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In this paper, we present some new identities for multiple polylogarithm functions by using the methods of iterated integral computations of logarithm functions. Then, by applying these formulas obtained, we establish several duality formulas for Arakawa–Kaneko zeta values and Kaneko–Tsumura $$\eta $$ -values. At the end of the paper, we study a variant of Kaneko–Tsumura $$\eta $$ -function with r-complex variables and establish two formulas about the values of this variant; these two formulas were proved previously by Yamamoto.

In this paper we consider iterated integral representations of multiple polylogarithm functions and prove some explicit relations of multiple polylogarithm functions. Then we apply the relations obtained to find numerous formulas of alternating multiple zeta values in terms of unit-exponent alternating multiple zeta values. In particular, we prove several conjectures given by Borwein-Bradley-Broadhurst [3] , and give some general results.

This is the second installment in a multi-part series starting with Corwin–Dan-Cohen [arXiv:1812.05707v3]. Building on previous work by Dan-Cohen–Wewers, Dan-Cohen, and F. Brown, we push the computational boundary of our explicit motivic version of Kim’s method in the case of the thrice punctured line over an open subscheme of Spec ⁡ Z \operatorname {Spec}\mathbb {Z} . To do so, we develop a refined version of the algorithm of Dan-Cohen–Wewers tailored specifically to this case. We also commit ourselves fully to working with the polylogarithmic quotient. This allows us to restrict our calculus with motivic iterated integrals to the so-called depth- 1 1 part of the mixed Tate Galois group studied extensively by Goncharov. An application was given in Corwin–Dan-Cohen [arXiv:1812.05707v3], where we verified Kim’s conjecture in an interesting new case.

We introduce the notion of a matching Rota-Baxter algebra motivated by the recent work on multiple pre-Lie algebras arising from the study of algebraic renormalization of regularity structures [10] , [19] . This notion is also related to iterated integrals with multiple kernels and solutions of the associative polarized Yang-Baxter equation. Generalizing the natural connection of Rota-Baxter algebras with dendriform algebras to matching Rota-Baxter algebras, we obtain the notion of matching dendriform algebras. As in the classical case of one operation, matching Rota-Baxter algebras and matching dendriform algebras are related to matching pre-Lie algebras which coincide with the aforementioned multiple pre-Lie algebras. More general notions and results on matching tridendriform algebras and matching PostLie algebras are also obtained.

We generalize the definition of overconvergent $p$-adic multiple polylogarithms and of $p$-adic cyclotomic multiple zeta values and we prove a bound on their norm. A byproduct of the proof is a characterization of these objects in terms of certain regularized $p$-adic iterated integrals. The generalization of the definition consists in replacing the underlying Frobenius structure by its iterations. The bound on the norms of overconvergent $p$-adic multiple polylogarithms that we obtain is a prerequisite for our subsequent papers on $p$-adic cyclotomic multiple zeta values., 25 pages. This is Part I-1 of "$p$-adic cyclotomic multiple zeta values and $p$-adic pro-unipotent harmonic actions"

Let F ∈ ℂ [ x , y ] be a polynomial, γ ( z ) ∈ π 1 ( F − 1 ( z ) ) a non-trivial cycle in a generic fiber of F and let ω be a polynomial 1-form, thus defining a polynomial deformation d F + e ω = 0 of the integrable foliation given by F . We study different invariants: the orbit depth k , the nilpotence class n , the derivative length d associated with the couple ( F , γ ) . These invariants bind the length l of the first nonzero Melnikov function of the deformation d F + e ω along γ . We analyze the variation of the aforementioned invariants in a simple but informative example, in which the polynomial F is defined by a product of four lines. We study as well the relation of this behavior with the length of the corresponding Godbillon–Vey sequence. We formulate a conjecture motivated by the study of this example.

We consider the genus-one curves which arise in the cuts of the sunrise and in the elliptic double-box Feynman integrals. We compute and compare invariants of these curves in a number of ways, including Feynman parametrization, lightcone and Baikov (in full and loop-by-loop variants). We find that the same geometry for the genus-one curves arises in all cases, which lends support to the idea that there exists an invariant notion of genus-one geometry, independent on the way it is computed. We further indicate how to interpret some previous results which found that these curves are related by isogenies instead., Comment: 24 pages, 2 figures

Let $$P_n^{(\alpha ,\beta )}$$ be the n-th monic Jacobi polynomial with $$\alpha ,\beta >-1$$. Given m numbers $$\omega _1, \ldots , \omega _m \in \mathbb {C} \setminus [-1,1]$$, let $$\varOmega _m=( \omega _1, \ldots , \omega _m)$$ and $$\mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m}$$ be the m-th iterated integral of $$\frac{(n+m)!}{n!}\;P^{(\alpha ,\beta )}_{n}$$ normalized by the conditions $$\begin{aligned} \frac{\mathrm{d}^k\,\mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m}}{\mathrm{d}z^k}(\omega _{m-k})=0, \; \text { for } \; k=0,1,\ldots , m-1. \end{aligned}$$The main purpose of the paper is to study the algebraic and asymptotic properties of the sequence of monic polynomials $$\{ \mathscr {P}_{n,m,\varOmega _m}^{(\alpha , \beta )}\}_n$$. In particular, we obtain the relative asymptotic for the ratio of the sequences $$\{ \mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m}\}_n$$ and $$\{P_n^{(\alpha ,\beta )}\}_n$$. We prove that the zeros of these polynomials accumulate on a suitable ellipse.

We study the algebra of certain q-series, called bi-brackets, whose coefficients are given by weighted sums over partitions. These series incorporate the theory of modular forms for the full modular group as well as the theory of multiple zeta values (MZV) due to their appearance in the Fourier expansion of regularized multiple Eisenstein series. Using the conjugation of partitions we obtain linear relations between bi-brackets, called the partition relations, which yield naturally two different ways of expressing the product of two bi-brackets similar to the stuffle and shuffle product of multiple zeta values. In a recent work, the author and K. Tasaka defined (shuffle) regularized multiple Eisenstein series G ⧢ , by using an explicit connection to the coproduct on formal iterated integrals introduced by Goncharov. These satisfy the shuffle product formula. Applying the same concept for the coproduct on quasi-shuffle algebras enables us to define (stuffle) regularized multiple Eisenstein series G ⁎ satisfying the stuffle product formula. We show that both G ⧢ and G ⁎ are given by linear combinations of products of MZV and bi-brackets.

In this paper we consider iterated integrals on P 1 ∖ { 0 , 1 , ∞ , z } and define a class of Q -linear relations among them, which arises from the differential structure of the iterated integrals with respect to z. We then define a new class of Q -linear relations among the multiple zeta values by taking their limits of z → 1 , which we call confluence relations (i.e., the relations obtained by the confluence of two punctured points). One of the significance of the confluence relations is that it gives a rich family and seems to exhaust all the linear relations among the multiple zeta values. As a good reason for this, we show that confluence relations imply both the regularized double shuffle relations and the duality relations.

When the elements of a random vector take any real values, formulas of product moments are obtained for continuous and discrete random variables using distribution/survival functions. The r...

The Hermite–Hadamard inequality was extended using iterated integrals by Retkes [Acta Sci Math (Szeged) 74:95–106, 2008]. In this paper we further extend the main results of the above paper for convex and also for s-convex functions in the second sense.

We investigate linear relations among a class of iterated integrals on the Riemann sphere minus four points $0,1,z$ and $\infty$. Generalization of the duality formula and the sum formula for multiple zeta values to the iterated integrals are given., Comment: 11 pages

We introduce a novel class of features for multidimensional time series, that are invariant with respect to transformations of the ambient space. The general linear group, the group of rotations and the group of permutations of the axes are considered. The starting point for their construction is Chen's iterated-integral signature., Comment: complete rewrite of Section 3.3

We prove a formula among finite multiple zeta values with four parameters. This formula is proved by using the iterated integral expression of the multiple polylogarithms. By specializing these parameters, we give some weighted sum formulas for finite multiple zeta values.

We introduce eight “flavors” of cyclic homology of a mixed complex and study their properties. In particular, we determine their behavior with respect to Chen’s iterated integrals.

We describe some particular finite sums of multiple zeta values which arise from J. Ecalle's "arborification", a process which can be described as a surjective Hopf algebra morphism from the Hopf algebra of decorated rooted forests onto a Hopf algebra of shuffles or quasi-shuffles. This formalism holds for both the iterated sum picture and the iterated integral picture. It involves a decoration of the forests by the positive integers in the first case, by only two colours in the second case., From a talk at the ESF exploratory workshop "New approaches to Multiple Zeta Values", ICMAT, Madrid, Sept-Oct. 2013

We study the zero location and the asymptotic behavior of iterated integrals of polynomials. Borwein–Chen–Dilcher’s polynomials play an important role in this issue. For these polynomials we find their strong asymptotics and give the limit measure of their zero distribution. We apply these results to describe the zero asymptotic distribution of iterated integrals of ultraspherical polynomials with parameters $$(2\alpha +1)/2$$ , $$\alpha \in \mathbb {Z}_{+}$$ .

We consider the on-shell mass and wave function renormalization constants $Z_m^{\rm OS}$ and $Z_2^{\rm OS}$ up to three-loop order allowing for a second non-zero quark mass. We obtain analytic results in terms of harmonic polylogarithms and iterated integrals with the additional letters $\sqrt{1-\tau^2}$ and $\sqrt{1-\tau^2}/\tau$ which extends the findings from Ref. [1] where only numerical expressions are presented. Furthermore, we provide terms of order ${\cal O}(\epsilon^2)$ and ${\cal O}(\epsilon)$ at two- and three-loop order which are crucial ingrediants for a future four-loop calculation. Compact results for the expansions around the zero-mass, equal-mass and large-mass cases allow for a fast high-precision numerical evaluation., Comment: 14 pages, 3 figures

Let X be a smooth projective curve of genus $$\ge 1$$ over $${{\mathbb {C}}}$$ , and $$e,\infty \in X({{\mathbb {C}}})$$ be distinct points. Let $$L_n$$ be the mixed Hodge structure of functions on $$\pi _1(X-\{\infty \},e)$$ given by iterated integrals of length $$\le n$$ (as defined by Hain). Building on a work of Darmon et al. (Publications Mathematiques de Besancon 2:19–46, 2012), we express the mixed Hodge extension $${\mathbb {E}}^\infty _{n,e}$$ given by the weight filtration on $$\frac{L_n}{L_{n-2}}$$ as the Abel–Jacobi image of a null-homologous algebraic cycle on $$X^{2n-1}$$ . This algebraic cycle is constructed using the different embeddings of $$X^{n-1}$$ into $$X^n$$ . As a corollary, we show that the extension $${\mathbb {E}}^\infty _{n,e}$$ determines the point $$\infty \in X-\{e\}$$ . When $$n=2$$ , our main result is a strengthening of a theorem of Darmon et al. (Publications Mathematiques de Besancon 2:19–46, 2012). In the final section we assume that $$X,e,\infty $$ are defined over a subfield K of $${{\mathbb {C}}}$$ . Generalizing a construction in (Darmon et al., Publications Mathematiques de Besancon 2:19–46, 2012), we use the extension $${\mathbb {E}}^\infty _{n,e}$$ to define a family of K-rational points on the Jacobian of X parametrized by $$(n-1)$$ -dimensional algebraic cycles on $$X^{2n-2}$$ defined over K.

Criteria for the boundedness of sublinear integral two-kernel operators of iterated type on cones of monotone functions in Lebesgue spaces on the real semiaxis are given.

In this paper we study polynomial Hamiltonian systems d F = 0 in the plane and their small perturbations: d F + ϵ ω = 0 . The first nonzero Melnikov function M μ = M μ ( F , γ , ω ) of the Poincare map along a loop γ of d F = 0 is given by an iterated integral [3] . In [7] , we bounded the length of the iterated integral M μ by a geometric number k = k ( F , γ ) which we call orbit depth. We conjectured that the bound is optimal. Here, we give a simple example of a Hamiltonian system F and its orbit γ having infinite orbit depth. If our conjecture is true, for this example there should exist deformations d F + ϵ ω with arbitrary high length first nonzero Melnikov function M μ along γ. We construct deformations d F + ϵ ω = 0 whose first nonzero Melnikov function M μ is of length three and explain the difficulties in constructing deformations having high length first nonzero Melnikov functions M μ .

This paper explains a method to calculate the coefficients of the Alekseev–Torossian associator as linear combinations of iterated integrals of Kontsevich weight forms of Lie graphs.

We introduce Schur multiple zeta functions which interpolate both the multiple zeta and multiple zeta-star functions of the Euler-Zagier type combinatorially. We first study their basic properties including a region of absolute convergence and the case where all variables are the same. Then, under an assumption on variables, some determinant formulas coming from theory of Schur functions such as the Jacobi-Trudi, Giambelli and dual Cauchy formula are established with the help of Macdonald's ninth variation of Schur functions. Moreover, we investigate the quasi-symmetric functions corresponding to the Schur multiple zeta functions. We obtain the similar results as above for them and, furthermore, describe the images of them by the antipode of the Hopf algebra of quasi-symmetric functions explicitly. Finally, we establish iterated integral representations of the Schur multiple zeta values of ribbon type, which yield a duality for them in some cases., Comment: 42 pages, 2 figures

In this paper we present some new identities for multiple polylogarithms (abbr. MPLs) and multiple harmonic star sums (abbr. MHSSs) by using the methods of iterated integral computations of logarithm functions. Then, by applying these formulas obtained, we establish some explicit relations between Kaneko-Yamamoto type multiple zeta values (abbr. K-Y MZVs), multiple zeta values (abbr. MZVs) and MPLs. Further, we find some explicit relations between MZVs and multiple zeta star values (abbr. MZSVs). Furthermore, we define an Apery-type variant of MZSVs ζ B ⋆ ( k ) (called multiple zeta B-star values, abbr. MZBSVs) which involve MHSSs and central binomial coefficients, and establish some explicit connections among MZVs, alternating MZVs and MZBSVs by using the method of iterated integrals. Finally, some interesting consequences and illustrative examples are presented.

We report on an implementation within GiNaC to evaluate iterated integrals related to elliptic Feynman integrals numerically to arbitrary precision within the region of convergence of the series expansion of the integrand. The implementation includes iterated integrals of modular forms as well as iterated integrals involving the Kronecker coefficient functions $g^{(k)}(z,\tau)$. For the Kronecker coefficient functions iterated integrals in $d\tau$ and $dz$ are implemented. This includes elliptic multiple polylogarithms., Comment: 37 pages, v2: version to be published

We study the multiple Eisenstein series introduced by Gangl, Kaneko and Zagier. We give a proof of (restricted) finite double shuffle relations for multiple Eisenstein series by revealing an explicit connection between the Fourier expansion of multiple Eisenstein series and the Goncharov co-product on Hopf algebras of iterated integrals.

This paper contains examples of shuffle relations among multiple Dedekind zeta values. Dedekind zeta values were defined by the author in his paper "Multiple Dedekind zeta functions". Here we concentrate on the cases of real or imaginary quadratic fields with up to double iteration. We give examples of integral shuffle relation in terms of iterated integrals over membranes and of infinite sum shuffle relation, sometimes called stuffle relation. Using both types of shuffles for the product $\zeta_{K,C}(2)\zeta_{K,C}(2)$, we find relations among multiple Dedekind zeta values for real and for imaginary quadratic fields., Comment: Defines more general multiple Dedekind zeta values; contains 43 diagrams; 25 pages

Numerous examples of functional relations for multiple polylogarithms are known. For elliptic polylogarithms, however, tools for the exploration of functional relations are available, but only very few relations are identified. Starting from an approach of Zagier and Gangl, which in turn is based on considerations about an elliptic version of the Bloch group, we explore functional relations between elliptic polylogarithms and link them to the relations which can be derived using the elliptic symbol formalism. The elliptic symbol formalism in turn allows for an alternative proof of the validity of the elliptic Bloch relation. While the five-term identity is the prime example of a functional identity for multiple polylogarithms and implies many dilogarithm identities, the situation in the elliptic setup is more involved: there is no simple elliptic analogue, but rather a whole class of elliptic identities., Comment: 39 pages, 5 appendices, added references and corrected typos

In this paper we show that the iterated integrals on products of one variable multiple polylogarithms from 0 to 1 are actually multiple zeta values if they are convergent. In the divergent case, we define regularized iterated integrals from 0 to 1. By the same method, we show that the regularized iterated integrals are also multiple zeta values. As an application, we give new series representations for multiple zeta values and calculate some interesting examples of iterated integrals., Comment: 18 pages

We present the results for two-loop massive kite master integrals with elliptics in terms of iterated integrals with algebraic kernels. The key ingredients are new integral representations for sunset subgraphs in $d=4-2\epsilon$ and $d=2-2\epsilon$ dimensions together with differential equations for considered kite master integrals in $A+B\epsilon$ form. The obtained results can be easily generalized to all orders in $\epsilon$-expansion and show that the class of functions defined as iterated integrals with algebraic kernels may be large enough for writing down results for a large class of massive Feynman diagrams., Comment: 21 page, 4 figures

Iterated integrals of paths arise frequently in the study of the Taylor's expansion for controlled differential equations. We will prove a factorial decay estimate, conjectured by M. Gubinelli, for the iterated integrals of non-geometric rough paths. We will explain, with a counter example, why the conventional approach of using the neoclassical inequality fails. Our proof involves a concavity estimate for sums over rooted trees and a non-trivial extension of T. Lyons' proof in 1994 for the factorial decay of iterated Young's integrals., Comment: 26 pages, 2 figures. Accepted version

We give a new proof of the duality of multiple zeta values, which makes no use of the iterated integrals. The same method is also applicable to Ohno's relation for ($q$-)multiple zeta values., 5 pages, to appear in Int. J. Number Theory

We consider a generalization of elliptic multiple zeta values, which we call twisted elliptic multiple zeta values. These arise as iterated integrals on an elliptic curve from which a rational lattice has been removed. At the cusp, twisted elliptic multiple zeta values are shown to degenerate to cyclotomic multiple zeta values in the same way as elliptic multiple zeta values degenerate to classical multiple zeta values. We investigate properties of twisted elliptic multiple zeta values and consider them in the context of the non-planar part of the four-point one-loop open-string amplitude., 28 + 19 pages, 6 figures, v2: explanations added, published version

In this paper, we prove certain algebraic identities, which correspond to differentiation of the shuffle relation, the stuffle relation, and the relations which arise from Mobius transformations of iterated integrals. These formulas provide fundamental and useful tools in the study of iterated integrals on a punctured projective line.

We present a general expression for any term of the Magnus series as an iterated integral of a linear combination of independent right-nested commutators with given coefficients. The relation with the Malvenuto--Reutenauer Hopf algebra of permutations is also discussed., 16 pages

We present a generalization of the symbol calculus from ordinary multiple polylogarithms to their elliptic counterparts. Our formalism is based on a special case of a coaction on large classes of periods that is applied in particular to elliptic polylogarithms and iterated integrals of modular forms. We illustrate how to use our formalism to derive relations among elliptic polylogarithms, in complete analogy with the non-elliptic case. We then analyze the symbol alphabet of elliptic polylogarithms evaluated at rational points, and we observe that it is given by Eisenstein series for a certain congruence subgroup. We apply our formalism to hypergeometric functions that can be expressed in terms of elliptic polylogarithms and show that they can equally be written in terms of iterated integrals of Eisenstein series. Finally, we present the symbol of the equal-mass sunrise integral in two space-time dimensions. The symbol alphabet involves Eisenstein series of level six and weight three, and we can easily integrate the symbol in terms of iterated integrals of Eisenstein series., Comment: 65 pages

We reconsider the two-loop electron self-energy in quantum electrodynamics. We present a modern calculation, where all relevant two-loop integrals are expressed in terms of iterated integrals of modular forms. As boundary points of the iterated integrals we consider the four cases $p^2=0$, $p^2=m^2$, $p^2=9m^2$ and $p^2=\infty$. The iterated integrals have $q$-expansions, which can be used for the numerical evaluation. We show that a truncation of the $q$-series to order ${\mathcal O}(q^{30})$ gives numerically for the finite part of the self-energy a relative precision better than $10^{-20}$ for all real values $p^2/m^2$., Comment: 45 pages, v2: version to be published

International audience; We consider small polynomial deformations of integrable systems of the form $dF=0, F\in\mathbb{C}[x,y]$ and the first nonzero term $M_\mu$ of the displacement function $\Delta(t,\epsilon)=\sum_{i=\mu}M_i(t)\epsilon^i$ along a cycle $\gamma(t)\in F^{-1}(t)$. It is known that $M_\mu$ is an iterated integral of length at most $\mu$. The bound $\mu$ epends on the deformation of $dF$.In this paper we give a universal bound for the length of the iterated integral expressing the first nonzero term $\mu$ depending only on the geometry of the unperturbed system $dF=0$. The result generalizes the result of Gavrilov and Iliev providing a sufficient condition for $M_\mu$ to be given by an abelian integral, i.e., by an iterated integral of length 1. We conjecture that our bound is optimal.

We give new instances where Chabauty–Kim sets can be proved to be finite, by developing a notion of “generalised height functions” on Selmer varieties. We also explain how to compute these generalised heights in terms of iterated integrals and give the 1st explicit nonabelian Chabauty result for a curve $X/\mathbb{Q}$ whose Jacobian has Mordell–Weil rank larger than its genus.

In this paper we show that certain Feynman integrals can be expressed as linear combinations of iterated integrals of modular forms to all orders in the dimensional regularisation parameter $\varepsilon$ . We discuss explicitly the equal mass sunrise integral and the kite integral. For both cases we give the alphabet of letters occurring in the iterated integrals. For the sunrise integral we present a compact formula, expressing this integral to all orders in $\varepsilon$ as iterated integrals of modular forms., 53 pages, v2: section on relation to Gamma_1(6) added, v3: section on the choice of the periods added

In this paper we prove some new identities for multiple zeta values and multiple zeta star values of arbitrary depth by using the methods of integral computations of logarithm function and iterated integral representations of series. By applying the formulas obtained, we prove that the multiple zeta star values whose indices are the sequences $$(\bar{1},\{1\}_m,\bar{1})$$ and $$(2,\{1\}_m,\bar{1})$$ can be expressed polynomially in terms of zeta values, polylogarithms and $$\ln (2)$$ . We also evaluate several restricted sums involving multiple zeta values.

We study the algebra $\mathcal{I}^{QM}$ of iterated integrals of quasimodular forms for $\operatorname{SL}_2(\mathbb{Z})$, which is the smallest extension of the algebra $QM_{\ast}$ of quasimodular forms, which is closed under integration. We prove that $\mathcal{I}^{QM}$ is a polynomial algebra in infinitely many variables, given by Lyndon words on certain monomials in Eisenstein series. We also prove an analogous result for the $M_{\ast}$-subalgebra $\mathcal{I}^{M}$ of $\mathcal{I}^{QM}$ of iterated integrals of modular forms., v2, minor changes, to appear in Algebra and Number Theory

In this paper we define multiple Dedekind zeta values (MDZV), using a new type of iterated integrals, called iterated integrals on a membrane. One should consider MDZV as a number theoretic generalization of Euler's multiple zeta values. Over imaginary quadratic fields MDZV capture, in particular, multiple Eisenstein series (Gangl, Kaneko and Zagier). We give an analogue of multiple Eisenstein series over real quadratic field and an alternative definition of values of multiple Eisenstein-Kronecker series (Goncharov). Each of them is a special case of multiple Dedekind zeta values. MDZV are interpolated into functions that we call multiple Dedekind zeta functions (MDZF). We show that MDZF have integral representation, can be written as infinite sum, and have analytic continuation. We compute explicitly the value of a multiple residue of certain MDZF over a quadratic number field at the point (1,1,1,1). Based on such computations, we state two conjectures about MDZV., Comment: This version has substantial improvements in the content and the style. There are more details about the analytic continuation together with new examples of multiple residues. 43 pages

A multiple zeta value can always be represented by its Drinfel’d integral. If we add some factors appeared in the integrand of the integral representation of the multiple zeta value, it would still represent a linear combination of multiple zeta values, but the depths and weights may decrease. In this paper, we shall investigate some of multiple zeta values obtained from Drinfel’d integral with additional factors aforementioned and study a class of deformation of multiple zeta values. Results are then obtained as analogues or generalizations of the sum formula of multiple zeta values.

For a polynomial P, we consider the sequence of iterated integrals of ln P(x). This sequence is expressed in terms of the zeros of P(x). In the special case of ln (1 + x2), arithmetic properties of certain coefficients arising are described. Similar observations are made for ln (1 + x3).

Let L be a finite Galois extension of a number field K. Let G:= Gal(L/K). Let z1,…, zN ∊ L* \ {1} and let m1 …, mN ∊ ℚl. Let us assume that the linear combination of l-adic polylogarithms (constructed in some given way) is a cocycle on GL and that the formal sum is G-invariant. Then we show that cn determines a unique cocycle sn on GK. We also prove a weak version of Zagier conjecture for l-adic dilogarithm. Finally we show that if c2 is “motivic” (m1,…, mN ∊ ℚ) then s2 is also “motivic”.

We study an elliptic analogue of multiple zeta values, the elliptic multiple zeta values of Enriquez, which are the coefficients of the elliptic KZB associator. Originally defined by iterated integrals on a once-punctured complex elliptic curve, it turns out that they can also be expressed as certain linear combinations of indefinite iterated integrals of Eisenstein series and multiple zeta values. In this paper, we prove that the $\mathbb{Q}$-span of these elliptic multiple zeta values forms a $\mathbb{Q}$-algebra, which is naturally filtered by the length and is conjecturally graded by the weight. Our main result is a proof of a formula for the number of $\mathbb{Q}$-linearly independent elliptic multiple zeta values of lengths one and two for arbitrary weight., Comment: 22 pages

International audience; Recently, one of the authors gave an algorithm for calculating the first nonzero Poincaré-Pontryagin function of a small polynomial perturbation of a polynomial Hamiltonian, under a generic hypothesis. We generalize this algorithm and show that any Poincaré-Pontryagin function of order l, denoted by Ml, can be written as a sum of an iterated integral of length at most l and of a combination of all previous Poincaré-Pontryagin functions, M1, M2, …, Ml-1, and their derivatives. This extends some results obtained recently and allows to identify the Bautin ideal with the ideal generated by iterated integrals.

We continue to study l-adic iterated integrals introduced in the first part. We shall show that the l-adic iterated integrals satisfy essentially the same functional equations as the classical complex iterated integrals. Next we are studying l-adic analogs of classical polylogarithms.