Remarques sur les différentielles des polylogarithmes uniformes
Annales de l'Institut Fourier, Volume 46 (1996) no. 5, p. 1327-1347
The purpose of the article is to study functional equations for the differentials of polylogarithms. One of the main ingredients is an infinitesimal analogue of a complex introduced by Goncharov. As a result, one obtains a 22-term relation for the differential of the trilogarithm.
On étudie des équations fonctionnelles pour les différentielles des polylogarithmes uniformes. Un des ingrédients est l’analogue infinitésimal d’un complexe introduit par Goncharov. On obtient en particulier une équation fonctionnelle à 22 termes pour la différentielle du trilogarithme.
@article{AIF_1996__46_5_1327_0,
     author = {Cathelineau, Jean-Louis},
     title = {Remarques sur les diff\'erentielles des polylogarithmes uniformes},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {46},
     number = {5},
     year = {1996},
     pages = {1327-1347},
     doi = {10.5802/aif.1551},
     mrnumber = {98k:19006},
     zbl = {0861.19003},
     language = {fr},
     url = {https://aif.centre-mersenne.org/item/AIF_1996__46_5_1327_0}
}
Cathelineau, Jean-Louis. Remarques sur les différentielles des polylogarithmes uniformes. Annales de l'Institut Fourier, Volume 46 (1996) no. 5, pp. 1327-1347. doi : 10.5802/aif.1551. https://aif.centre-mersenne.org/item/AIF_1996__46_5_1327_0/

[1] J. Aczél, The state of the second part of Hilbert's fifth problem, Bull. Amer. Math. Soc., 20 (1989), 153-163. | MR 981872 | MR 90h:39017 | Zbl 0676.39004

[2] S. Bloch, Higher regulators, algebraic K-theory and zeta functions of elliptic curves, Lect. notes, Irvine, 1977.

[3] S. Bloch, Applications of the dilogarithm function in algebraic K-theory and algebraic geometry, Proc. Int. Symp. Alg. Geom., Kyoto, (1977), 1-14. | Zbl 0416.18016

[4] J.-L. Chatelineau, Sur l'homologie de SL2 à coefficients dans l'action adjointe, Math. Scand., 63 (1988), 51-86. | MR 994970 | Zbl 0682.55013

[5] J.-L. Cathelineau, θ-Structures in Algebraic K-Theory and Cyclic Homology, K-Theory, 4 (1991), 591-606. | MR 1123180 | MR 92k:19003 | Zbl 0735.19005

[6] J.-L. Cathelineau, Homologie du groupe linéaire et polylogarithmes (d'après Goncharov et d'autres), Séminaire Bourbaki, 772 (1992-1993), Astérisque, 216 (1993), 311-341. | Numdam | MR 1246402 | MR 95c:19004 | Zbl 0845.19003

[7] J. Dupont, C. H. Sah, Scissors congruences II, J. Pure Appl. Alg., 25 (1982), 159-195. | MR 662760 | MR 84b:53062b | Zbl 0496.52004

[8] Ph. Elbaz-Vincent, K3 indécomposable des anneaux et homologie de SL2, Thèse de doctorat, Université de Nice Sophia-Antipolis (1995).

[9] A.B. Goncharov, Geometry of configurations, polylogarithms and motivic cohomology, Adv. Math., 114 (1995), 197-318. | MR 1348706 | MR 96g:19005 | Zbl 0863.19004

[10] A.B. Goncharov, Polylogarithms and motivic Galois groups, Proc. of the Seattle conf. on motives, Seattle july 1991, AMS Proc. Symp. in Pure Math., 2, 55 (1994), 43-96. | MR 94m:19003 | Zbl 0842.11043

[11] S. Lichtenbaum, Groups related to scissors congruence groups, Contemp. Math., 83 (1989), 151-157. | MR 90e:20030 | Zbl 0674.55012

[12] J. Oesterlé, Polylogarithmes, Sém. Bourbaki, 762 (1992-1993), Astérisque, 216 (1993), 49-67. | Numdam | MR 94m:11135 | Zbl 0799.11056

[13] A.A. Suslin, Algebraic K-theory of fields, Proc. Int. Cong. of Math., 1986, Berkeley, 222-243. | MR 89k:12010 | Zbl 0675.12005

[14] A.A. Suslin, K3 of a field and the Bloch group, Proc. Steklov Inst. of Math., 4 (1991), 217-239. | Zbl 0741.19005

[15] Z. Wojtkowiak, A construction of analogs of the Bloch-Wigner function, Math. Scand., 65 (1989), 140-142. | MR 92b:11084 | Zbl 0698.33002

[16] D. Zagier, Polylogarithms, Dedekind zeta functions and the algebraic K-theory of fields, Proc. Texel Conf. on Arithm. Alg. Geometry 1989, Birkhäuser, Boston (1991), 391-430. | Zbl 0728.11062