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Finite and \(p\)-adic polylogarithms. (English) Zbl 1062.11041
The author proves a slightly modified version of an unpublished formula conjectured by M. Kontsevich between the \(n\)-th finite polylogarithms \(\text{li}_n(z)\) and the infinitesimal polylogarithm introduced by J.-L. Cathelineau.
More precisely, let \(p\) be a prime number and \(n \geq 1\) be an integer. The \(n\)-th finite polylogarithm is the polynomial function \(\text{li}_n(z) = \sum_{k=1}^{p-1} z^k/k^n \in \mathbb F_p[x]\). It was introduced for \(n=1\) by M. Kontsevich, who proved that it satisfies a 4-term functional equation known as the fundamental equation of information theory, which is also satified by the so-called infinitesimal dilogarithm \(-(x\log| x| +(1-x)\log| 1-x|\). Infinitesimal polylogarithms were defined by J.-L. Cathelineau [Ann. Inst. Fourier 46, 1327–1347 (1996; Zbl 0861.19003)], who found that they also satisfy interesting functional equation. Inspired by the works of P. Elbaz-Vincent and H. Gangl [Compos. Math. 130, 161–210 (2002; Zbl 1062.11042)], M. Kontsevich conjectured that the finite polylogarithms may be a reduction of an infinitesimal version of the locally analytic \(p\)-adic polylogarithm \(\text{Li}_n|\mathbb C_p \mapsto \mathbb C_p\), as defined by R. Coleman [Invent. Math. 69, 171–208 (1982; Zbl 0516.12017)].
The main result of the paper under review is the following theorem: Set \(D = z(1-z)d/dz\); let \(W\) be the group of Witt vectors on \(\mathbb F_p\) and put \(X = \{z \in W \mid | z| = | z-1| = 1 \}\). For every \(n>1\), let \(F_n(z) = \sum_{k=0}^{n-1} a_k \log ^k(z) \text{Li}_{n-k}(z)\), with \(a_0=-n\) and \(a_k = (-1)^k/(k-1)! \, + \, (-1)^{k+1}n/k!\,\) for \(k>1\). Then for \(p>n+1\) one has {(i)} \(Df_n(X) \subset p^{n-1}W\) and {(ii)} \(p^{1-n}DF_n(z) \equiv \text{li}_{n-1}(z^{1/p}) \mod p\). Furthermore the choice of the coefficients \(a_k\) is the the unique choice in \(\mathbb Q\) for which the theorem holds for all \(p>n+1\).
This theorem is used by P. Elbaz-Vincent and H. Gangl [loc. cit.] to deduce functional equations of finite polylogarithms from those of complex polylogarithms.

MSC:
11G55 Polylogarithms and relations with \(K\)-theory
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
33B30 Higher logarithm functions
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