Abstract
Some combinatorial aspects of relations between multiple zeta values of depths 2 and 3 and period polynomials are discussed.
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S. Baumard, L. Schneps, Period polynomial relations between double zeta values, Ramanujan Journal 32 (2013), 83–100.
D. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Physics Letters. B 393 (1997), 403–412.
F. Brown, On the decomposition of motivic multiple zeta values, in Galois-Teichmullër Theory and Arithmetic Geometry, Advanced Studies in Pure Mathematics, Vol. 63, Mathematical Society of Japan, Tokyo, 2012, pp. 31–58.
F. Brown, Mixed Tate motives over ℤ, Annals of Mathematics 175 (2012), 949–976.
F. Brown, Motivic periods and ℙ1 {0, 1, ∞}, in Proceedings of the International Congress of Mathematicians-Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 295–318.
F. Brown, Zeta elements in depth 3 and the fundamental Lie algebra of a punctured elliptic curve, Forum of Mathematics. Sigma 5 (2017), 1–56.
F. Brown, Depth-graded motivic multiple zeta values, https://arxiv.org/abs/1301.3053.
S. Carr, H. Gangl and L. Schneps, On the Broadhurst-Kreimer generating series for multiple zeta values, in Feynman Amplitudes, Periods and Motives, Contemporary Mathematics, Vol. 648, American Mathematical Society, Providence, RI, 2015, pp. 57–77.
J. Ecalle, ARI/GARI, la dimorphie et l’arithmétique des multizêtas: un premier bilan, Journal de Théorie des Nombres Bordeaux 15 (2003), 411–478.
B. Enriquez and P. Lochak, Homology of depth-graded motivic Lie algebras and koszulity, Journal de Théorie des Nombres Bordeaux 28 (2016), 829–850.
H. Gangl, M. Kaneko and D. Zagier, Double zeta values and modular forms, in Automorphic Forms and Zeta Functions, World Scientific, Hackensack, NJ, 2006, pp. 71–106.
C. Glanois, Motivic unipotent fundamental groupoid of \({\mathbb{G}_m}\backslash {\mu _N}\) for N = 2, 3, 4, 6, 8 and Galois descents, Journal of Number Theory 160 (2016), 334–384
A. B. Goncharov, Periods and mixed motives, https://arxiv.org/abs/math/0202154.
A. B. Goncharov, The dihedral Lie algebras and Galois symmetries of \(\pi _1^{\left( l \right)}\left(\mathbb{P}{{^1} - \left( {\left\{ {0,\infty } \right\} \cup {\mu _N}} \right)} \right)\), Duke Mathematical Journal 110 (2001), 397–487.
A. B. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Mathematical Journal 128 (2005), 209–284.
M. E. Hoffman, The algebra of multiple harmonic series, Journal of of Algebra 194 (1997), 477–495.
K. Ihara, M. Kaneko and D. Zagier, Derivation and double shuffle relations for multiple zeta values, Compositio Mathematica 142 (2006), 307–338.
M. Kaneko and K. Tasaka, Double zeta values, double Eisenstein series, and modular forms of level 2, Mathematische Annalen 357 (2013), 1091–1118.
W. Kohnen and D. Zagier, Modular forms with rational periods, in Modular Forms (Durham, 1983), Ellis Horwood Series in Mathematics and its Applications: Statistics and Operational Research, Horwood, Chichester, 1984, pp. 197–249.
D. Ma, Period polynomial relations between formal double zeta values of odd weight, Mathematische Annalen 365 (2016), 345–362.
D. Ma, Period polynomial relations of binomial coefficients and binomial realization of formal double zeta space, International Journal of Number Theory 13 (2017), 761–774.
D. Ma, Inverse ofa matrix related to double zeta values of odd weight, Journal of Number Theory 166 (2016), 166–180.
D. Ma, Connections between double zeta values relative to μn, Hecke operators Tn, and newforms of level Γ0(N) for N = 2, 3, https://arxiv.org/abs/1511.06102.
D. Ma, Relations among multiple zeta values and modular forms of low level, Ph.D. Thesis.
E. Panzer, The parity theorem for multiple polylogarithms, Journal of Number Theory 172 (2017), 93–113.
K. Tasaka, On linear relations among totally odd multiple zeta values related to period polynomials, Kyushu Journal of Mathematics 70 (2016), 1–28.
H. Tsumura, Combinatorial relations for Euler-Zagier sums, Acta Arithmetica 111 (2004), 27–42.
S. Yamamoto, Interpolation of multiple zeta and zeta-star values, Journal of Algebra 385 (2013), 102–114.
D. Zagier, Periods of modular forms and Jacobi theta functions, Inventiones Mathematicae 104 (1991), no. 3, 449–465.
D. Zagier, Values of zeta functions and their applications, in First European Congress of Mathematics (Paris, 1992), Vol. II, Progress in Mathematics, Vol. 120, Birkhäuser, Basel, 1994, 497–512.
D. Zagier, Periods of modular forms, traces of Hecke operators, and multiple zeta values, Sūrikaisekikenkyūsho Kôkyûroku 843 (1993), 162–170.
D. Zagier, Evaluation of the multiple zeta values ζ(2, …, 2, 3, 2,…, 2), Annals of Mathematics 175 (2012), 977–1000.
Acknowledgements
The authors are grateful to Francis Brown for initial advice and useful comments. The second author wishes to express his thanks to Herbert Gangl for drawing his attention to Yamamoto’s \({1 \over 2}\)-interpolated multiple zeta values. The second author also thanks Max Planck Institute for Mathematics, where the paper was written, for the invitation and hospitality. This work is partially supported by Japan Society for the Promotion of Science, Grant-in-Aid for JSPS Fellows (No. 16H07115).
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Ma, D., Tasaka, K. Relationships between multiple zeta values of depths 2 and 3 and period polynomials. Isr. J. Math. 242, 359–400 (2021). https://doi.org/10.1007/s11856-021-2139-8
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DOI: https://doi.org/10.1007/s11856-021-2139-8