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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The author proves a slightly modified version of an unpublished formula conjectured by M. Kontsevich between the $$n$$-th finite polylogarithms $$\text{li}_n(z)$$ and the infinitesimal polylogarithm introduced by J.-L.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
##### Citations:
Zbl 0861.19003; Zbl 0516.12017; Zbl 1062.11042
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