Multiple polylogarithms, cyclotomy and modular complexes

By A. Goncharov. 2011

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The multiple ζ-values were invented and studied by Euler [E] and then forgotten. They showed up again in such different subjects as quantum groups [Dr] (the Drinfeld associator), Zagier’s studies [Z1-2], the Kontsevich integrals for Vassiliev knot invariants, mixed Tate motives over Spec Z [G1-2], and, recently, in computations in quantum field theory [B], [Kr]. The multiple polylogarithms were studied in [G1-4]. In this paper we investigate them at N -th roots of unity: x1 = . . . = x N m = 1. Notice that Li1(x) = − log(1 − x), so if ζN is a primitive N -th root of 1, then Li1(ζN ) is a logarithm of a cyclotomic unit in Z[ζN , N−1]. In general the supply of numbers we get coincides with the linear combinations of multiple Dirichlet L-values

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7837ef17c8907aa56d9580714d78032ac295c793

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Citations in the corpus

Iterated integrals of modular forms and noncommutative modular symbols (2005, background): “egrals. For the harmonic shuﬄe relations between multiple zeta values, A. Goncharov established this in the Ch. 9 of [Go6] (for convergent integrals), and in 7.5 of [Go6] elaborating the last page of [Go4] (for regularized integrals). Conversely, integral shuﬄe relations can be deduced from harmonic”
Soft triple-real radiation for Higgs production at N3LO (2013, methodology): “Multiple polylogarithms are defined recursively via the iterated integral [97, 98]”
Feynman integrals and hyperlogarithms (2015, background): “The multiple polylogarithm [90, 180] (MPL) of weight n1 + .”
Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series (2018, background): “of the symbol map and the coaction that extends to the elliptic case. Our starting point is a formula for the total diﬀerential of eMPLs that is very similar to the total diﬀerential of ordinary MPLs [4]. The formula for the total diﬀerential shows that eMPLs satisfy a diﬀerential equation without a”
p-adic multiple zeta values I. (2004, background): “p-adic multiple zeta values I 259 Proof. At least we have two proofs [ Gon ].”
Galois symmetries of fundamental groupoids and noncommutative geometry (2002, background): “2;0|t 2)·IC(0;1|t 2) = IC(a 2 : 1|t 2)⊗IC(a 1 : a 2|t 1)·a−t2 2 Part b) follows from a) if we notice that aN = 1 provides at = 1 modulo N-torsion. The proposition is proved. Remark. In Theorem 4.5 of [G9] the reader can ﬁnd a diﬀerent way to write the formulas for the coproduct in depth 2 case. It i”
Towards multiple elliptic polylogarithms (2007, background): “There is a very precise conjectural picture [9, 11 ] of the algebraic rational relations that occur among them.”
From polygons and symbols to polylogarithmic functions (2011, methodology): “Multiple polylogarithms can be defined recursively, for n ≥ 0, via the iterated integral [1, 2]”