanishing orders of motivic L-functions can be explained within the formalism of
loc. cit. as follows: The functor:
F : MMoS −→ sheaves of C-vector spaces on “spec oS ” with action by Θ
47
postulated there should be exact and induce isomorphisms:
∼
F : ExtνMMo (Q(0), M )C −→ ? H ν (“spec oS ”, F (M ))Θ∼0 .
S
Hence we get by (33):
Prediction: The functor
FR : MMfoS −→ locally free R-modules on X with action ψ opposite to φ
should be exact and the induced map:
FR : ExtνMMf (Q(0), M )lR −→ ExtνR−Mod (FR (Q(0)), FR (M )) = H ν (X, FR (M ))
oS
should give isomorphisms for ν = 0, 1:
(34)
∼
ν
FR : ExtνMMf (Q(0), M )lR −→ H (X, FR (M ))Θ∼0 .
oS
Let FlR (M ) be the local system corresponding to (FR (M ), ψ) according to proposition 3.6. It follows that the functor
FlR : MMfoS −→ local systems of lR-vector spaces on X
is exact as well and hence induces homomorphisms:
FlR : ExtνMMf (Q(0), M )lR −→ ExtνX (FlR (Q(0)), FlR (M )) = H ν (X, FlR (M ))
oS
since FlR (Q(0)) = lR(0). Using (31) for n = 0 we get a commutative diagram:
∼
H 0 (X, FlR (M ))
−→ H 0 (X,FR (M ))Θ=0
x
FlR
FR
∼
y
HomMMfo (Q(0), M )lR −→ H 0 (X, FR (M ))Θ∼0
S
where the lower arrow is an isomorphism by (34) and (28). Hence all arrows must
be isomorphisms
∼
H 0 (X, FxlR (M ))
FlR o
−→
FR
∼
H 0 (X, FR (M ))Θ=0
HomMMfo (Q(0), M )lR −→ H 0 (X, FR (M ))Θ∼0 .
S
Applying this to M = M 00 ⊗ M
(35)
0∗
for M 0 , M 00 in MMfoS we obtain an isomorphism:
∼
FlR : Hom(M 0 , M 00 )lR −→ HomX (FlR (M 0 ), FlR (M 00 ))
if FlR commutes with ⊗ and duals which we assume by analogy with the function
field case. This insight can be improved since in contrast to FR the functor FlR has
48
a natural Q-structure: One has to show that if in the construction of 2.8 we replace
RX by lRX etc. one obtains the functor FlR . Replacing RX by QX etc. a functor
FQ :
MMfoS
−→ LSX =
(
category of finite rank local
systems of Q-vector spaces on X
)
is obtained. The isomorphism:
FlR (M ) = FQ (M ) ⊗Q lR functorially in M
is then immediate as FlR (M ) has finite rank for all M .
Remark: The existence of FQ implies in particular that there is a local system on
X of Q-vector spaces Q(1) := FQ (Q(1)) with Q(1) ⊗Q lR = lR(1). Using 3.11 a) it
follows that Λ ⊂ log Q∗+ . In particular for the dynamical system attached to spec ZZ
we must have Λ = log Q∗+ .
Exactness of FlR and the isomorphism (35) lead to the following conclusion:
4.3 Consequence: The ⊗-functor of neutral Q-linear Tannakian categories
FQ : MMfoS → LSX is exact and fully faithful.
Fixing a point x0 of X we get the fibre functor “stalk in x0 ” on LSX and composing
with FQ gives a fibre functor on MMfoS . Define the affine group schemes π1alg (X, x0 )
and MG resp. G as Tannaka duals of LSX and MMfoS resp. MfoS with respect
to these fibre functors where MfoS is the full subcategory of semisimple motives in
MMfoS . Then FQ induces a morphism
f : π1alg (X, x0 ) −→ MG
which yields the first of the canonical homomorphisms:
(36)
π1 (X, x0 ) −→ MG(Q) −→ G(Q) −→ Gal(kS /k)
where kS is the maximal extension of k which is unramified at all the finite primes
of k not in S.
The composition π1 (X, x0 ) → Gal(kS /k) is also obtained from our basic assumption
that the association of Frobenius systems to schemes in 2.8 is functorial.
If every sub-local system of a local system of the form FQ (M ) for M in MMfoS is
isomorphic to FQ (M 0 ) for a submotive M 0 of M then using [DeM] Prop. 2.21 and
4.3 we get the following:
4.4 Partly motivated prediction: The morphism:
f : π1alg (X, x0 ) −→ MG
is faithfully flat.
49
Although we have no idea how to deduce the above condition on sub-local systems
from our formalism it seems reasonable to us by analogy with the theory of “geometric” l-adic representations of global Galois groups i.e. those that are everywhere
de Rham and almost everywhere unramified respectively crystalline. This class too
is closed under subobjects.
ss
In a similar vein one may expect that FQ maps MfoS to the category LSX
of semisimple local systems in LSX . Compare proposition 4.7 below for an analogous
assertion. Then using [DeM] Remark 2.29 and 4.3 we would get:
4.5 Partly motivated prediction: The morphism
f ss : π1alg (X, x0 )ss −→ G
ss
is faithfully flat. Here π1alg (X, x0 )ss is the Tannaka
induced by FQ : MfoS → LSX
ss
dual of LSX
and hence the maximal pro-reductive quotient of π1alg (X, x0 ).
4.6 For a finite extension E/Q consider the functor of E-linear categories
FQ (E) : MMfoS (E) −→ LSX (E)
induced by FQ c.f. [De2] (2.1). It is exact and fully faithful by 4.3. Hence it induces
an injective homomorphism of abelian groups:
fQ (E) : P (E) = Pic(MMfoS (E)) ,→ Hom(π1 (X, x0 ), E ∗ )
where P (E) is the group under ⊗ of isomorphism classes of E-rank one motives in
MMfoS (E). To every algebraic Hecke character χ of k with values in E which is
unramified at the prime ideals p ∈
/ S one can associate its motive M (χ) – at least for
absolute Hodge cycles – and one obtains a homomorphism from the group of these
Hecke characters to P (E) which conjecturally [De2] § 8 is an isomorphism. After the
choice of an embedding E ⊂ C these algebraic Hecke characters can be identified
with a subgroup of Homcont (IA∗k /k ∗ IA∗k,S , C∗ ) which we denote by an index E. Here,
Q
∗
we have set IA∗k,S = p∈S
/ okp . Thus fQ (E) induces an injection:
(37)
In particular
(38)
f : Homcont (IA∗k /k ∗ IA∗k,S , C∗ )E ,→ Hom(π1 (X, x0 ), C∗ ) .
f (k kα ) = ρα
for any α ∈ ZZ
since k kn corresponds to the class of the motive Q(n) in P (Q) which is mapped to
QX (n) by FQ . Now use proposition 3.8. Note that we can increase the domain of
definition of f in Homcont (IA∗k /k ∗ IA∗k,S , C∗ ) as a map to Hom(π1 (X, x0 ), C∗ ) by setting
f (k kα ) = ρα for all α ∈ C.
4.7 In this subsection we explain why the infinite dimensional complex cohomologies H ∗ (X0 /lL) with lR-action of varieties X0 /IFq constructed in [D1] § 4 cannot be
obtained in general simply as leafwise cohomologies within a framework of Frobenius
50
systems similar to the above which was based on 2.5 and 2.8. The obstruction comes
from the phenomenon of non-liftability. Possibly H ν (X/lL) may be obtained from
Frobenius systems attached to local liftings similar to the construction of crystalline
cohomology via local liftings.
Let MIFq be the Tannakian category of motives over IFq c.f. [M] and for simplicity
assume the Tate conjecture over finite fields which is required for most of the results
of loc. cit. on MIFq . It is then immediate that the theory H ∗ ( /lL) factorizes over
MIFq . As noted in 2.7 the Frobenius system Sq1 = (lR/ log qZZ, τ, 0) corresponds naturally to spec IFq . Assume that we have a full Tannakian subcategory M of MIFq
such that the following condition holds:
(A) There is a ⊗-functor
FR : M −→
(
locally free R-modules on Sq1 of finite
rank with action opposite to τ
)
with FR (Q(0)) = R and rank M = rankR FR (M ) such that
(39)
∗
H ∗ (M/lL) ∼
= H (Sq1 , FR (M ))C
as lR≥0 -modules
∗
0
0
for all M in M where H is a suitable subquotient of H ∗ and we have H = Hsm
.
Note that the dynamical formalism if it were applicable to all smooth projective
varieties over IFq would imply (A) for M = MIFq .
As we will now see condition (A) implies that M has a fibre functor to the category
of finite dimensional lR-vector spaces. In particular M cannot equal MIFq ! Let
FlR : M −→ local systems of lR-vector spaces on Sq1 of finite rank
be the ⊗-functor corresponding to FR via proposition 3.6. Then we have:
Fr =id
Hom(Q(0), M )C ∼
= Ml q ⊗Ql ,ι C for l /| q, ι : Ql ,→ C by [M] Cor. 1.16
∼
= H 0 (M/lL)Θq =0 by [D1] (2.4)
Θ=0
∼
by assumption (A)
= H 0 (Sq1 , FR (M ))C
∼
= H 0 (Sq1 , FlR (M ))C by proposition 3.7
∼
= HomS 1 (FlR (Q(0)), FlR (M ))C since FlR (Q(0)) = lR(0) .
q
As a tensor functor FlR commutes with ⊗ and duals and we finally get that FlR
induces an isomorphism:
(40)
∼
FlR : Hom(M1 , M2 )lR −→ HomSq1 (FlR (M1 ), FlR (M2 ))
for all M1 , M2 in M. In particular for any point x0 ∈ Sq1 we get a fibre functor FlR,x0
on M with values in finite dimensional lR-vector spaces.
In fact the dynamical formalism would give an even stronger statement because by
similar arguments as above we would get:
51
(B) The ⊗-functor FlR has a Q-structure i.e. there is a ⊗-functor
FQ : M −→ local systems of Q-vector spaces on Sq1 of finite rank
such that FlR ∼
= FQ ⊗ lR.
Granting this (40) implies that FQ induces an isomorphism:
(41)
∼
FQ : Hom(M1 , M2 ) −→ HomSq1 (FQ (M1 ), FQ (M2 ))
for all M1 , M2 in M, so that in particular FQ,x0 is a fibre functor of M over Q.
According to the next result which was obtained in discussions with A.J. Scholl there
is a uniquely determined largest category M – the category of ordinary motives over
IFq – for which assertions (A) and (B) are true.
Call a Weil q-number λ ordinary if kλkv ∈ q ZZ for all valuations v of Q(λ). Only the
real places and the v | p need to be considered here. A simple motive M in MIFq
is called ordinary if End M = Q(πM ) where πM is the Frobenius endomorphism of
M . By [M] Th. 2.16 this is equivalent to πM being an ordinary Weil q-number.
An arbitrary motive in MIFq is called ordinary if it is the sum of simple ordinary
motives. Let Mord
IFq denote the full subcategory of MIFq of ordinary motives. It is a
Tannakian subcategory.
Proposition: There is a fully faithful ⊗-functor:
1
FQ : Mord
IFq −→ local systems of Q-vector spaces on Sq of finite rank.
Its essential image consists of the semisimple local systems for which the eigenvalues
of monodromy are ordinary Weil q-numbers. The category Mord
IFq is neutral over Q
and it contains every full neutral Tannakian subcategory of MIFq . Setting FlR =
FQ ⊗Q lR and defining FR as corresponding to FlR via proposition 3.6 assertions (A)
∗
∗
and (B) hold for M = Mord
IFq with H = Hsm in (A) i.e.
H ∗ (M/lL) ∼
= H ∗ (Sq1 , FR (M ))C,sm as lR-modules .
Idea of proof: One first checks that Mord
IFq is generated as a Tannakian category
1
by the motives H (A) for ordinary abelian varieties A/IFq . Then one extends the
isogeny version of [De1] Th. 7 c.f. (E) on p. 242 to ordinary motives. Maximality
of Mord
IFq among neutral subcategories of MIFq follows e.g. from [M] Lemma 3.15.
4.8 Fix a point x0 on the connected Frobenius system X = XS for spec oS and
define NS by the exact sequence:
l
1 −→ NS −→ π1 (XS , x0 ) −→ ΛS −→ 0 .
The group of periods ΛS ⊂ lR contains log N (Jk,S ). If νoS (α) ≥ 1 for some α ∈
1
1
C∗ then H (X, C)Θ∼α 6= 0 by (30) and hence H (X, C)Θ=α 6= 0. If this implies
H 1 (X, C)Θ=α 6= 0 which we assume then by (32) we have H 1 (X, C(α)) 6= 0 and by
52
Corollary 3.15 therefore HomΛS (ΓS , C(α)) 6= 0 where ΓS = NSab / torsion. Given
abelian groups Λ ⊂ lR and Γ with an action of Λ on Γ we define the spectrum of
(Γ, Λ) to be the set of α in C such that HomΛ (Γ, C(α)) 6= 0 i.e. such that there
exists a nonzero homomorphism ϕ : Γ → C with ϕλ = eαλ ϕ for all λ in Λ. By
the above discussion it might be worthwile to look for a pair (Γ, Λ) with Γ and Λ
countable, Λ ⊃ log N (Jk,S ) whose spectrum consists of all α with νoS (α) ≥ 1. Less
ambitiously one might try to find a pair whose spectrum is at most countable and
contains any nonzero α with νoS (α) ≥ 1. Note that even for countable Γ, Λ the
spectrum may equal C. Where to look for such a pair (Γ, Λ) related to oS ? The
work of Connes in [C2] seems to be relevant to this question. Another suggestion
comes from Iwasawa theory and an old idea of Ihara. From now on let k = Q
for simplicity and assume that in fact ΛS = log Q∗+,S where Q∗+,S = N (JQ,S ) is
the group of nonzero rational numbers with p-valuation zero for all p ∈ S. Set
Q
ẐZS = p∈S ZZp and consider the canonical injection Q∗+,S ,→ ẐZ∗S . Let QS be the
maximal extension of Q which is unramified at all the finite primes not in S and
let Q∞,S be the extension generated by all pn - th roots of unity for p in S, n ≥ 1.
Set GS = Gal(QS /Q) and HS = Gal(QS /Q∞,S ). The canonical homomorphism
π1 (XS , x0 ) → GS from (36) should make the following diagram commute:
exp ◦ l
π1 (XS , x0 ) −→→ Q∗+,S
i
y
ydiag
κ
GS −→→ ẐZ∗S
(42)
where κ is the cyclotomic character. Note in this regard that exp ◦ l maps the
∗
/ S. Correspondingly
conjugacy class of the closed orbit γp =
b p to p ∈ Q+,S for p ∈
κ maps the conjugacy class of a Frobenius element Frp to p = (p, . . . , p) ∈ ZbZ∗S .
Moreover the diagram is functorial in S. We would thus get a commutative diagram:
1 −→
NS
1 −→
y
HS
exp ◦ l
−→ π1 (XS , x0 ) −→ Q∗+,S −→ 1
−→
iy
GS
κ
y
−→ ẐZ∗S −→ 1 .
If I understood Ihara correctly he expects some abstract non-profinite subgroups
of Galois groups to play an important role in the understanding of the Riemann
zeta function analogous to his ideas in the function field case [I]. One might thus
be tempted to imagine that for S 6= ∅ the pair (ΓS , ΛS ) for spec ZZS can be realized
modulo torsion by the abelianization of an abstract subgroup of HS which is invariant
under conjugation inside GS with the elements of Q∗+,S ⊂ ẐZ∗S . Is i actually injective?
Its image is dense by 4.4.
4.9 The most important stept to realize the dynamical approach to L-functions is of
course to find a suitable Frobenius system for spec ZZ. We have two comments in this
direction. Consider quite generally a free abelian group Λ ⊂ lR with basis λ1 , λ2 , . . .
which operates by diffeomorphism φλ on a manifold M . Then X = M ×Λ lR is a
53
Frobenius system if and only if
0) the operation by Λ on M × lR is free and properly discontinuous.
The length function induces a bijection:
l
∼
{closed orbits of the lR-action on X} −→ {λ1 , λ2 , . . .}
if and only if the following conditions on the Λ-action on M hold:
1) For every λi there is a unique Λ-orbit which is pointwise fixed by φλi .
2) If φN λi (x) = x for some N ≥ 1, x ∈ M then φλi (x) = x.
3) For every λ not of the form N λi for some N ∈ ZZ the diffeomorphism φλ has no
fixed points on M .
In [Si] W. Singhof constructed for any free abelian subgroup Λ ⊂ lR with a given basis
λ1 , λ2 , . . . a connected non compact surface M and a Λ-operation by diffeomorphisms
such that 0)–3) are satisfied.
In particular for Λ = log Q∗+,S we do indeed get a three dimensional Frobenius system
XS = MS ×Λ lR whose closed orbits for the lR-action are in bijection with the primes
not in S and such that l(γp ) = log p if γp =
b p. However as Singhof’s construction
does not depend on the arithmetic nature of the λ1 , λ2 , . . . it is not very probable
unfortunately that this particular Frobenius system will be very useful for the study
of spec ZZS .
Another approach to finding XS in the form XS = MS ×Q∗+,S lR∗+ where we have
exponentiated the action is suggested by K-theory. Combining [D1] (7.28) and 2.5
we see that we should have a natural map
i
i
Im (K2j−i (spec ZZS )Q −→ HM
(spec Q, Q(j))) −→ H (XS , R(j))Θ=0
where
i
HM
(spec Q, Q(j)) = {u ∈ K2j−i (Q)Q | ψ a (u) = aj u for all integers a ≥ 1}
and ψ a is the a-th Adams operator. On the other hand
i
i
H (XS , R(j))Θ=0 = {v ∈ H (XS , R) | a∗ (v) = aj v for all a ∈ lR∗+ }
a
. Thus one might imagine that the action of Q∗+,S
if we let a ∈ lR∗+ act via a∗ = φlog
∗
on MS is given by a lifting of Adams operators to maps of spaces: For integers a, b ≥
1 prime to S the operator ψ b should be invertible on MS and ab−1 ∈ Q∗+,S should
act on MS by ψ a (ψ b )−1 . Clearly this idea is too simple: First Adams operators will
not be invertible on the level of spaces. But note that the “right” Frobenius system
for spec ZZS is probably not complete as we have assumed and the two phenomena
may be related. Secondly the spaces on which Adams operators can be realized are
certainly not Banach manifolds. However one can make sense of the cohomologies
H 1 (M ×Λ lR, C(α)) for α ∈ C even if M is only e.g. a spectrum with an action by Λ.
In the case above ideally they would be nonzero if and only if νZZS (α) ≥ 1. Another
instance where ψ a and a∗ show a similar behaviour is in the following situation. For
a variety X0 /IFp the operator Θ acts on H ∗ (X0 /lL) with eigenvalues
54
Log p λ + 2πiν
, ν ∈ ZZ where λ runs through the eigenvalues of Frobenius on l-adic
log p
cohomology H ∗ (X0 ⊗ IFp , Ql ) c.f. [D1] § 4. Since a ∈ lR∗+ acts by a∗ = aΘ on
H ∗ (X0 /lL) the eigenvalues of p∗ on this cohomology are the numbers:
2πiν
pLog p λ+ log p = λ .
More precisely one easily sees that p∗ equals the action induced by Frobenius on
H ∗ (X0 /lL). Thus “in characteristic p the prime p ∈ lR∗+ acts on cohomology via
Frobenius”. Similarly in characteristic p the action of the Adams operator ψ p and
of Frobenius agree on algebraic K-theory in many instances.
55
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Christopher Deninger
Mathematisches Institut
Einsteinstr. 62
48149 Münster
Germany
e-mail: [email protected]
57
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