Non-abelian unipotent periods Monodromy of iterated integrals Zdzisl aw Wojtkowiak x0. Introduction. 0.1. Let X be a smooth, algebraic variety de�ned over a number �eld k. Let � : k ,! C be an inclusion. We set XC = X �C: Let X(C) be the set of C-points of XC with its k complex topology. There is the canonical isomorphism n � n pcomp : HB(X(C)) � C �! HDR(X)�C k between Betti (singular) cohomology and algebraic De Rham cohomology. The period P n matrix (pij) is de�ned by equations !i = pji�j where f!ig and f�ig are bases of HDR(X) R n � and HB(X(C)) or pji = �j� !i where f�j g is the dual base of Hn(X(C)): Let us assume that X is an abelian variety. Let G be the largest subgroup of 1 GL(HB(X(C))�Q)�Gm which �xes all tensors clB(Z), where Z is an algebraic cycle on n some X (see [D2]). Let P be the functor of k-algebras, such that any element of P (A) is 1 1 an isomorphism p : HB(X(C))�A ! HDR(X)�A mapping clB(Z)�1 to clDR(Z)�1 for n any algebraic cycle Z on any X . The isomorphism pcomp belongs to P(C); the functor P is represented by an algebraic variety over k, which is a Gk-torsor under the natural 1 1 1 action. It is a subtorsor of the GL(HB(X(C) � k)-torsor Iso(HB(X(C)) � k;HDR(X)): 1 1 Let T be a smallest subtorsor de�ned over k of the torsor Iso(HB(X(C))�k;HDR(X)); which contains pcomp as a C-point and let G be the corresponding subgroup (de�ned over 1 1 k) of GL(HB(X(C))� k): Let Z(pcomp) be the Zariski closure of pcomp in Iso(HB(X(C)� 1 k;HDR(X)) i.e. the smallest Zariski closed subset de�ned over k, which contains pcomp as a C-point. Then we have Z(pcomp) � T � P 1 and G � G: In order to calculate Z(pcomp) and to show that Z(pcomp) = P one need to show that certain number are transcendental. On the other hand to calculate T and G seems to be 1 1 an easier task. The requirement that T is a subtorsor of Iso(H (X(C)) � k;H (X)) is B DR very strong and usually relatively weak informations about numbers fpijg are necessary 1 to calculate T and G. We give an obvious example. If X = P then in order to show that Q Z(pcomp) � Gm we must know that 2�i is transcendental. But already the fact that 2�i th is not a k -root of a rational number for any k 2 N implies that T � Gm and G = Gm: In this note we shall discuss periods for fundamental groups. We shall concentrate on analogues of T and G for fundamental groups. On the other hand we have no analogue of 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. �[1] @�[1] 4. The Gauss-Manin connection associated with the morphism X ! X of cosim- plicial schemes. 5. Torsors associated to the canonical unipotent connection with logarithmic singulari- ties. 1 6. Partial informations about GDR(P (C) n f0; 1;1g): 1 7. Homotopy relative tangential base points on P (C) n fa1; : : : ; an+1g: 1 8. Generators of �1(P (C) n fa1; : : : ; an+1g; x): 1 9. Monodromy of iterated integrals on P (C) n fa1; : : : ; an+1g: 10. Calculations. 11. Con�guration spaces. 12. The Drinfeld-Ihara Z=5-cycle relation. 2 13. Functional equations of iterated integrals. 2 14. Subgroups of Aut(� ). A.1. Malcev completion. The dependence of sections 0 3 � � � � 1 7 � � � � � � 4 8 � � 2 � � 9 � � 14 5 � � 10 � � � � 6 11 � � 12 0.2. Below we shall brie�y discuss the contents of the paper. Let us assume that X is a smooth, quasi-projective, algebraic variety de�ned over a number �eld k. Let x be a k- point of X. In [W1] we de�ned a�ne, connected, pro-unipotent group schemes over k and DR B Q respectively; � (X; x) | the algebraic De Rham fundamental group and � (X(C); x) 1 1 | the Betti fundamental group. We have also the inclusion (of Q-points into C-points) B DR �x : � 1 (X(C); x)(Q) ! �1 (X; x)(C) such that the induced homomorphism on C-points B DR ’x : � 1 (X(C); x)(C) ! �1 (X; x)(C) is an isomorphism. The a�ne, pro-algebraic scheme over k B DR Iso := Iso(� (X(C); x) � k; � (X; x)) 1 1 3 DR is an Aut(� 1 (X(C); x))-torsor. Let Z(’x) be the Zariski closure of ’x in Iso. Let T (’x) be the smallest subtorsor of Iso, de�ned over k which contains ’x as a C-point. Let DR GDR(’x) � Aut(� 1 (X; x)) be the corresponding subgroup. One can hope that (0:1:) Z(’x) = T(’x) DR We shall denote by GDR(X) the image of GDR(’x) in Out(� 1 (X; x)). The calculation of the homomorphism ’x is equivalent to the calculation of the mon- odromy of all iterated integrals on X. We shall see that the monodromy representa- � 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, �(�) �2�i then \�(S) � S " in the l-adic case and \�(S) � S " for iterated integrals. Let � � ’ : Gk := Gal(k=k) ! Out(�1(X �k; x)(l)) be the natural homomorphism. Very opti- k mistically we can state the following conjecture: (0:2) Lie(’(Gk)) � k � Lie(GDR(X)) � Ql: 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�i). 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�cient for our applications. In Sections 9 and 10 we describe 1 the monodromy of iterated integrals on P (C) n fa1; : : : ; an+1g and in more details on 1 P (C)nf0; 1;1g. In Sections 11{13 we study monodromy of iterated integrals on con�gu- ration spaces. We give a proof of Drinfeld-Ihara Z=5-cycle relation which is di�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 ! and it is also fundamental 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�nitions. In Section 2 we de�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�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- 1 culate some part of the group GDR(X) for X = P (C) n f0; 1;1g. This part corresponds to �! � �! �! the Galois representation on �1(X; 01) [�0; �0] where �0 := [�1(X; 01); �1(X; 01)] (see [I3]). About this part of GDR(X), let us call it G, we have the following result. � The group G contains a group H = fftjft(X) = t �X; ft(Y ) = t �Y jt 2 C g if and only if all numbers �(2k+1) are irrational. This leads to a de�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 n f0; 1g. Thanks are due to Professor Y. Ihara for showing us his proof of 5-cycle relation, which help us to �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 x1. Torsors. Let G1 and G2 be two groups. We say that a set T is a (G1;G2)-bitorsor if T is equipped with a free, transitive, left action of G1 and with a free, transitive right action of G2 and if the actions of G1 and G2 commute. We say that a subset S � T is a subtorsor of T if there exist subgroups H1 � G1 and H2 � G2 such that S is a (H1;H2)-bitorsor under the natural actions of H1 and H2. We say that a subset S � T is a left (resp. right) subtorsor of T if there is a subgroup H1 � G1 (resp. H2 � G2 ) such that the natural action of H1 (resp. H2) on S is free and transitive. Main example. Let G1 and G2 be two groups. Assume that G1 and G2 are isomorphic. Then the set of isomorphisms from G1 to G2, which we denote by Iso(G1;G2); is an (Aut(G1); Aut(G2))-bitorsor. For any non-empty subset S � Iso(G1;G2), the intersection of all subtorsors (resp. right subtorsors, resp. left subtorsors) of Iso(G1;G2), which contain S, is a subtorsor (resp. right subtorsor, resp. left subtorsor) of Iso(G1;G2); which we denote by T(S) (resp. Tr(S); resp. Tl(S)): 1.1. Unipotent, a�ne, algebraic groups and torsors. Let k be a �eld of characteristic zero. We say that X is an algebraic variety (de- �ned) over k if X is an algebraic scheme over Spec k: If A is a k-algebra, we set XA := X � SpecA: The set of A-points of X we denote by X(A): We say that G is an a�ne, Spec k algebraic group (de�ned) over k, if G is an a�ne, algebraic group scheme over Spec k: 1.1.1. Let G be an a�ne, unipotent, connected, algebraic group over k. Then there is an a�ne, algebraic group Aut(G) over k such that for any k-algebra A we have Aut(G)(A) = Aut(GA): 6 Let G1 and G2 be two connected, a�ne, unipotent, algebraic groups over k. Then there is a smooth, a�ne, algebraic variety Iso(G1;G2) over k, such that for any k-algebra A we have Iso(G1;G2)(A) := Iso(G1A;G2A): Proof of 1.1.1. Let G be a Lie algebra of G. The exponential map exp: G ! G is an isomorphism of a�ne, algebraic groups if we equip G with a group law given by the Baker-Campbell-Hausdor� formula. The automorphisms of the group G coincides with the automorphisms of the Lie algebra G. One can easily give an ideal de�ning AutLie(G) in k[GL(G)]: In the similar way one constructs Iso(G1;G2): 1.1.2. In [S] page 149 there is a de�nition of a (left) G-torsor (a principal homogeneous space of G) if G is a linear, algebraic group over k. This de�nition extends immediately for bitorsors of algebraic groups. We say that an a�ne, algebraic variety T over k is a (G1;G2)-bitorsor, if there are morphisms G1 �T ! T and T �G2 ! T over Spec k; which de�ne free, transitive actions of G1 and G2 on T and if these actions commute. Observe that if T has a k-point then G1 and G2 are isomorphic. The de�nitions of a subtorsor, a right subtorsor and a left subtorsor we left to a reader, as well as a proof of the following lemma. 0 0 Lemma 1.1.2. Let T1 be an (H1;H2)-subtorsor of T and let T2 be an (H 1;H2)-subtorsor 0 0 of T . Assume that T1 \ T2 =6 ;: Then the intersection T1 \ T2 is an (H1 \ H2;H 1 \ H2)- subtorsor of T . The similar statements hold for right and left subtorsors of T . 1.1.3. Main example. Let G1 and G2 be two connected, unipotent, a�ne, algebraic groups over k. Assume that there is an isomorphism G 1k� ! G2k�: Then the algebraic variety Iso(G1;G2) is an (Aut(G1); Aut(G2))-bitorsor, if we equip Iso(G1;G2) with the obvious actions of Aut(G1) and Aut(G2). Let k � C be a sub�eld of C. Let � : G1(C) ! G2(C) be an isomorphism. Then � is a C-point of Iso(G1;G2): We denote by Z(�) the Zariski closure of � in Iso(G1;G2) 7 i.e. the smallest algebraic subset of Iso(G1;G2) (de�ned) over k, which contains � as a C-point. The connected, unipotent, a�ne, algebraic group Gi is isomorphic as an algebraic m variety over k to the a�ne space A k ; hence � can be view as a C-point (�ij)1�i;j�n of 2 m A k : Let k(�) be a sub�eld of C generated over k by all �i;j : m Lemma 1.1.3. The �eld k(�) does not depend on the choice of isomorphisms Gi � A k and the transcendental degree of the �eld k(�) over k is equal to the dimension of Z(�): Proof. This follows (is) Lemma 1.7 in [D2]. De�nition-Proposition 1.1.4. Let T (�) (resp. Tr(�); resp. Tl(�)) be the intersection of all subtorsors T (resp. right-subtorsors Tr, resp. left subtorsors Tl) de�ned over k of Iso(G1;G2); which contain � as a C-point. Then T(�) (resp. Tr(�) resp. Tl(�) ) is a bi bi (G r (�);Gl (�))-subtorsor (resp. right Gr(�)-subtorsor, resp. left Gl(�)-subtorsor) of bi bi Iso(G1;G2) for some G r (�) � Aut(G2) and Gl (�) � Aut(G1) (resp. Gr(�) � Aut(G2); resp. Gl(�) � Aut(G1)): Proof. The intersection of a family of algebraic varieties coincides with an intersection of a �nite number of them. Hence it follows from Lemma 1.1.2 that T (�); Tr(�) and Tl(�) exist and are unique. The groups are also unique because they are intersections of the corresponding subgroups of Aut(Gi): bi �1 bi If T (�) has a k-point f, then G l (�) = f �Gr (�)�f: If Tl(�) (resp. Tr(�)) has a k- bi bi point, then T (�) = Tl(�) and G l (�) = Gl(�) (resp. T (�) = Tr(�) and Gr (�) = Gr(�): Lemma 1.1.5. Let G be a unipotent, connected, a�ne, algebraic group over k. Then ab Aut(G) is an extension of an algebraic subgroup of GL(G ) by a connected, unipotent, bi bi a�ne, algebraic group. Hence the groups G r (�);Gl (�);Gr(�);Gl(�) are extensions of ab algebraic subgroups of GL(G ) by connected, unipotent, a�ne, algebraic groups. (i) Proof. Let G be the Lie algebra of G and let (G )i be a �ltration of G by the lower central series. Any automorphism of the Lie algebra G preserves the �ltration and the induced � � (i) (1) G (i+1) ab G (2) automorphism of G is determined by the automorphism of G = G : Hence 8 ab AutLie(G) is an extension of a closed subgroup of GL(G ) be a unipotent group. The lemma follows from the identi�cation of Aut(G) with AutLie(G) by the exponential map � exp : G ! G: � 1 k� Lemma 1.1.6. Assume that Gr(�) is an extension of Gm (or G such that H (Gal( k); G)= 0) by a connected, unipotent, a�ne, algebraic group N. Then Tr(�) has a k-point. � 1 k� Proof. It follows from [S] Proposition 4.1 that H (Gal( k);N) = 0. It follow from � 1 k� [S] Proposition 2.2 and the assumption of the lemma that H (Gal( k); Gr(�)) = 0. Proposition 1.1 from [S] implies that Tr(�)(k) =6 ;: i (i) Let � (G) be a �ltration of a group G by the lower central series. Let us set G := � G i+1 (i) (i) � G: The isomorphism � : G1(C) ! G2(C) induces isomorphisms � : G1 (C) ! (i) (i) (k) G (C). Let k < i: The projections G ! G for j = 1; 2 induce 2 j j i (i) (i) (k) (k) � : Iso(G ;G ) ! Iso(G ;G ); k 1 2 1 2 i (i) (k) �(j) : Aut(G ) ! Aut(G ) for j = 1; 2: k j j Lemma 1.1.8. We have i (i) (k) i (i) (k) i) � (Z(� )) = Z(� ); ii) � (T (� )) = T(� ); k k i (i) (k) i (i) (k) iii) � k(Tl(� )) = Tl(� ); �k(Tr(� )) = Tr(� ); i (i) (k) iv) �(1) k(Gl(� )) = Gl(� ); i bi (i) bi (k) �(1) (G (� )) = G (� ); k l l i (resp:bi) (i) (resp:bi) (k) �(2) k(Gr (� )) = Gr (� ): Proof. In the point i) ( ) means the Zariski closure and we skipped its proof because we do not need this fact later. i 0 i 0 (i) Let us set p = � k and p = �(2)k: Observe that the image p (Gr(� )) of the group (i) 0 (k) Gr(� ) by the morphism p is a closed subgroup of Aut(G 2 ) de�ned over k. This implies (i) (k) (k) 0 (i) that p(Tr(� )) is a closed subvariety of Iso(G 1 ;G2 ) and a p (Gr(� ))-torsor de�ned (k) over k. This torsor contains � as a C-point, so we have (k) 0 (i) (k) 0 (i) Tr(� ) � p (Tr(� )) and Gr(� ) � p (Gr(� )): 9 0 0 (i) (i) Let P (resp. P ) be the projection p (resp. p ) restricted to Tr(� ) (resp. Gr(� )): Then �1 (k) 0�1 (k) (i) P (Tr(� )) is P (Gr(� ))-torsor de�ned over k, which contain � as a C-point. �1 (k) (i) 0�1 (k) (i) Hence we get P (Tr(� )) = Tr(� ) and P (Gr(� )) = Gr(� ) This implies that (i) (k) 0 (i) (k) p(Tr(� )) = Tr(� ) and p (Gr(� )) = Gr(� ): All other statements are proved in the same way. 1.2. A�ne, pro-algebraic, pro-unipotent groups and torsors. (i) (i) 1.2.1 We assume that G = limG where the groups G are a�ne, connected, unipotent, � i � (i) G i+1 algebraic groups over k. We assume further that G = � G: Finally we assume that the Lie algebra G of G is �nitely presented i.e. for i big enough the number of relations, � G i+1 de�ning � G; of degree less than i + 1 does not depend on i. 1.2.2. The condition that G is �nitely presented implies that for i big enough the mor- phisms (i+1) (i) Aut(G ) ! Aut(G ) are surjective. We set (i) Aut(G) := lim Aut(G ): � i Similarly, if G1 and G2 satisfy 1.2.1 and if there is an isomorphism G1;L ! G2;L for some 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(G1;G2) := lim Iso(G 1 ;G2 ): � i Observe that Iso(G1;G2) is an (Aut(G1); Aut(G2))-bitorsor de�ned over k. 1.2.3. Examples of groups satisfying 1.2.1. 10