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Odd zeta motive and linear forms in odd zeta values

  • Clément Dupont (a1)


We study a family of mixed Tate motives over  $\mathbb{Z}$ whose periods are linear forms in the zeta values  $\unicode[STIX]{x1D701}(n)$ . They naturally include the Beukers–Rhin–Viola integrals for  $\unicode[STIX]{x1D701}(2)$ and the Ball–Rivoal linear forms in odd zeta values. We give a general integral formula for the coefficients of the linear forms and a geometric interpretation of the vanishing of the coefficients of a given parity. The main underlying result is a geometric construction of a minimal ind-object in the category of mixed Tate motives over  $\mathbb{Z}$ which contains all the non-trivial extensions between simple objects. In a joint appendix with Don Zagier, we prove the compatibility between the structure of the motives considered here and the representations of their periods as sums of series.



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[Apé79] Apéry, R., Irrationalité de 𝜁(2) et 𝜁(3) , Astérisque 61 (1979), 1113.
[BR01] Ball, K. and Rivoal, T., Irrationalité d’une infinité de valeurs de la fonction zêta aux entiers impairs , Invent. Math. 146 (2001), 193207.
[Beu79] Beukers, F., A note on the irrationality of 𝜁(2) and 𝜁(3) , Bull. Lond. Math. Soc. 11 (1979), 268272.
[Blo86] Bloch, S., Algebraic cycles and higher K-theory , Adv. Math. 61 (1986), 267304.
[Bor77] Borel, A., Cohomologie de SL n et valeurs de fonctions zeta aux points entiers , Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 4 (1977), 613636.
[Bro16] Brown, F., Irrationality proofs for zeta values, moduli spaces and dinner parties , Mosc. J. Comb. Number Theory 6 (2016), 102165.
[CFR08a] Cresson, J., Fischler, S. and Rivoal, T., Phénomènes de symétrie dans des formes linéaires en polyzêtas , J. Reine Angew. Math. 617 (2008), 109151.
[CFR08b] Cresson, J., Fischler, S. and Rivoal, T., Séries hypergéométriques multiples et polyzêtas , Bull. Soc. Math. France 136 (2008), 97145.
[Del71] Deligne, P., Théorie de Hodge II , Publ. Math. Inst. Hautes Études Sci. 40 (1971), 557.
[Del74] Deligne, P., Théorie de Hodge III , Publ. Math. Inst. Hautes Études Sci. 44 (1974), 577.
[Del89] Deligne, P., Le groupe fondamental de la droite projective moins trois points , in Galois groups over Q , Berkeley, CA, 1987, Mathematical Sciences Research Institute Publications, vol. 16 (Springer, New York, NY, 1989), 79297.
[DG05] Deligne, P. and Goncharov, A. B., Groupes fondamentaux motiviques de Tate mixte , Ann. Sci. Éc. Norm. Supér. (4) 38 (2005), 156.
[Dup17] Dupont, C., Relative cohomology of bi-arrangements , Trans. Amer. Math. Soc. 369 (2017), 81058160.
[Fis04] Fischler, S., Irrationalité de valeurs de zêta (d’après Apéry, Rivoal, …) , Astérisque 294 (2004), vii, 2762.
[Foa77] Foata, D., Distributions eulériennes et mahoniennes sur le groupe des permutations , in Higher combinatorics: Proc. NATO Advanced Study Institute, Berlin, 1976, NATO Advanced Study Institutes Series, Series C, vol. 31 (Reidel, Dordrecht, 1977), 2749.
[Foa10] Foata, D., Eulerian polynomials: from Euler’s time to the present , in The legacy of Alladi Ramakrishnan in the mathematical sciences (Springer, New York, 2010), 253273.
[Gon02] Goncharov, A. B., Periods and mixed motives, Preprint (2002), arXiv:math.AG/0202154.
[GM04] Goncharov, A. B. and Manin, Yu. I., Multiple 𝜁-motives and moduli spaces M 0, n , Compos. Math. 140 (2004), 114.
[Hat02] Hatcher, A., Algebraic topology (Cambridge University Press, Cambridge, 2002).
[Hub00] Huber, A., Realization of Voevodsky’s motives , J. Algebraic Geom. 9 (2000), 755799.
[Hub04] Huber, A., Corrigendum to: “Realization of Voevodsky’s motives” [J. Algebraic Geom.] 9 (2000) 755–799; mr1775312 , J. Algebraic Geom. 13 (2004), 195207.
[Lev93] Levine, M., Tate motives and the vanishing conjectures for algebraic K-theory , in Algebraic K-theory and algebraic topology Lake Louise, AB, 1991, NATO Advanced Study Institutes Series, Series C, vol. 407 (Kluwer Academic, Dordrecht, 1993), 167188.
[Riv00] Rivoal, T., La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs , C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), 267270.
[RV96] Rhin, G. and Viola, C., On a permutation group related to 𝜁(2) , Acta Arith. 77 (1996), 2356.
[RV01] Rhin, G. and Viola, C., The group structure for 𝜁(3) , Acta Arith. 97 (2001), 269293.
[Voe00] Voevodsky, V., Triangulated categories of motives over a field , in Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, vol. 143 (Princeton University Press, Princeton, NJ, 2000), 188238.
[Zud04] Zudilin, W., Arithmetic of linear forms involving odd zeta values , J. Théor. Nombres Bordeaux 16 (2004), 251291.
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Odd zeta motive and linear forms in odd zeta values

  • Clément Dupont (a1)


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