This paper establishes the equivalence of two criteria for functional equations of complex and \(l\)-adic iterated integrals. The first, given by Z. Wojtkowiak in a series of papers [“The basic structure of polylogarithmic functional equations” in “Structural properties of polylogarithms”, L. Lewin (ed.), AMS Math. Surv. Monogr. 37, 205–231 (1991; Zbl 0745.33009) and [Nagoya Math. J. 177, 117–153 (2005; Zbl 1161.11363)] is given in terms of induced morphisms on fundamental groups. More precisely, as in Theorem 1.1 of the paper, if \(K \subset \mathbb{C}\) is a subfield; \(X:= \mathbb{P}^1_K \backslash\{a_1, \ldots, a_M, \infty\}\); \(Y:= \mathbb{P}^1_K\backslash\{b_1, \ldots, b_N, \infty\}\); and \(\nu\) is any \(K\)-rational basepoint on \(X\); and algebraic morphisms \(f_i:X \rightarrow Y\) for \(1\leq i \leq m\) and homomorphisms \(\psi_i: \text{gr}^n_{\Gamma}(\pi_1(Y(\mathbb{C}), f_i(\nu)) \rightarrow \mathbb{Z}\) are given along with constants \(c_1, \ldots, c_m \in \mathbb{Z}\) satisfying \[ \sum_{i=1}^{m}c_i \psi_i \circ \text{gr}^n_{\Gamma}(f_{i*}) = 0 \] where \(f_{i*}: \pi_1(X(\mathbb{C}), \nu) \rightarrow \pi_1(Y(\mathbb{C}), f_{i}(\nu))\) denote the induced homomorphisms and \(\text{gr}^n_{\Gamma}\) is the \(n\)th graded piece with respect to the lower central filtration, then \[ \sum_{i=1}^m c_i {\mathcal{L}}_Y^{\psi}(f_i(z), f_i(x)) \] is zero modulo lower degree terms. Here \({\mathcal{L}}_Y^{\psi}(f_i(z), f_i(x))\) denote either complex or \(l\)-adic iterated integrals on \(Y\), according to the setting.
The second criterion is due to D. Zagier [Arithmetic algebraic geometry, Proc. Conf., Texel/Neth. 1989, Prog. Math. 89, 391–430 (1991; Zbl 0728.11062)] in the case of classical polylogarithms but is generalized by the authors to arbitrary iterated integrals. It is given in terms of a generalized Bloch group (which as mentioned in the introduction is a “certain tensor of symmetric and wedge products of multiplicative groups of fields”), and here is expressed in terms of the dual of a multi-linearized version of the classical Kummer pairing, say \(\widehat{\kappa_{\otimes n}}\): in the above notation, \[ \sum_{i=1}^m c_i \widehat{\kappa_{\otimes n}}(\psi_i)(f_i) = 0. \] The precise error terms are also given, and in the \(l\)-adic case Wojtkowiak’s earlier results are extended to the case when \(X\) is an arbitrary nonsingular variety. Furthermore, the authors give a mechanism for computing a functional equation from a family of morphisms on the fundamental group of varieties, and in some \(l\)-adic examples show that non-trivial arithmetic relations arise between certain generalized Soulé characters.
For the entire collection see [Zbl 1237.11001].