For $x\in ℂ$, $\left|x\right|\<1$, $s\in \mathbb{N}$, let ${\mathrm{Li}}_s\left(x\right)$ be the $s$-th polylogarithm of $x$. We prove that for any non-zero algebraic number $\alpha$ such that $\left|\alpha \right|\<1$, the $\mathbb{Q}\left(\alpha \right)$-vector space spanned by $1,{\mathrm{Li}}_1\left(\alpha \right),{\mathrm{Li}}_2\left(\alpha \right),\cdots$ has infinite dimension. This result extends a previous one by Rivoal for rational $\alpha$. The main tool is a method introduced by Fischler and Rivoal, which shows the coefficients of the polylogarithms in the relevant series to be the unique solution of a suitable Padé approximation problem.

We prove a conjecture due to Goncharov and Manin which states that the periods of the moduli spaces ${𝔐}_{0,n}$ of Riemann spheres with $n$ marked points are multiple zeta values. We do this by introducing a differential algebra of multiple polylogarithms on ${𝔐}_{0,n}$ and proving that it is closed under the operation of taking primitives. The main idea is to apply a version of Stokes’ formula iteratively to reduce each period integral to multiple zeta values. We also give a geometric interpretation of the double shuffle...

In this paper, we give an explicit description of the de Rham and $p$-adic polylogarithms for elliptic curves using the Kronecker theta function. In particular, consider an elliptic curve $E$ defined over an imaginary quadratic field $𝕂$ with complex multiplication by the full ring of integers ${𝒪}_{𝕂}$ of $𝕂$. Note that our condition implies that $𝕂$ has class number one. Assume in addition that $E$ has good reduction above a prime $p\ge 5$ unramified in ${𝒪}_{𝕂}$. In this case, we prove that the specializations of the $p$-adic elliptic...

Motivated by a renewed interest for the “additive dilogarithm” appeared recently, the purpose of this paper is to complete calculations on the tangent complex to the Bloch-Suslin complex, initiated a long time ago and which were motivated at the time by scissors congruence of polyedra and homology of ${\mathrm{SL}}_2$. The tangent complex to the trilogarithmic complex of Goncharov is also considered.

Nous montrons une version explicite du théorème de Beilinson pour la courbe modulaire $X_1\left(N\right)$. Ce résultat est la première étape d’un travail reliant, d’une part, la valeur en $2$ de la fonction $L$ d’une forme primitive de poids $2$, et d’autre part, la fonction dilogarithme associée à la courbe modulaire correspondante, dans l’esprit de la conjecture de Zagier pour les courbes elliptiques. Comme corollaire de notre théorème, dans le cas où $N$ est premier, nous répondons à une question de Schappacher et Scholl...