Non-abelian unipotent periods Monodromy of iterated integrals

63 Pages·2012·0.34 MB·English

Non-abelian unipotent periods
Monodromy of iterated integrals
Zdzisl aw Wojtkowiak
0. Introduction.
x
0.1. Let X be a smooth, algebraic variety de(cid:12)ned over a number (cid:12)eld k. Let (cid:27) : k , C
!
be an inclusion. We set X = X C: Let X(C) be the set of C-points of X with its
C C
(cid:2)k
complex topology. There is the canonical isomorphism
p : Hn(X(C)) C (cid:25) Hn (X) C
comp B (cid:10) (cid:0)! DR (cid:10)
k
between Betti (singular) cohomology and algebraic De Rham cohomology. The period
matrix(p )isde(cid:12)nedbyequations! = p (cid:27) where ! and (cid:27) arebasesofHn (X)
ij i ji j f ig f ig DR
and Hn(X(C)) or p = ! where (cid:27)P(cid:3) is the dual base of H (X(C)):
B ji (cid:27)(cid:3) i f jg n
j
Let us assume thatR X is an abelian variety. Let G be the largest subgroup of
GL(H1(X(C)) Q) G which (cid:12)xes all tensors cl (Z), where Z is an algebraic cycle on
B (cid:10) (cid:2) m B
some Xn (see [D2]). Let P be the functor of k-algebras, such that any element of P(A) is
an isomorphism p : H1(X(C)) A H1 (X) A mapping cl (Z) 1 to cl (Z) 1 for
B (cid:10) ! DR (cid:10) B (cid:10) DR (cid:10)
any algebraic cycle Z on any Xn. The isomorphism p belongs to P(C); the functor
comp
P is represented by an algebraic variety over k, which is a G -torsor under the natural
k
action. It is a subtorsor of the GL(H1(X(C) k)-torsor Iso(H1(X(C)) k;H1 (X)):
B (cid:10) B (cid:10) DR
LetT beasmallestsubtorsorde(cid:12)ned overk ofthetorsorIso(H1(X(C)) k;H1 (X));
B (cid:10) DR
which contains p as a C-point and let be the corresponding subgroup (de(cid:12)ned over
comp
G
k) of GL(H1(X(C)) k): Let Z(p ) be the Zariski closure of p in Iso(H1(X(C)
B (cid:10) comp comp B (cid:10)
k;H1 (X)) i.e. the smallest Zariski closed subset de(cid:12)ned over k, which contains p as
DR comp
a C-point.
Then we have
Z(p ) T P
comp
(cid:26) (cid:26)
1
and
G:
G (cid:26)
In order to calculate Z(p ) and to show that Z(p ) = P one need to show that
comp comp
certain number are transcendental. On the other hand to calculate T and seems to be
G
an easier task. The requirement that T is a subtorsor of Iso(H1(X(C)) k;H1 (X)) is
B (cid:10) DR
very strong and usually relatively weak informations about numbers p are necessary
ij
f g
to calculate T and . We give an obvious example. If X = P1 then in order to show that
Q
G
Z(p ) G we must know that 2(cid:25)i is transcendental. But already the fact that 2(cid:25)i
comp m
(cid:25)
is not a kth-root of a rational number for any k N implies that T G and = G :
m m
2 (cid:25) G
In this note we shall discuss periods for fundamental groups. We shall concentrate on
analogues of T and for fundamental groups. On the other hand we have no analogue of
G
P and G.
The plan of the paper.
0. Introduction.
1. Torsors.
2. Torsors associated to non-abelian unipotent periods.
3. Canonical connection with logarithmic singularities.
4. The Gauss-Manin connection associated with the morphism X(cid:1)[1] X@(cid:1)[1] of cosim-
!
plicial schemes.
5. Torsors associated to the canonical unipotent connection with logarithmic singulari-
ties.
6. Partial informations about (P1(C) 0;1; ):
DR
G nf 1g
7. Homotopy relative tangential base points on P1(C) a ;:::;a :
1 n+1
nf g
8. Generators of (cid:25) (P1(C) a ;:::;a ;x):
1 1 n+1
nf g
9. Monodromy of iterated integrals on P1(C) a ;:::;a :
1 n+1
nf g
10. Calculations.
11. Con(cid:12)guration spaces.
12. The Drinfeld-Ihara Z=5-cycle relation.
2
13. Functional equations of iterated integrals.
14. Subgroups of Aut((cid:25)2).
A.1. Malcev completion.
The dependence of sections
0 3
1 7
(cid:12) (cid:12)
(cid:12) (cid:12)
4 8
(cid:12) (cid:12)
2(cid:12) (cid:12)
(cid:12)
(cid:12) 9
(cid:12)
14 5 (cid:12)
(cid:12)
(cid:12) 10
(cid:12)
(cid:12)
(cid:12)
6(cid:12) 11
(cid:12) (cid:12)
(cid:12) (cid:12)
12
(cid:12)
(cid:12)
0.2. Below we shall brie(cid:13)y discuss the contents of the paper. Let us assume that X is
a smooth, quasi-projective, algebraic variety de(cid:12)ned over a number (cid:12)eld k. Let x be a k-
point of X. In [W1] we de(cid:12)ned a(cid:14)ne, connected, pro-unipotent group schemes over k and
Q respectively; (cid:25)DR(X;x) | the algebraic De Rham fundamental group and (cid:25)B(X(C);x)
1 1
| the Betti fundamental group. We have also the inclusion (of Q-points into C-points)
(cid:8) : (cid:25)B(X(C);x)(Q) (cid:25)DR(X;x)(C)
x 1 ! 1
such that the induced homomorphism on C-points
’ : (cid:25)B(X(C);x)(C) (cid:25)DR(X;x)(C)
x 1 ! 1
is an isomorphism.
The a(cid:14)ne, pro-algebraic scheme over k
Iso := Iso((cid:25)B(X(C);x) k;(cid:25)DR(X;x))
1 (cid:2) 1
3
is an Aut((cid:25)DR(X(C);x))-torsor. Let Z(’ ) be the Zariski closure of ’ in Iso. Let T(’ )
1 x x x
be the smallest subtorsor of Iso, de(cid:12)ned over k which contains ’ as a C-point. Let
x
G (’ ) Aut((cid:25)DR(X;x)) be the corresponding subgroup. One can hope that
DR x (cid:26) 1
(0:1:) Z(’ ) = T(’ )
x x
We shall denote by (X) the image of G (’ ) in Out((cid:25)DR(X;x)).
GDR DR x 1
The calculation of the homomorphism ’ is equivalent to the calculation of the mon-
x
odromy of all iterated integrals on X. We shall see that the monodromy representa-
(cid:22)
tion of iterated integrals have a lot of properties similar to the action of Gal(k=k) on
fundamental groups. For example if S is a loop on a curve around a missing point,
then \(cid:27)(S) S(cid:31)((cid:27))" in the l-adic case and \(cid:18)(S) S(cid:0)2(cid:25)i" for iterated integrals. Let
(cid:24) (cid:24)
(cid:22) (cid:22)
’ : G := Gal(k=k) Out((cid:25) (X k;x) ) be the natural homomorphism. Very opti-
k 1 (l)
! (cid:2)
k
mistically we can state the following conjecture:
(0:2) Lie(’(G )) k Lie( (X)) Q :
k DR l
(cid:10) (cid:25) G (cid:10)
We also point out that (0.1) and (0.2) will imply that values of the Riemann zeta function
at odd integers are all transcendental over Q( 2(cid:25)i).
(cid:0)
0.3. The calculations of non-abelian unipotent periods are in fact the calculations of
monodromy of iterated integrals. This causes that the paper contains in fact two dif-
ferent papers. In one part (sections 3,7,8,9,10,11,12,13) we are studying monodromy of
iterated integrals. In Section 3 are established some general properties of the canonical
unipotent connection. In Section 7 we present a \naive" approach to the Deligne tangen-
tial base point, which is su(cid:14)cient for our applications. In Sections 9 and 10 we describe
the monodromy of iterated integrals on P1(C) a ;:::;a and in more details on
1 n+1
n f g
P1(C) 0;1; . In Sections 11{13 we study monodromy of iterated integrals on con(cid:12)gu-
nf 1g
ration spaces. We give a proof of Drinfeld-Ihara Z=5-cycle relation which is di(cid:11)erent from
the proof in [Dr]. The proof should be (is) analogous to the proof in [I2]. We point out
that the main point in our proof is the functoriality property of the universal unipotent
4
connection. This property can be shortly written as f ! = f(cid:3)! and it is also fundamental
(cid:3)
in our results on functional equations of polylogarithms and iterated integrals (see [W4]).
In \the second paper" contained in this paper we discuss torsors and corresponding
groups associated to non-abelian unipotent periods. In Section 1 we give some general
results and de(cid:12)nitions. In Section 2 we de(cid:12)ne a torsor and a corresponding group asso-
ciated to non-abelian unipotent periods and we state some conjectures related to Galois
representations on fundamental groups. In Section 5 we de(cid:12)ne a torsor associated to the
monodromy of iterated integrals. Using results from Section 4 we show that this torsor
and the corresponding group coincides with the ones from Section 2. In Section 6 we cal-
culate some part of the group (X) for X = P1(C) 0;1; . This part corresponds to
DR
G nf 1g
the Galois representation on (cid:25)1(X;(cid:0)0!1) [(cid:25)0;(cid:25)0] where (cid:25)0 := [(cid:25)1(X;(cid:0)0!1);(cid:25)1(X;(cid:0)0!1)] (see [I3]).
About this part of (X), let us call(cid:14)it G, we have the following result.
DR
G
The group G contains a group H = f f (X) = t X;f (Y) = t Y t C(cid:3) if and only
t t t
f j (cid:1) (cid:1) j 2 g
if all numbers (cid:16)(2k+1) are irrational. This leads to a de(cid:12)nition of a new group associated
to unipotent periods, which should be relatively easily calculated. This new group is not
considered in this paper, see however Corollary 6.4 and Theorem 6.7 iii).
Acknowledgment. We would like to thank Professor Deligne, who once showed the
one-form considered in Section 3 in the case of C 0;1 . Thanks are due to Professor
n f g
Y. Ihara for showing us his proof of 5-cycle relation, which help us to (cid:12)nd an analogous
one for unipotent periods. We would like to express our thanks to Professor Hubbuck for
his invitation to Aberdeen, where Section 12 was written and where in May 1993 we had
a possibility to give seminar talks on 5-cycle relation for unipotent periods.
We would like to thank very much Professor Y. Ihara for his invitation to Kyoto.
We would like to thank Professors Oda, Matsumoto, Tamagawa for useful discussions and
comments during my seminar talks.
Finally thanks are due to Professor L. Lewin, who once invited us to write a chapter
in the book on polylogarithms and suggested to include also results about monodromy of
iterated integrals. This encourage us very much to continue to work on this subject (see
preprints [W2] and [W3] which some parts are included in the present paper).
5
1. Torsors.
x
Let G and G be two groups. We say that a set T is a (G ;G )-bitorsor if T is equipped
1 2 1 2
with a free, transitive, left action of G and with a free, transitive right action of G and
1 2
if the actions of G and G commute.
1 2
We say that a subset S T is a subtorsor of T if there exist subgroups H G and
1 1
(cid:26) (cid:26)
H G such that S is a (H ;H )-bitorsor under the natural actions of H and H .
2 2 1 2 1 2
(cid:26)
We say that a subset S T is a left (resp. right) subtorsor of T if there is a subgroup
(cid:26)
H G (resp. H G ) such that the natural action of H (resp. H ) on S is free and
1 1 2 2 1 2
(cid:26) (cid:26)
transitive.
Main example. Let G and G be two groups. Assume that G and G are isomorphic.
1 2 1 2
Then the set of isomorphisms from G to G , which we denote by Iso(G ;G ); is an
1 2 1 2
(Aut(G );Aut(G ))-bitorsor.
1 2
For any non-empty subset S Iso(G ;G ), the intersection of all subtorsors (resp.
1 2
(cid:26)
right subtorsors, resp. left subtorsors) of Iso(G ;G ), which contain S, isa subtorsor (resp.
1 2
right subtorsor, resp. left subtorsor) of Iso(G ;G ); which we denote by T(S)(resp. T (S);
1 2 r
resp. T (S)):
l
1.1. Unipotent, a(cid:14)ne, algebraic groups and torsors.
Let k be a (cid:12)eld of characteristic zero. We say that X is an algebraic variety (de-
(cid:12)ned) over k if X is an algebraic scheme over Speck: If A is a k-algebra, we set X :=
A
X SpecA: The set of A-points of X we denote by X(A): We say that G is an a(cid:14)ne,
(cid:2)
Speck
algebraic group (de(cid:12)ned) over k, if G is an a(cid:14)ne, algebraic group scheme over Speck:
1.1.1. Let G be an a(cid:14)ne, unipotent, connected, algebraic group over k. Then there is an
a(cid:14)ne, algebraic group Aut(G) over k such that for any k-algebra A we have
Aut(G)(A) = Aut(G ):
A
6
Let G and G be two connected, a(cid:14)ne, unipotent, algebraic groups over k. Then
1 2
there is a smooth, a(cid:14)ne, algebraic variety Iso(G ;G ) over k, such that for any k-algebra
1 2
A we have
Iso(G ;G )(A) := Iso(G ;G ):
1 2 1A 2A
Proof of 1.1.1. Let be a Lie algebra of G. The exponential map exp: G is
G G !
an isomorphism of a(cid:14)ne, algebraic groups if we equip with a group law given by the
G
Baker-Campbell-Hausdor(cid:11) formula. The automorphisms of the group coincides with the
G
automorphisms of the Lie algebra . One can easily give an ideal de(cid:12)ning Aut ( ) in
Lie
G G
k[GL( )]: In the similar way one constructs Iso(G ;G ):
1 2
G
1.1.2. In [S] page 149 there is a de(cid:12)nition of a (left) G-torsor (a principal homogeneous
space of G) if G is a linear, algebraic group over k. This de(cid:12)nition extends immediately
for bitorsors of algebraic groups.
We say that an a(cid:14)ne, algebraic variety T over k is a (G ;G )-bitorsor, if there are
1 2
morphisms G T T and T G T over Speck; which de(cid:12)ne free, transitive actions
1 2
(cid:2) ! (cid:2) !
of G and G on T and if these actions commute.
1 2
Observe that if T has a k-point then G and G are isomorphic.
1 2
The de(cid:12)nitions ofasubtorsor, arightsubtorsor and aleftsubtorsor we lefttoareader,
as well as a proof of the following lemma.
Lemma 1.1.2. Let T be an (H ;H )-subtorsor of T and let T be an (H0;H0)-subtorsor
1 1 2 2 1 2
of T. Assume that T T = : Then the intersection T T is an (H H ;H0 H0)-
1 \ 2 6 ; 1 \ 2 1 \ 2 1 \ 2
subtorsor of T. The similar statements hold for right and left subtorsors of T.
1.1.3. Main example. Let G and G be two connected, unipotent, a(cid:14)ne, algebraic
1 2
groups over k. Assume that there is an isomorphism G G : Then the algebraic
1k(cid:22) ! 2k(cid:22)
variety Iso(G ;G ) is an (Aut(G );Aut(G ))-bitorsor, if we equip Iso(G ;G ) with the
1 2 1 2 1 2
obvious actions of Aut(G ) and Aut(G ).
1 2
Let k C be a sub(cid:12)eld of C. Let (cid:2) : G (C) G (C) be an isomorphism. Then
1 2
(cid:26) !
(cid:2) is a C-point of Iso(G ;G ): We denote by Z((cid:2)) the Zariski closure of (cid:2) in Iso(G ;G )
1 2 1 2
7
i.e. the smallest algebraic subset of Iso(G ;G ) (de(cid:12)ned) over k, which contains (cid:2) as a
1 2
C-point.
The connected, unipotent, a(cid:14)ne, algebraic group G is isomorphic as an algebraic
i
variety over k to the a(cid:14)ne space Am; hence (cid:2) can be view as a C-point ((cid:2) ) of
k ij 1(cid:20)i;j(cid:20)n
Am2: Let k((cid:2)) be a sub(cid:12)eld of C generated over k by all (cid:2) :
k i;j
Lemma 1.1.3. The (cid:12)eld k((cid:2)) does not depend on the choice of isomorphisms G Am
i (cid:25) k
and the transcendental degree of the (cid:12)eld k((cid:2)) over k is equal to the dimension of Z((cid:2)):
Proof. This follows (is) Lemma 1.7 in [D2].
De(cid:12)nition-Proposition 1.1.4. Let T((cid:2)) (resp. T ((cid:2)); resp. T ((cid:2))) be the intersection
r l
of all subtorsors T (resp. right-subtorsors T , resp. left subtorsors T ) de(cid:12)ned over k of
r l
Iso(G ;G ); which contain (cid:2) as a C-point. Then T((cid:2)) (resp. T ((cid:2)) resp. T ((cid:2)) ) is a
1 2 r l
(Gbi((cid:2));Gbi((cid:2)))-subtorsor (resp. right G ((cid:2))-subtorsor, resp. left G ((cid:2))-subtorsor) of
r l r l
Iso(G ;G ) for some Gbi((cid:2)) Aut(G ) and Gbi((cid:2)) Aut(G ) (resp. G ((cid:2)) Aut(G );
1 2 r (cid:26) 2 l (cid:26) 1 r (cid:26) 2
resp. G ((cid:2)) Aut(G )):
l 1
(cid:26)
Proof. The intersection of a family of algebraic varieties coincides with an intersection of
a (cid:12)nite number of them. Hence it follows from Lemma 1.1.2 that T((cid:2));T ((cid:2)) and T ((cid:2))
r l
exist and are unique. The groups are also unique because they are intersections of the
corresponding subgroups of Aut(G ):
i
If T((cid:2)) has a k-point f, then Gbi((cid:2)) = f(cid:0)1 Gbi((cid:2)) f:If T ((cid:2)) (resp. T ((cid:18))) has a k-
l (cid:14) r (cid:14) l r
point, then T((cid:2)) = T ((cid:2)) and Gbi((cid:2)) = G ((cid:2)) (resp. T((cid:2)) = T ((cid:2))and Gbi((cid:2)) = G ((cid:2)):
l l l r r r
Lemma 1.1.5. Let G be a unipotent, connected, a(cid:14)ne, algebraic group over k. Then
Aut(G) is an extension of an algebraic subgroup of GL(Gab) by a connected, unipotent,
a(cid:14)ne, algebraic group. Hence the groups Gbi((cid:2));Gbi((cid:2));G ((cid:2));G ((cid:2)) are extensions of
r l r l
algebraic subgroups of GL(Gab) by connected, unipotent, a(cid:14)ne, algebraic groups.
Proof. Let be the Lie algebra of G and let ( (i)) be a (cid:12)ltration of by the lower central
i
G G G
series. Any automorphism of the Lie algebra preserves the (cid:12)ltration and the induced
G
automorphism of (i) (i+1) is determined by the automorphism of ab = (1) (2): Hence
G G G G G
(cid:14) (cid:14)
8
Aut ( ) is an extension of a closed subgroup of GL( ab) be a unipotent group. The
Lie
G G
lemma follows from the identi(cid:12)cation of Aut(G) with Aut ( ) by the exponential map
Lie
G
(cid:25)
exp : G:
G !
(cid:22)
Lemma 1.1.6. AssumethatG ((cid:2))isanextensionofG (orGsuchthatH1(Gal(k k);G)=
r m
0) by a connected, unipotent, a(cid:14)ne, algebraic group N. Then T ((cid:2)) has a k-point(cid:14).
r
(cid:22)
Proof. It follows from [S] Proposition 4.1 that H1(Gal(k k);N) = 0. It follow from
(cid:22)
[S] Proposition 2.2 and the assumption of the lemma tha(cid:14)t H1(Gal(k k);G ((cid:2))) = 0.
r
Proposition 1.1 from [S] implies that T ((cid:2))(k) = : (cid:14)
r
6 ;
Let (cid:0)i(G) be a (cid:12)ltration of a group G by the lower central series. Let us set G(i) :=
G (cid:0)i+1G: The isomorphism (cid:2) : G (C) G (C) induces isomorphisms (cid:2)(i) : G(i)(C)
1 ! 2 1 !
G(cid:14)(i)(C). Let k < i: The projections G(i) G(k) for j = 1;2 induce
2 j ! j
(cid:26)i : Iso(G(i);G(i)) Iso(G(k);G(k));
k 1 2 ! 1 2
(cid:26)(j)i : Aut(G(i)) Aut(G(k)) for j = 1;2:
k j ! j
Lemma 1.1.8. We have
i) (cid:26)i(Z((cid:2)(i))) = Z((cid:2)(k)); ii) (cid:26)i(T((cid:2)(i))) = T((cid:2)(k));
k k
iii) (cid:26)i(T ((cid:2)(i))) = T ((cid:2)(k)); (cid:26)i(T ((cid:2)(i))) = T ((cid:2)(k));
k l l k r r
iv) (cid:26)(1)i(G ((cid:2)(i))) = G ((cid:2)(k));
k l l
(cid:26)(1)i(Gbi((cid:2)(i))) = Gbi((cid:2)(k));
k l l
(cid:26)(2)i(G(resp:bi)((cid:2)(i))) = G(resp:bi)((cid:2)(k)):
k r r
Proof. In the point i) ( ) means the Zariski closure and we skipped its proof because we
do not need this fact later.
Let us set p = (cid:26)i and p0 = (cid:26)(2)i: Observe that the image p0(G ((cid:2)(i))) of the group
k k r
G ((cid:2)(i)) by the morphism p0 is a closed subgroup of Aut(G(k)) de(cid:12)ned over k. This implies
r 2
that p(T ((cid:2)(i))) is a closed subvariety of Iso(G(k);G(k)) and a p0(G ((cid:2)(i)))-torsor de(cid:12)ned
r 1 2 r
over k. This torsor contains (cid:2)(k) as a C-point, so we have
T ((cid:2)(k)) p0(T ((cid:2)(i))) and G ((cid:2)(k)) p0(G ((cid:2)(i))):
r r r r
(cid:26) (cid:26)
9
Let P (resp. P0) be the projection p(resp. p0)restricted to T ((cid:2)(i))(resp. G ((cid:2)(i))):Then
r r
P(cid:0)1(T ((cid:2)(k))) is P0(cid:0)1(G ((cid:2)(k)))-torsor de(cid:12)ned over k, which contain (cid:2)(i) as a C-point.
r r
Hence we get P(cid:0)1(T ((cid:2)(k))) = T ((cid:2)(i)) and P0(cid:0)1(G ((cid:2)(k))) = G ((cid:2)(i)) This implies that
r r r r
p(T ((cid:2)(i))) = T ((cid:2)(k)) and p0(G ((cid:2)(i))) = G ((cid:2)(k)): All other statements are proved in
r r r r
the same way.
1.2. A(cid:14)ne, pro-algebraic, pro-unipotent groups and torsors.
1.2.1 We assume that G = limG(i) where the groups G(i) are a(cid:14)ne, connected, unipotent,
i(cid:0)
algebraic groups over k. We assume further that G(i) = G (cid:0)i+1G: Finally we assume that
the Lie algebra of G is (cid:12)nitely presented i.e. for i big e(cid:14)nough the number of relations,
G
de(cid:12)ning (cid:0)i+1 ; of degree less than i+1 does not depend on i.
G
G
(cid:14)
1.2.2. The condition that G is (cid:12)nitely presented implies that for i big enough the mor-
phisms
Aut(G(i+1)) Aut(G(i))
!
are surjective. We set
Aut(G) := limAut(G(i)):
i(cid:0)
Similarly, if G and G satisfy 1.2.1 and if there is an isomorphism G G for some
1 2 1;L 2;L
!
extension L of k, then the morphisms
(i+1) (i+1) (i) (i)
Iso(G ;G ) Iso(G ;G )
1 2 ! 1 2
are surjective for i big enough. We set
(i) (i)
Iso(G ;G ) := limIso(G ;G ):
1 2 1 2
i(cid:0)
Observe that Iso(G ;G ) is an (Aut(G );Aut(G ))-bitorsor de(cid:12)ned over k.
1 2 1 2
1.2.3. Examples of groups satisfying 1.2.1.
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