SMF - Publications - Annales scientifiques de l'ENS - Parutions

Résumé :
Valeurs zêta multiples et périodes des espaces de modules
Nous démontrons une conjecture de Goncharov et Manin qui prédit que les périodes des espaces de modules M_0,n des courbes de genre avec points marqués sont des valeurs zêta multiples. Nous introduisons une algèbre différentielle de fonctions polylogarithmes multiples sur M_0,n dans laquelle il existe des primitives. L'idée principale est d'appliquer une version de la formule de Stokes récursivement pour réduire chaque intégrale de périodes à une combinaison linéaire de valeurs zêta multiples. Nous donnons également une interprétation géométrique des double relations de mélange pour les valeurs zêta multiples. En considérant des applications naturelles entre les espaces des modules, on déduit des formules de produit générales entre leurs périodes. Les doubles relations de mélange s'obtiennent comme deux cas particuliers de cette construction.

Mots-clefs : Espace des modules, multizêtas, intégrales itérées, polylogarithmes, associateurs, associaèdres

Abstract:
We prove a conjecture due to Goncharov and Manin which states that the periods of the moduli spaces M_0,n of Riemann spheres with marked points are multiple zeta values. We do this by introducing a differential algebra of multiple polylogarithms on M_0,n and proving that it is closed under the operation of taking primitives. The main idea is to apply a version of Stokes' formula iteratively to reduce each period integral to multiple zeta values. We also give a geometric interpretation of the double shuffle relations, by showing that they are two extreme cases of general product formulae for periods which arise by considering natural maps between moduli spaces.

Keywords: Moduli spaces, multiple zeta values, iterated integrals, polylogarithms, associators, associahedra

Class. math. : 14G32, 11G55, 32G34

ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

Bibliographie:

1: Aomoto, Kazuhiko; Fonctions hyperlogarithmiques et groupes de monodromie unipotents; J. Fac. Sci. Univ. Tokyo Sect. IA Math. 25 (1978) 149–156; Math Reviews MR509582; Zentralblatt 416.32020
2: Aomoto, Kazuhiko; Addition theorem of Abel type for hyper-logarithms; Nagoya Math. J. 88 (1982) 55–71; Math Reviews MR683242; Zentralblatt 545.33014
3: Aomoto, Kazuhiko; Special values of hyperlogarithms and linear difference schemes; Illinois J. Math. 34 (1990) 191–216; Math Reviews MR1046562; Zentralblatt 684.33010
4: Arnold, V. I.; The cohomology ring of the group of dyed braids; Mat. Zametki 5 (1969) 227–231; Math Reviews MR0242196
5: Beilinson, A. A. and Goncharov, Alexander B. and Schechtman, V. V. and Varchenko, A. N.; Aomoto dilogarithms, mixed Hodge structures and motivic cohomology of a pair of triangles in the plane; in Grothendieck Festschrift; 86 (1990) 135–171
6: Borel, A. and Serre, J.-P.; Corners and arithmetic groups; Comment. Math. Helv. 48 (1973) 436–491; Math Reviews MR0387495
7: Brown, Francis C. S.; Polylogarithmes multiples uniformes en une variable; C. R. Math. Acad. Sci. Paris 338 (2004) 527–532; Math Reviews MR2057024; Zentralblatt 1048.11053
8: Brown, Francis C. S.; Single-valued hyperlogarithms and unipotent differential equations
9: Chen, Kuo Tsai; Extension of function algebra by integrals and Malcev completion of; Advances in Math. 23 (1977) 181–210; Math Reviews MR0458461
10: Chen, Kuo Tsai; Iterated path integrals; Bull. Amer. Math. Soc. 83 (1977) 831–879; Math Reviews MR0454968; Zentralblatt 389.58001
11: Deligne, Pierre; Équations différentielles à points singuliers réguliers; Springer, 1970; Math Reviews MR0417174; Zentralblatt 244.14004
12: Deligne, Pierre; Théorie de Hodge. III; Publ. Math. I.H.É.S. 44 (1974) 5–77; Math Reviews MR0498552; Zentralblatt 237.14003
13: Deligne, Pierre; Le groupe fondamental de la droite projective moins trois points; in Galois groups over (Berkeley, CA, 1987); Math. Sci. Res. Inst. Publ. 16 (1989) 79–297; Math Reviews MR1012168; Zentralblatt 742.14022
14: Deligne, Pierre and Goncharov, Alexander B.; Groupes fondamentaux motiviques de Tate mixte; Ann. Sci. École Norm. Sup. 38 (2005) 1–56; Math Reviews MR2136480; Zentralblatt 1084.14024
15: Devadoss, Satyan L.; Tessellations of moduli spaces and the mosaic operad; in Homotopy invariant algebraic structures (Baltimore, MD, 1998); Contemp. Math. 239 (1999) 91–114; Math Reviews MR1718078; Zentralblatt 968.32009
16: Devadoss, Satyan L.; Combinatorial equivalence of real moduli spaces; Notices Amer. Math. Soc. 51 (2004) 620–628; Math Reviews MR2064149; Zentralblatt 1093.14509
17: Dixon, A. C.; On a certain double integral; Proc. London Math. Soc. 2 (1905) 8–15
18: Drinfeld, V. G.; On quasitriangular quasi-Hopf algebras and on a group that is closely connected with; Leningrad Math. Journal 2 (1991) 829–860; Math Reviews MR1080203
19: Écalle, Jean; Singularités non abordables par la géométrie; Ann. Inst. Fourier (Grenoble) 42 (1992) 73–164; Math Reviews MR1162558; Zentralblatt 940.32013
20: Falk, Michael and Randell, Richard; The lower central series of a fiber-type arrangement; Invent. Math. 82 (1985) 77–88; Math Reviews MR808110; Zentralblatt 574.55010
21: Fischler, Stéphane; Groupes de Rhin-Viola et intégrales multiples; J. Théor. Nombres Bordeaux 15 (2003) 479–534; Math Reviews MR2140865; Zentralblatt 1074.11040
22: Getzler, E.; Operads and moduli spaces of genus Riemann surfaces; in The moduli space of curves (Texel Island, 1994); Progr. Math. 129 (1995) 199–230; Math Reviews MR1363058; Zentralblatt 851.18005
23: Goncharov, Alexander B.; The dihedral Lie algebras and Galois symmetries of; Duke Math. J. 110 (2001) 397–487; Math Reviews MR1869113
24: Goncharov, Alexander B.; Multiple -values, Galois groups, and geometry of modular varieties; in European Congress of Mathematics, Vol. I (Barcelona, 2000); Progr. Math. 201 (2001) 361–392; Math Reviews MR1905330; Zentralblatt 1042.11042
25: Goncharov, Alexander B.; Multiple polylogarithms and mixed Tate motives; arXiv:math.AG/0103059
26: Goncharov, Alexander B.; Periods and mixed motives; arXiv:math.AG/0202154
27: Goncharov, Alexander B. and Manin, Yuri I.; Multiple -motives and moduli spaces; Compos. Math. 140 (2004) 1–14; Math Reviews MR2004120
28: Gonzalez-Lorca, J.; Série de Drinfel'd, monodromie et algèbres de Hecke; Thèse de doctorat, École Normale Supérieure (1998)
29: Griffiths, Phillip and Schmid, Wilfried; Recent developments in Hodge theory: a discussion of techniques and results; in Discrete subgroups of Lie groups and applicatons to moduli (Internat. Colloq., Bombay, 1973); (1975) 31–127; Math Reviews MR0419850; Zentralblatt 355.14003
30: Grothendieck, A.; On the de Rham cohomology of algebraic varieties; Publ. Math. I.H.É.S. 29 (1966) 95–103; Math Reviews MR0199194; Zentralblatt 145.17602
31: Hain, Richard M.; The geometry of the mixed Hodge structure on the fundamental group; in Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985); Proc. Sympos. Pure Math. 46 (1987) 247–282; Math Reviews MR927984; Zentralblatt 654.14006
32: Hain, Richard M.; Classical polylogarithms; in Motives (Seattle, WA, 1991); Proc. Sympos. Pure Math. 55 (1994) 3–42; Math Reviews MR1265550; Zentralblatt 807.19003
33: Hain, Richard M. and MacPherson, Robert; Higher logarithms; Illinois J. Math. 34 (1990) 392–475; Math Reviews MR1046570; Zentralblatt 737.14014
34: Hoang Ngoc, M. and Petitot, M. and Van Den Hoeven, J.; Shuffle algebra and polylogarithms; in Formal Power Series and Algebraic Combinatorics 1998, Toronto
35: Hoffman, Michael E.; Quasi-shuffle products; J. Algebraic Combin. 11 (2000) 49–68; Math Reviews MR1747062; Zentralblatt 959.16021
36: Kapranov, Mikhail M.; The permutoassociahedron, Mac Lane's coherence theorem and asymptotic zones for the KZ equation; J. Pure Appl. Algebra 85 (1993) 119–142; Math Reviews MR1207505; Zentralblatt 812.18003
37: Knizhnik, V. G. and Zamolodchikov, A. B.; Current algebra and Wess-Zumino model in two dimensions; Nuclear Phys. B 247 (1984) 83–103; Math Reviews MR853258
38: Kohno, Toshitake; Série de Poincaré-Koszul associée aux groupes de tresses pures; Invent. Math. 82 (1985) 57–75; Math Reviews MR808109; Zentralblatt 574.55009
39: Kolchin, E. R.; Differential algebra and algebraic groups; Academic Press, 1973 Pure and Applied Mathematics, Vol. 54; Math Reviews MR0568864; Zentralblatt 264.12102
40: Lappo-Danilevskii, I.; Mémoires sur la théorie des systèmes des équations différentielles linéaires; Chelsea, New York, 1953
41: Magid, Andy R.; Lectures on differential Galois theory; Amer. Math. Soc., 1994; Math Reviews MR1301076; Zentralblatt 855.12001
42: Boutet de Monvel, L.; Polylogarithmes; http://www.math.jussieu.fr/~boutet
43: Orlik, Peter and Terao, Hiroaki; Arrangements of hyperplanes; Springer, 1992; Math Reviews MR1217488; Zentralblatt 757.55001
44: Poincaré, Henri; Sur les groupes d'équations linéaires; Acta Mathematica 4 (1884)
45: Racinet, Georges; Doubles mélanges des polylogarithmes multiples aux racines de l'unité; Publ. Math. Inst. Hautes Études Sci. 95 (2002) 185–231; Math Reviews MR1953193; Zentralblatt 1050.11066
46: Radford, David E.; A natural ring basis for the shuffle algebra and an application to group schemes; J. Algebra 58 (1979) 432–454; Math Reviews MR540649; Zentralblatt 409.16011
47: Rhin, Georges and Viola, Carlo; On a permutation group related to; Acta Arith. 77 (1996) 23–56; Math Reviews MR1404975; Zentralblatt 864.11037
48: Stasheff, James Dillon; Homotopy associativity of -spaces. I; Trans. Amer. Math. Soc. 108 (1963) 275-292; Math Reviews MR0158400
49: Terasoma, Tomohide; Selberg integrals and multiple zeta values; Compositio Math. 133 (2002) 1–24; Math Reviews MR1918286; Zentralblatt 1003.11042
50: Voisin, Claire; Hodge theory and complex algebraic geometry. II; Cambridge University Press, 2003; Math Reviews MR1997577; Zentralblatt 1032.14002
51: Waldschmidt, Michel; Valeurs zêta multiples. Une introduction; J. Théor. Nombres Bordeaux 12 (2000) 581–595; Math Reviews MR1823204; Zentralblatt 976.11037
52: Yoshida, Masaaki; Fuchsian differential equations; Friedr. Vieweg Sohn, 1987; Math Reviews MR986252; Zentralblatt 618.35001
53: Zhao, Jianqiang; Multiple polylogarithms: analytic continuation, monodromy, and variations of mixed Hodge structures; in Contemporary trends in algebraic geometry and algebraic topology (Tianjin, 2000); Nankai Tracts Math. 5 (2002) 167–193; Math Reviews MR1945360; Zentralblatt 1058.32005
54: Zlobin, S. A.; Integrals that can be represented as linear forms of generalized polylogarithms; Mat. Zametki [Math. Notes] 71 (2002) 782–787 [711–716]; Math Reviews MR1936201; Zentralblatt 1049.11077
55: Zudilin, W.; Well-poised hypergeometric transformations of Euler-type multiple integrals; J. London Math. Soc. 70 (2004) 215–230; Math Reviews MR2064759; Zentralblatt 1065.11054