aa r X i v : . [ m a t h . N T ] J un MULTIPLE ELLIPTIC POLYLOGARITHMS
FRANCIS BROWN AND ANDREY LEVIN
Abstract.
We study the de Rham fundamental group of the configuration space E ( n ) of n + 1 marked points on a complex elliptic curve E , and define multipleelliptic polylogarithms. These are multivalued functions on E ( n ) with unipotentmonodromy, and are constructed by a general averaging procedure. We showthat all iterated integrals on E ( n ) , and in particular the periods of the unipotentfundamental group of the punctured curve E\{ } , can be expressed in terms ofthese functions. Introduction
Motivation.
Iterated integrals on the moduli space M ,n of curves of genus 0with n ordered marked points can be expressed in terms of multiple polylogarithms.These are defined for n , . . . , n r ∈ N by(1.1) Li n ,...,n r ( x , . . . , x r ) = X 1) at x i = 1, one obtains the multiple zeta values(1.2) ζ ( n , . . . , n r ) = X 2. These are of particular interest since they are the periodsof the fundamental group of P \{ , , ∞} , and generate the periods of all mixed Tatemotives over Z .The goal of this paper is to construct the elliptic analogues of the multiple poly-logarithms and to set up the necessary algebraic and analytic background required tostudy multiple elliptic zeta values. The former are iterated integrals on the configu-ration space E ( n ) of n + 1 marked points on a complex elliptic curve, i.e., the fiber ofthe map M ,n +1 → M , , where M ,m denotes the moduli space of curves of genus1 with m marked points. They generalize the classical elliptic polylogarithms studiedin [13], and are the universal periods of unipotent variations of mixed elliptic Hodgestructures.In a sequel to this paper, we shall study the multiple elliptic zeta values, obtainedby specializing multiple elliptic polylogarithms to the zero section of the universalelliptic curve. They define multivalued functions on M , which degenerate to ordinarymultiple zeta values at the cusp. The existence of these functions sheds light onthe structural relations between ordinary multiple zeta values, and in particular, therelation between double zetas and period polynomials for cusp forms [9].1.2. The rational case. Firstly we recall the definition of iterated integrals [6]. Let M be a smooth real manifold, and let ω , . . . , ω n denote smooth 1-forms on M . Let γ : [0 , → M be a smooth path, and write γ ∗ ω i = f i ( t ) dt for some smooth functions f i : [0 , → R , where 1 ≤ i ≤ n . The iterated integral of ω , . . . , ω n is defined by(1.3) Z γ ω . . . ω n = Z ≤ t n ≤ ... ≤ t ≤ f ( t ) . . . f n ( t n ) dt . . . dt n . Now let M = P \{ , , ∞} , and let ω = dzz and ω = dz − z . Let 0 < z < 1, and denotethe straight path from 0 to z by γ z . The initial point γ (0) does not in fact lie in M ,but the following iterated integral still makes sense nonetheless, and gives(1.4) Z γ z ω . . . ω | {z } n r − ω . . . ω . . . ω | {z } n − ω = Li n ,...,n r (1 , . . . , , z ) . This is easily proved by a series expansion of the forms ω . The periods of the fun-damental torsor of paths of M from 0 to 1 (with tangential basepoints 1, − z → 1. In the case n r ≥ 2, this yields(1.5) Z ω . . . ω | {z } n r − ω . . . ω . . . ω | {z } n − ω = ζ ( n , . . . , n r ) , as first observed by Kontsevich. Similarly, one can define the regularized iteratedintegral from 0 to 1 of any word in the one-forms ω , ω , and in every case, it is easyto show that it is a linear combination of multiple zeta values.To verify that all (homotopy invariant) iterated integrals on M are expressible interms of multiple polylogarithms requires Chen’s reduced bar construction. By Chen’sgeneral theory, the iterated integrals on a manifold M are described by the zerothcohomology of the bar construction on the de Rham complex of M . To write thisdown explicitly for M = P \{ , , ∞} , we can use the rational model Q ⊕ ( Q ω ⊕ Q ω ) ֒ → Ω · DR ( P \{ , , ∞} ; Q )which is a quasi-isomorphism of differential graded algebras. From this one deducesthat H ( B (Ω · DR ( P \{ , , ∞} ; Q ))) ∼ = T c ( Q ω ⊕ Q ω ), where B is the bar complex,and T c denotes the tensor coalgebra. This is the Q -vector space generated by wordsin the forms ω , ω (equipped with the shuffle product and the coproduct for which ω , ω are primitive), and leads to integrals of the form (1 . π un ( P \{ , , ∞} , + , z ) are multiplepolylogarithms (1 . π un ( P \{ , , ∞} , + , − ) are multiple zetas(1 . r ≥ 1, let(1.6) M ,r +3 ( C ) = { ( t , . . . , t r ) ∈ C r : t i = 0 , , t i = t j } denote the moduli space of genus 0 curves with r + 3 marked points. One can likewisewrite down a rational model for the de Rham complex in terms of the one-forms dt i − dt j t i − t j , dt i − t i , dt i t i which satisfy certain quadratic relations due to Arnold. Forgetting a marked pointdefines a fibration M ,r +3 → M ,r +2 , and by general properties of the bar constructionof fibrations, one can likewise write down all homotopy invariant iterated integrals on M ,r +3 [4] and show that they are expressible in terms of the functions(1.7) I n ,...,n r ( t , . . . , t r ) = Li n ,...,n r (cid:0) t t , . . . , t r − t r , t r (cid:1) . ULTIPLE ELLIPTIC POLYLOGARITHMS 3 The purpose of this paper is to generalize the above picture for P \{ , , ∞} to thecase of a punctured complex elliptic curve E × . In particular, we compute the periodsof π un ( E × , ̺, ξ ), where ρ, ξ ∈ E × are finite basepoints (the case of tangential basepointsis similar and will be postponed to a later paper). There are two parts: first, to writedown the iterated integrals generalizing the left hand side of (1 . 4) using Chen’s generaltheory, and the second is to construct multiple elliptic polylogarithm functions whichcorrespond to the right-hand side of (1 . E ( n ) to construct the functions on E × .1.3. The elliptic case. Let E be an elliptic curve, viewed as the analytic manifold C / Z τ ⊕ Z , where τ ∈ C satisfies Im ( τ ) > 0. We first require a model for the de Rhamcomplex on E × . For this, we construct a universal family of smooth one-forms(1.8) ν, ω (0) , ω (1) , ω (2) , . . . ∈ A ( E × ) , where ω (0) , ν are closed and form a basis of H ( E × ) which is compatible with its Hodgestructure. The forms ω ( i ) , i ≥ dω ( i ) = ν ∧ ω ( i − for i ≥ . Note that a priori E × has no natural Q -structure on its de Rham complex. However,the forms (1 . 8) have good modularity and rationality properties as a function of themoduli τ , and there are good reasons to take X = graded Q -algebra spanned by the ν, ω ( i ) , i ≥ Q -model for the C ∞ -de Rham complex on E × , (indeed, X ⊗ Q C ֒ → A · ( E × ) is aquasi-isomorphism). We also define a higher-dimensional model for the configurationspace E ( n ) of n + 1 points on E . It is an elliptic version of Arnold’s theorem describingthe cohomology of the configuration space of n points in P .The next stage is to write down the bar construction of X , which defines a Q -structure on the iterated integrals on E × . The bar construction has a filtration by thelength, and the associated graded is just the tensor coalgebra on ω (0) and ν :(1.9) gr ℓ H ( B ( X )) ∼ = T c ( Q ω (0) ⊕ Q ν ) . The Massey products ω ( i ) , for i ≥ 1, give a canonical way to lift an element of (1 . 9) to H ( B ( X )), and thus enable us to write down explicitly all the iterated integrals on E × .These are indexed by any word in the two one-forms ω (0) and ν . The Hodge filtrationon the space of iterated integrals is related to the number of ν ’s. This completes thealgebraic description of the iterated integrals on E × .The main problem is then to write down explicit formulae for these iterated integrals,and for this we write the elliptic curve via its Jacobi uniformization E ∼ = C × /q Z , where q = exp(2 πiτ ). In order to construct multivalued functions on E × , the basicidea is to average a multivalued function on C × with respect to multiplication by q aswas done for the classical polylogarithms [2, 13]. However, applying this idea naivelyto the multiple polylogarithms in one variable (1 . 4) does not lead to elliptic functions.Instead, the correct approach is to view the multiple polylogarithms in r variables(1 . . M ,r +3 ∼ = C × × . . . × C × | {z } r \ diagonalsand average with respect to the group q Z × . . . × q Z ( r factors). Since polylogarithmshave logarithmic singularities at infinity, the straightforward average diverges, and so FRANCIS BROWN AND ANDREY LEVIN instead one must take a weighted average with respect to some auxilliary parameters u i to dampen the singularities. In short, one considers the functions(1.10) X m ,...,m r ∈ Z u m . . . u m r r I n ,...,n r ( q m t , . . . , q m r t r )which converge uniformly under some conditions on the u i . A considerable part of thispaper is devoted to studying the structure of the poles of (1 . 10) in the u i variables,which are related to the geometry of M ,r +3 and the asymptotics of the polylogarithms(1 . 7) at infinity. Finally, writing u i = exp(2 iπα i ) for 1 ≤ i ≤ r , the multiple ellipticpolylogarithms are defined to be the coefficients of (the finite part of) (1 . 10) withrespect to the α i . The analysis involved in this averaging procedure is quite generaland should apply to a class of functions of finite determination and moderate growthon certain toric varieties.The functions obtained in this way are multivalued functions on E ( n ) . By allowingsome of the arguments t i of (1 . 10) to degenerate to 1, we obtain multivalued functionson E × . By computing the differential equations satisfied by these functions, we seethat they are iterated integrals in the forms (1 . 8) and, using the description of the barconstruction of X , we deduce that all the iterated integrals on E × are of this form.An important caveat is that the rational structure that we define on the de Rhamfundamental group on E × is the correct Q -structure from the point of view of averagingfunctions in genus zero, but does not coincide with the canonical Q -structure in thespecial case when E is defined over Q (see § Plan of the paper. First, § E ( n ) . The second, consistingof sections 6, 7 and 8, concerns the procedure for averaging multiple polylogarithms.Since this is quite involved, we give a separate overview of the method in § . § . § 4, we define some differential graded algebras X n and proveby a Leray spectral sequence argument that they are Q -models for the de Rham com-plex on E ( n ) . In § X n to study Chen’s reduced bar constructionon E ( n ) , and hence obtain an algebraic description for the iterated integrals on E ( n ) .Some of the results of this section require some generalities on the bar construction ofdifferential graded algebras which we decided to relegate to a separate paper [3].In § 6, we study the general averaging procedure for functions on M ,n ( C ). This re-quires constructing a certain partial compactification of M ,n and analyzing the asymp-totics of series in the neighbourhood of boundary divisors. We apply this formalismto the classical multiple polylogarithms in § 7. In § 8, which is logically independentfrom the rest of the paper, we compute the asymptotics of the Debye polylogarithms atinfinity in terms of a certain coproduct. The Debye multiple polylogarithms (definition1) are essentially generating series of multiple polylogarithms and are useful for sim-plifying many formulae. In § 9, we treat the case of the classical elliptic polylogarithms(depth 1) and the double elliptic polylogarithms (depth 2) in detail. The two partsof the story recombine in § 10 where we prove that all iterated integrals on E × , withrespect to finite basepoints, are obtained by averaging.1.5. Related work. One of many motivations for this paper is the study of mixedelliptic motives. Since the Beilinson-Soul´e conjectures are currently unavailable in thiscase, our goal was to tease out the elementary consequences of such a theory, and in ULTIPLE ELLIPTIC POLYLOGARITHMS 5 particular, write down the underlying numbers and functions in the belief that theywill find applications in other parts of mathematics. We learned at a conference inBristol in 2011 that there has been recent progress in constructing categories of mixedelliptic motives [12, 17], and universal elliptic motives [10] with similar goals. In thispaper we completely neglected the Betti side, which is dual to the bar construction,and its relation to quantum groups and stable derivations. This is treated in [5, 8, 18].Somewhat further afield, it may also be helpful to point out related work in the profi-nite setting [16], and possible diophantine applications [11]. Acknowledgements . The first author is partially supported by ERC grant 257368.The second author is partially supported by ‘the National Research University HigherSchool of Economics’ academic fund number 11-01-0133 for 2012-2013, and also AGLaboratory GU-HSE, RF government grant, ag. 11 11.G34.31.0023. We thank theFondation des Sciences Math´ematiques de Paris, the Institut Poncelet in Moscow, theMPIM, and Humboldt University for support and hospitality.2. Preliminaries: the rational case Standard coordinates on M ,n +3 . Let M ,n +3 denote the moduli space ofcurves of genus 0 with n + 3 ordered marked points. By placing three of the markedpoints at 0 , , ∞ , it can be identified with an affine hyperplane complement:(2.1) M ,n +3 ∼ = { ( t , . . . , t n ) ∈ P \{ , , ∞} : t i = t j } . We refer to the coordinates t i as simplicial coordinates. We will often write t n +1 = 1.There is a smooth compactification M ,n +3 , such that the complement M ,n +3 \ M ,n +3 is a normal crossing divisor. Its irreducible components are indexed by partitions S ∪ T of the set of marked points, which we denote by S | T , or, T | S , where S ∩ T = ∅ and | S | , | T | ≥ 2. The corresponding divisor is isomorphic to M , | S | +1 × M , | T | +1 .2.2. Multiple polylogarithms. These are defined for n , . . . , n r ∈ N by(2.2) Li n ,...,n r ( x , . . . , x r ) = X 1, and therefore defines a family of holomorphicfunctions in the neighbourhood of the origin. We can also write these functions insimplicial coordinates, using standard notations:(2.3) I n ,...,n r ( t , . . . , t r ) = Li n ,...,n r (cid:16) t t , . . . , t r − t r , t r (cid:17) . We denote the quantities N = n + . . . + n r and r by the weight and the depth,respectively. There is an iterated integral representation(2.4) I n ,...,n r ( t , . . . , t r ) = ( − r Z ≤ σ ≤ ...σ r ≤ dσ σ − ρ . . . dσ N σ N − ρ N , where ( ρ , . . . , ρ N ) = ( t − , n − , t − , n − , . . . , t − r , n r − ), N = n + . . . + n r , and0 n denotes a string of n M ,n +3 ( C ) ⊂ ( C × ) n . The following differential equation is easily verified from (2 . dI ,..., ( t , . . . , t n ) = n X k =1 (cid:2) dI (cid:16) t k t k +1 (cid:17) − dI (cid:16) t k t k − (cid:17)(cid:3) I ,..., ( t , . . . , b t k , . . . , t n )(2.5) FRANCIS BROWN AND ANDREY LEVIN where by convention we take dI ( t / 0) = 0, t n +1 = 1 and t = 0. The differentialequations for the multiple polylogarithms I n ,...,n r ( t , . . . , t r ) in the general case areeasily computed and left to the reader, since they are not required in the sequel. Definition 1. The generating series of multiple Debye polylogarithms is:Λ r ( t , . . . , t r ; β , . . . , β r ) = t − β . . . t − β r r X m ,...,m r ≥ I m ,...,m r ( t , . . . , t r ) β m − . . . β m r − r One easily verifies from d Li n ( t ) = t − Li n − ( t ), valid for n ≥ 2, that d Λ ( t ; β ) = t − β d Li ( t ) . In general, they satisfy a differential equation which is entirely analogous to equation(2 . 5) for the multiple 1-logarithm. Namely, d Λ r ( t , . . , t r ; β , . . , β r ) == r X k =1 d Λ (cid:16) t k t k +1 , β k (cid:17) Λ r − ( t , . . , b t k , . . , t r ; β , . . , β k + β k +1 , . . , β r ) − r X k =2 d Λ (cid:16) t k t k − , β k (cid:17) Λ r − ( t , . . , b t k , . . , t r ; β , . . , β k − + β k , . . , β r )(2.6)where β r +1 = 0, so the last term in the first line is d Λ ( t r ; β r )Λ r − ( t , . . , t r − ; β , . . , β r − ).2.3. Asymptotics of regular nilpotent connections. Let X be a smooth projec-tive complex variety, let D ⊂ X be a simple normal crossing divisor, and let U = X \ D .Let V ⊂ X be a simply connected open set and let z , . . . , z n denote local coordinateson V such that V ∩ D = ∪ ki =1 { z i = 0 } , for some k ≤ n . Definition 2. Let f be a multivalued holomorphic function on U (i.e. f is holomorphicon a covering of U ). We say that f has locally unipotent mondromy (or is locallyunipotent) on V if it admits a finite expansion:(2.7) f ( z , . . . , z n ) = X I =( i ,...,i k ) log i ( z ) . . . log i k ( z k ) f I ( z , . . . , z n ) , where f I ( z , . . . , z n ) is holomorphic on V . We say that a multivalued function f on X \ D is unipotent if it is everywhere locally unipotent.The main class of functions which are studied in this paper, and in particular themultiple polylogarithms, are unipotent. The expansion (2 . 7) can be characterized bya property that f has ‘moderate’ growth near D (in particular, no poles), and thatfor sufficiently large N , ( M i − id ) . . . ( M i N − id ) f = 0 for all i , . . . , i N ∈ { , . . . , k } ,where M i denotes analytic continuation around a small loop encircling z i = 0.3. Differential forms on E ( n ) Basic notations. Let e ( z ) denote the function e ( z ) = exp(2 πiz ). In accordancewith [19], Greek letters ξ and η denote coordinates on C , the letter α will denotea formal variable, and τ will denote a point of the upper half-plane H = { τ ∈ C :Im ( τ ) > } . Put z = e ( ξ ), w = e ( η ) and q = e ( τ ). Then z, w ∈ C ∗ , and 0 < | q | < ULTIPLE ELLIPTIC POLYLOGARITHMS 7 Uniformization. We represent a complex elliptic curve E = C / ( τ Z + Z ), where τ ∈ H , and q = e ( τ ), via its Jacobi uniformization E ∼ −→ C ∗ /q Z . Let ξ (resp. z = e ( ξ )) denote the coordinate on E (resp. C ∗ ). The punctured curve E × = E\{ } is isomorphic to C ∗ \{ q Z } /q Z . For n ≥ 1, let E ( n ) denote the configurationspace of n + 1 distinct points on E modulo translation by E . Thus E ( n ) ∼ = { ( ξ , . . . , ξ n ) ∈ ( E\{ } ) n : ξ i = ξ j for i = j } , and has an action of S n +1 which permutes the marked points. Setting t i = e ( ξ i ) for1 ≤ i ≤ n , and setting t n +1 = 1, gives an isomorphism E ( n ) ∼ = { ( t , . . . , t n ) ∈ C ∗ , t i / ∈ q Z t j for 1 ≤ i < j ≤ n + 1 } /q Z n . The set on the right-hand side is the largest open subset of M ,n +3 ( C ) stable undertranslation by q Z n . The symmetry group S n +1 of E ( n ) can thus be identified with thesubgroup of Aut( M ,n +3 ( C )) ∼ = S n +3 which fixes the marked points 0 and ∞ .In order to fix branches when considering multivalued functions, and to ensureconvergence when averaging functions on M ,n +3 ( C ), we must fix certain domains in E ( n ) . Let D be the standard open fundamental domain for Z + τ Z (the parallelogramwith corners 0 , , τ, τ ), and let(3.1) U = { ( ξ , . . . , ξ n ) ∈ D n : Im ( ξ n ) < . . . < Im ( ξ ) } Elliptic functions. Let θ ( ξ, τ ) denote “two thirds of the Jacobi triple formula”:(3.2) θ ( ξ, τ ) = q / ( z / − z − / ) ∞ Y j =1 (1 − q j z ) ∞ Y j =1 (1 − q j z − ) = θ ( ξ, τ ) η ( τ ) , where θ ( ξ, τ ) is the standard odd elliptic theta function and η ( τ ) is the Dedekind η -function q / Q ∞ j =1 (1 − q j ). Recall from [19] that the Eisenstein summation of adouble series ( a m,n ) m,n ∈ Z is defined by: X m,ne a m,n = lim N →∞ lim M →∞ N X n = − N M X m = − M a m,n Define the Eisenstein functions E j ( ξ, τ ) and the Eisenstein series e j ( τ ), for j ≥ 1, by E j ( ξ, τ ) = X m,n e ξ + m + nτ ) j , e j ( τ ) = X m,n ′ e m + nτ ) j , where ′ means that we omit ( m, n ) = (0 , 0) in the summation. Lemma 3. It follows from the definitions that for j ≥ , ∂∂ξ E j ( ξ, τ ) = − jE j +1 ( ξ, τ ) , ∂∂ξ log( θ ( ξ, τ )) = E ( ξ, τ ) , and E ( α, τ ) = 1 /α − P ∞ k =0 e k +1 ( τ ) α k . The series e j ( τ ) vanish for odd indices j . The Weierstrass function ℘ is equal to E − e , and ℘ ′ = − E . The coefficients ofthe Weierstrass equation ℘ ′ = 4 ℘ − g ℘ − g are given by g = 60 e , g = 140 e . FRANCIS BROWN AND ANDREY LEVIN The Kronecker function. See also [13, 14, 20] for further details. Proposition-Definition 4. The following three definitions are equivalent: i ) F ( ξ, η, τ ) = θ ′ (0) θ ( ξ + η ) θ ( ξ ) θ ( η ) ,ii ) F ( ξ, η, τ ) = − πi z − z + 11 − w + X m,n> ( z m w n − z − m w − n ) q mn ! ,iii ) F ( ξ, α, τ ) = 1 α exp − X j ≥ ( − α ) j j ( E j ( ξ, τ ) − e j ( τ )) . The equivalence of ( i ) and ( ii ) is proved in [19]. The equivalence of ( i ) and ( iii )follows by computing the logarithmic derivative of F , from the relationship between E and log( θ ) (lemma 3), and the Taylor expansion of E at a point α . The followingproperties of the Kronecker function F will be important for the sequel. Proposition 5. F ( ξ, η, τ ) has the following properties:i) Quasi-periodicity with respect to ξ ξ + 1 and ξ ξ + τ : F ( ξ + 1 , η, τ ) = F ( ξ, η, τ ) F ( ξ + τ, η, τ ) = w − F ( ξ, η, τ ) ii) The mixed heat equation: πi ∂F∂τ = ∂ F∂ξ∂η . iii) The Fay identity: F ( ξ , η , τ ) F ( ξ , η , τ ) = F ( ξ , η + η , τ ) F ( ξ − ξ , η , τ )+ F ( ξ , η + η , τ ) F ( ξ − ξ , η , τ ) . Proof. The quasi-periodicity is immediate from the first definition of F . The mixedheat equation follows from the second definition of F . The last statement is a conse-quence of the third representation of F , and the Fay trisecant equation (see [15]). (cid:3) The following formula is an easy corollary of iii ):(3.3) F ( ξ, α ) F ′ ( ξ, α ) − F ′ ( ξ, α ) F ( ξ, α ) = F ( ξ, α + α )( E ( α ) − E ( α )) , where F ′ denotes the derivative of F with respect to its second argument.3.5. Massey products on E ( n ) . We use the Eisenstein-Kronecker series F to writedown some explicit one-forms on E ( n ) . First consider a single elliptic curve E × withcoordinate ξ as above. Write ξ = s + rτ , where r, s ∈ R and τ is fixed, and let ω = dξ and ν = 2 πidr . The classes [ ω ] , [ ν ] form a basis for H ( E × ; C ). Lemma 6. The form Ω( ξ ; α ) = e ( αr ) F ( ξ ; α ) dξ is invariant under elliptic transfor-mations ξ ξ + τ and ξ ξ + 1 , and satisfies d Ω( ξ ; α ) = να ∧ Ω( ξ ; α ) .Proof. Straightforward calculation using proposition 5i). (cid:3) We can view Ω( ξ ; α ) as a generating series of one-forms on E × . Let ξ , . . . , ξ n denotethe usual holomorphic coordinates on E × × . . . × E × and set ν i = 2 iπdr i . Definition 7. Let ξ = 0 and define one forms ω ( k ) i,j ∈ A ( E ( n ) ) for all 0 ≤ i ≤ j ≤ n and k ≥ , 0) by the generating series:(3.4) Ω( ξ i − ξ j ; α ) = X k ≥ ω ( k ) i,j α k − . ULTIPLE ELLIPTIC POLYLOGARITHMS 9 We clearly have ω ( k ) i,i = 0 and ω ( k ) i,j + ( − k ω ( k ) j,i = 0 for all i, j, k . The leading terms ω (0) i,j are equal to dξ i − dξ j and therefore satisfy the relations:(3.5) ω (0) i,j + ω (0) j,k = ω (0) i,k for all i, j, k . The higher terms ω ( k ) i,j can be viewed as Massey products via the equation:(3.6) d ω ( k +1) i,j = ( ν i − ν j ) ∧ ω ( k ) i,j for k ≥ , which follows from lemma 6. The Fay identity implies thatΩ( ξ i − ξ ℓ ; α ) ∧ Ω( ξ j − ξ ℓ ; β ) + Ω( ξ j − ξ i ; β ) ∧ Ω( ξ i − ξ ℓ ; α + β )(3.7) + Ω( ξ j − ξ ℓ ; α + β ) ∧ Ω( ξ i − ξ j ; α ) = 0which gives rise to infinitely many quadratic relations between the ω ( k ) i,j . Finally, thedefinition of F shows that the residues of these forms are given by(3.8) Res ξ i = ξ j ω ( k ) i,j = 2 iπ δ k , where δ denotes the Kronecker delta. Now consider the projection E ( n ) → E ( n − givenby ( ξ , . . . , ξ n ) → ( ξ , . . . , ξ n − ). Its fibers E F n are isomorphic to the punctured ellipticcurve E × \{ ξ , . . . , ξ n − } with coordinate ξ n . Let ω ( k ) i,j (resp. ν n ) denote the relativeforms obtained by restricting ω ( k ) i,j (resp. ν n ) to the fiber. Clearly ω (0) n,i = dξ n for all i . Lemma 8. The 1-forms { ν n , dξ n , ω ( k ) n,i for k ≥ , all i } ⊂ A ( E F n ) , and the 2-forms (3.9) { ν n ∧ dξ n , ν n ∧ ω ( k ) n,i for k ≥ , all i } ⊂ A ( E F n ) are linearly independent over C .Proof. Since the ω ’s are of type (1 , 0) and ν n is not, it follows from (3 . 8) that the forms dξ n , ω (1) n, , . . . , ω (1) n,n − , ν n are linearly independent. Consider a non-trivial relation X ≤ i 9) are linearly independent. (cid:3) A rational model for the de Rham complex on E ( n ) We construct a differential graded algebra X n over Q which is defined by generatorsand quadratic relations, along with a quasi-isomorphism X n ⊗ Q C ֒ → A n , where A n = A • ( E ( n ) ) is the C ∞ -de Rham complex on the configuration space of n + 1 points on E .We show that X n carries a mixed Hodge structure and give a presentation for H • ( E ( n ) )which is an elliptic analogue of Arnold’s theorem in the genus 0 case.4.1. Differential graded algebras and fibrations. Let k be a field of characteristiczero. Recall that a (positively-graded) DGA over k is a graded-commutative algebra A = L n ≥ A n with a differential d : A → A of degree +1 which satisfies the Leibnizrule. It is said to be connected if A ∼ = k . We shall consider algebras A which areeither finite-dimensional in each degree, or else carry a second grading (called theweight grading) for which they are finite-dimensional in every bidegree. Let A T be such a DGA with differential d T , and let A B ⊂ A T be a sub-DGA. Define(4.1) A F = A T /A ≥ B A T , which inherits a differential d F from d T . We call the triple A B , A T , A F a fibration if A T is a free A B -module. The indices T, B, F stand for the total space, base, andfiber. Now suppose that we are given a splitting i F : A F → A T of A B -modules. When A B , A T , A F is a fibration, the map i F defines an isomorphism of A B -modules:(4.2) A T ∼ = A B ⊗ A B A F = L i ≥ A iB ⊗ A B A F , which does not necessarily respect the differential or algebra structure.4.2. The model X n . We consider the differential graded algebra X n generated bysymbols corresponding to the forms considered in § X n and their images in A n = A • ( E ( n ) ).This will be justified when we show that X n → A n is injective (corollary 16). Definition 9. Let X n be the Q -differential graded algebra generated by elements ω ( k ) i,j for k ≥ ≤ i ≤ j ≤ nν i for 1 ≤ i ≤ n in degree 1, modulo the graded-commutative ideal generated by the relations (3 . 5) andthe coefficients of (3 . dν i = 0, dω (0) i,j = 0, and (3 . 6) in allother cases. It is a simple calculation to check that the differential ideal generated bythe Fay identity (3 . 7) is equal to the (usual) ideal it generates.There is an obvious map X n − → X n . Let X + n − be the ideal in X n generated by theimages of elements of X n − of positive degree, and let X F n = X n /X + n − . Denote theimages of ω ( k ) i,j and ν i under the natural map X n → X F n by ω ( k ) i,j and ν i , respectively. Lemma 10. X F n is isomorphic to the Q -differential graded algebra generated by ω ( k ) n,i and ν n in degree 1, subject to the relations: ω (0) n,i = ω (0) n,j for all i, j ; ν n ∧ ν n = 0 ; and (4.3) ω ( k ) n,i ∧ ω ( ℓ ) n,j = 0 ∀ i, j, k, ℓ . The differential is given by d ω (0) n,i = d ν n = 0 and d ω ( k +1) n,i = ν n ∧ ω ( k ) n,i for k ≥ . Proof. All the relations are obvious except for (4 . . 7) that Ω( ξ n − ξ i , α ) ∧ Ω( ξ n − ξ j , β ) ≡ X + n − . (cid:3) In particular, X F n is concentrated in degrees 0, 1, and 2. Let i F n : X F n → X n denotethe splitting of the quotient map X n → X F n defined by: i F n ( ν n ) = ν n , i F n ( ω ( k ) n,i ) = ω ( k ) n,i − ω ( k )0 ,i . Mixed Hodge structure on X n . The complex of C ∞ forms on E ( n ) with log-arithmic singularities carries a Hodge and weight filtration. The weight filtration on1-foms is concentrated in degrees 1 and 2. But it turns out that there is a refinedweight filtration on X n . To define it, set W X n = 0 and W ℓ X n = h ν i , ω ( k ) i,j : k < ℓ i for all ℓ ≥ , and extend it by multiplication to X n . It is well-defined because the relations impliedby (3 . 7) are homogeneous for the weight. This filtration is induced by the grading forwhich ν i has weight 1 and ω ( k ) ij has weight k + 1. The Hodge filtration is given by F X n = X n ⊃ F X n = h ω ( k ) i,j i ⊃ F X n = 0 ULTIPLE ELLIPTIC POLYLOGARITHMS 11 and extends to X n in the same way. One easily verifies that this defines a mixed Hodgestructure on X n such that d : X n → X n is homogeneous for the weight. Likewise X F n inherits a mixed Hodge structure which is compatible with the map i F n .4.4. Quadratic Algebras. We give a sufficient criterion for an algebra defined byquadratic relations to be a fibration. We shall only apply this in the case of X n . Definition 11. Let V be a finite dimensional vector space over a field k . Let R ⊆ V V be a subspace (the space of relations). The associated quadratic algebra is Y · = V · V / h R i , where h R i ⊆ V · V is the ideal generated by R . We have Y = k , Y = V .Now suppose that V B ⊆ V is a subspace, and let V F = V /V B . Choose a splitting V = V B ⊕ V F , which induces a splitting V V = V V B ⊕ ( V B ⊗ k V F ) ⊕ V V F . Let π F : V V → V V F denote projection onto the last component. Assume that the space of relations splits: R = R B ⊕ R F , where R B ⊆ V V B , and R F ⊆ ( V B ⊗ k V F ) ⊕ V V F . Let Y · B = V · V B / h R B i . Proposition 12. Suppose that (4.4) π F : R F −→ V V F is an isomorphism.In this case, the relations R F define the graph of a map α : V V F → V B ⊗ k V F , where α = id − π − F . Extend α to a map V ⊗ F → V V F → V B ⊗ k V F . Consider the twodifferent ways of composing α with itself, namely α (1)3 : V ⊗ F α ⊗ id VF −→ V B ⊗ k V F ⊗ k V F id VB ⊗ α −→ V ⊗ B ⊗ k V F −→ V V B ⊗ k V F and α (2)3 : V ⊗ F id VF ⊗ α −→ V F ⊗ k V B ⊗ k V F ∼ = V B ⊗ k V F ⊗ k V F id VB ⊗ α −→ V ⊗ B ⊗ k V F −→ V V B ⊗ k V F satisfy the associativity condition (4.5) α (1)3 − α (2)3 ∈ R B ⊗ k V F . Then Y B → Y is injective, and a fibration, with fibers Y F , where Y F = k , and Y F ∼ = V F ,and Y kF = 0 for k ≥ . Thus there is an isomorphism of Y B -modules: Y ∼ = Y B ⊗ k Y F ∼ = Y B ⊕ ( Y B ⊗ k V F ) . Proof. There is an obvious natural map i : Y B ⊕ ( Y B ⊗ k V F ) −→ Y . We construct an inverse to i by defining by induction a sequence of linear maps α n : V n V F −→ Y n − B ⊗ k V F for n ≥ , such that i ◦ α n ( ξ ) ≡ ξ mod h R i . For this, let α be the map α = id − π − F definedabove, and let α n be the map obtained by composing α with itself n − . α n is well-defined. It is clear from the definition that i ◦ α ≡ id mod R , and from this we deduce that i ◦ α n ≡ id mod h R i for all n byinduction. Now write V n V = L ni =0 V i V B ⊗ k V n − i V F . If we set α : k → k and α : V F → V F to be the identity maps, we deduce a map ρ = n M i =0 π iB ⊗ α n − i : V n V −→ Y B ⊗ k ( Y B ⊗ k V F ) , where π iB : V i V B → Y B is the natural map. Since α n ( h R i ) = 0 for all n , and since R = R B ⊕ R F , the map ρ passes to the quotient to define a map ρ : Y −→ Y B ⊕ ( Y B ⊗ k V F )which satisfies ρ ◦ i = id by definition and i ◦ ρ is an isomorphism since i ◦ α n ≡ id mod h R i . Thus i is an isomorphism. (cid:3) Remark . In the previous discussion, we can also replace V with a graded vectorspace which is of finite dimension in every degree, and R by a graded subspace.4.5. Structure of X n . We show that X n − , X n , X F n is a fibration of DGA’s. Lemma 14. There is an isomorphism of graded-commutative algebras X n ∼ = ^ ( Q ν ⊕ . . . ⊕ Q ν n ) ⊗ Q Z n , where Z n is the subalgebra of X n spanned by the elements ω ( k ) i,j . Likewise, X F n ∼ = ^ ( Q ν n ) ⊗ Q Z F n , where Z F n is the subalgebra of X F n spanned by ω ( k ) n,i . Note that these isomorphisms donot respect the differential structures on X n and X F n .Proof. All defining relations of X n have Hodge filtration ≥ 1, so X n /F X n is iso-morphic to the free graded-commutative algebra spanned by ν , . . . , ν n . An identicalargument gives the corresponding isomorphism for X F n . (cid:3) Lemma 15. The map X n − → X n is injective, and X n is a free X n − -module.Proof. We must prove that X n − ֒ → X n and X n ∼ = X n − ⊗ Q X F n as X n − -modules.By lemma 14 this is equivalent to showing that Z n − ֒ → Z n and Z n ∼ = Z n − ⊗ Q Z F n as Z n − -modules. Since Z n is quadratic, it is enough to verify the criteria of proposition12. The quadratic relations R are defined by (3 . 7) and so R F is generated by(4.6) ( i, n ; α ) ∧ ( j, n, β ) + ( j, n ; α + β ) ∧ ( i, j ; α ) + ( j, i ; β ) ∧ ( i, n ; α + β ) = 0where i, j ≤ n − i, n ; α ) denotes Ω( ξ i − ξ n ; α ), etc. Since every term ω ( k ) n,i ∧ ω ( ℓ ) n,j for k, ℓ ≥ . . 4) is verified. The cases where k or ℓ = 0 are trivial to check. To verify(4 . . 6) four times to get: (cid:2)(cid:2) ( i, n ; α ) ∧ ( j, n ; β ) (cid:3) ∧ ( k, n ; γ ) (cid:3) = ( j, i ; β ) ∧ ( k, i ; γ ) ∧ ( i, n ; α + β + γ )+ ( k, j ; γ ) ∧ ( i, j ; α ) ∧ ( j, n ; α + β + γ )+ ( i, k ; α ) ∧ ( j, k, β ) ∧ ( k, n ; α + β + γ )Since the right-hand side is antisymmetric, the left hand side clearly does not dependon the bracketing, and the analogue of proposition 12 holds in the infinite graded case(remark 13), where the grading is given by the weight grading of § (cid:3) Let us write A F n = A n / A + n − and let φ denote the natural map X n → A n . Thechoice of coordinate ξ n on the fiber of E ( n ) → E ( n − gives an isomorphism(4.7) A n − ⊗ A n − A F n ∼ = A n . ULTIPLE ELLIPTIC POLYLOGARITHMS 13 Corollary 16. The map φ is injective.Proof. By lemma 10, X F n is concentrated in degrees at most two, so it follows fromlemma 8 that X F n → A F n is injective. The injectivity of X → A is a special case.The lemma follows by induction on n using the previous lemma and (4 . (cid:3) Proof that X n is a model. We now show that φ : X n ⊗ Q C → A n is a quasi-isomorphism. First we compute H ( X F n ) and the Gauss-Manin connection on it. Lemma 17. We have H ( X F n ) = Q , H k ( X F n ) = 0 if k ≥ , and gr W H ( X F n ) ∼ = Q [ ν n ] ⊕ Q [ ω (0) n, ] , gr W H ( X F n ) ∼ = M ≤ i ≤ n − Q [ ω (1) n,i − ω (1) n, ] , where H ( X F n ) ∼ = gr W H ( X F n ) ⊕ gr W H ( X F n ) .Proof. For all k ≥ X kF n = 0 and so H k ( X F n ) = 0. By (4 . 3) and ν n ∧ ν n = 0, anytwo-form in X F n can be written X k,i c kn,i ν n ∧ ω ( k ) n,i = d (cid:0) X k,i c kn,i ω ( k +1) n,i (cid:1) where c kn,i ∈ Q , so is exact. Thus H ( X F n ) = 0 and clearly H ( X F n ) ∼ = X F n = Q . Since ν and ω (0) n, are closed, it suffices by lemma 10 to consider a one-form η = X k ≥ , ≤ i 8) follow from the fact that ω (0) n, and ν n are exact. Lemma 18. H ( φ ) : gr W · H ( X F n ) ⊗ Q C → gr W · H ( A F n ) is an isomorphism.Proof. The differential graded algebra A F n computes the de Rham cohomology of thefiber of the map E ( n ) → E ( n − , which is isomorphic to E minus n points. Furthermore,it carries a Hodge and weight filtration which induce the corresponding filtrations on H ( E\{ n points } ). The Gysin sequence gives:0 → H ( E ; C ) → H ( E\{ n points } ; C ) → C ( − n − → , where the third map is given by the residue. Therefore gr W H ( A F n ) ∼ = H ( E ) andgr W H ( A F n ) ∼ = C ( − n − . The lemma follows from the fact that [ φ ( ν n )] , [ φ ( ω (0) n, )] isa basis of H ( E ) and φ ( ω (1) n,i ) has residue 2 πi at ξ n = ξ i , by (3 . (cid:3) Theorem 19. φ : X n ⊗ Q C ֒ → A n is a quasi-isomorphism.Proof. The case n = 1 follows from the previous lemma. The case n > (cid:3) In conclusion, X n is a model for the de Rham complex on E ( n ) and provides auniversal Q -structure on its cohomology.4.7. A simplified model. Although we shall not use it, one can consider a finitely-generated DGA model Y n for the cohomology of a configuration of elliptic curves. Definition 20. Let Y n be the commutative graded Q -algebra defined by generators ω i , ν i for 1 ≤ i ≤ n and ω ij for 1 ≤ i ≤ j ≤ n in degree one such that ω ij − ω ji = 0(4.9) ω i ∧ ν i = 0 ω ij ∧ ω i + ω ji ∧ ω j = 0 ω ij ∧ ν i + ω ji ∧ ν j = 0 ω iℓ ∧ ω jℓ + ω jℓ ∧ ω ij + ω ji ∧ ω il = 0and define a differential d : Y n → Y n by dω i = dν i = 0 and(4.10) dω ij = ω i ∧ ν j + ω j ∧ ν i . There is a surjective map X n → Y n which sends ω (0) i, to ω i and ω (1) i,j to ω ij , ν i to ν i and all ω ( k ) i,j for k ≥ 2, to zero. We can define a mixed Hodge structure on Y n in thesame way as for X n , i.e., ω i , ν i have weight one and ω ij weight two. Theorem 21. The map X n → Y n is a quasi-isomorphism, i.e., H • ( Y n ) ∼ = H • ( E ( n ) ) .Proof. (Sketch) Follow the steps of the proof that X n → A n is a quasi-isomorphism:first define Y F n to be Y n /Y + n − and check that it is a fibration. Then apply the Lerayspectral sequence argument, noting that (4 . 10) exactly corresponds to the Gauss-Maninconnection on H ( X F n ) ∼ = H ( Y F n ). (cid:3) The model Y n kills all higher Massey products in X n . Since Y n is finitely generatedit may be useful for explicit implementation of the algebra H • ( E ( n ) ). We have quasi-isomorphisms Y n ←← X n ֒ → A n but note that there is no map from Y n to A n .5. Bar construction of the de Rham complex of E ( n ) The model X n enables us to put a Q -structure on the bar construction of A n . ULTIPLE ELLIPTIC POLYLOGARITHMS 15 Reminders on the bar construction. See [7], § A bea DGA as in § A has an augmentation map ε : A → k . Let s : A → A denote the map s ( a ) = ( − deg a a . Let A + = ker ε denote the augmentationideal of A and let T ( A + ) = k ⊕ A + ⊕ A + ⊗ ⊕ . . . denote the tensor algebra on A + ,where all tensor products are over k . We write the element a ⊗ . . . ⊗ a n ∈ A ⊗ n usingthe bar notation [ a | . . . | a n ]. Recall that T ( A + ) is a commutative Hopf algebra for theshuffle product (Σ( r, s ) denotes the set of r, s shuffles):[ a | . . . | a r ] x [ a r +1 | . . . | a r + s ] = X σ ∈ Σ( r,s ) ε ( σ )[ a σ (1) | . . . | a σ ( r + s ) ] , where the sign ε ( σ ) also depends on the degrees of the a i ’s but is always equal to 1 ifall a i are of degree 1. The coproduct ∆ : T ( A + ) → T ( A + ) ⊗ k T ( A + ) is given by∆([ a | . . . | a r ]) = r X i =1 [ a | . . . | a i ] ⊗ [ a i +1 | . . . | a r ] . The length filtration is the increasing filtration associated to the tensor grading F n T ( A + ) = M ≤ i ≤ n A + ⊗ i . The bar complex is the double complex with terms ( A + ⊗ p ) q (elements of total degree q and length p in T ( A + )), and with one differential ( − p d i : ( A + ⊗ p ) q → ( A + ⊗ p ) q +1 : d i ([ a | . . . | a p ]) = X ≤ i ≤ p ( − i [ sa | . . . | sa i − | da i | a i +1 | . . . | a p ]and a second differential d e : ( A + ⊗ p ) q → ( A + ⊗ p − ) q where: d e ([ a | . . . | a p ]) = X ≤ i
When A is connected, this construction simplifies. The aug-mentation ideal A + = L n ≥ A n is the set of elements of positive degree. It is clearfrom (5 . 1) that only elements of A of degree 1 contribute to V ( A ) = H ( B ( A )), solet T ( A ) denote the tensor algebra generated by elements of degree 1. Note that( A + ⊗ p ) q = 0 if p > q and the total degree (5 . 1) is non-negative. The total differential − D = d i + d e reduces to D : T ( A ) → T ( A ): D ([ w | . . . | w r ]) = r X i =1 [ w | . . . | w i − | dw i | w i +1 | . . . | w r ] + r − X i =1 [ w | . . . | w i ∧ w i +1 | . . . | w r ]where we changed the overall sign for convenience. We can simply write(5.2) V ( A ) = ker (cid:0) D : T ( A ) → T ( A ) (cid:1) . We say that elements ξ ∈ T ( A ) satisfying Dξ = 0 are integrable . Definition 22. Define the bar construction of E ( n ) to be V ( X n ) = H ( B ( X n )), where X n is our model ( § Q , filtered by the length.By (5 . V ( X n ) is the subalgebra of T ( X n ) given by the integrable words in X n .Since X n carries a mixed Hodge structure, so too does V ( X n ). Description of V ( X F n ) . We first give an explicit description of the bar con-struction of an elliptic curve with punctures. By the general theory, the length-gradedgr ℓ V ( X F n ) ∼ = M ℓ ≥ H ( E F n ) ⊗ ℓ , since there is no integrability condition for one-dimensional varieties. The right-handside is just the set of words in the generators of H ( E F n ). These generators are repre-sented by the closed one-forms ω (0) n , ν n and η i := ω (1) n,i − ω (1) n, , for 1 ≤ i < n . Proposition 23. Any word in the letters [ ω (0) n ] , [ ν n ] , [ η i ] of length ℓ can be canonicallylifted to an integrable element in V ( X F n ) using the forms ω ( k ) n,i for k < ℓ .Proof. Use Chen’s formal power series connections [6]. Let S = Q hh x , x , y , . . . , y n − ii denote the ring of non-commutative formal power series in symbols x , x , y , . . . , y n − and let α = − ad( x ). Consider the formal 1-form [5, 14]: J = ν x + α Ω( ξ n ; α ) x + n − X i =1 (cid:0) Ω( ξ n − ξ i ; α ) − Ω( ξ n ; α ) (cid:1) y i ∈ X F n ⊗ Q S It is well-defined since all polar terms in α cancel, and is of the form J = ν n x + ω (0) n x + η y + . . . + η n − y n − + higher order terms in x ’s, y ’sOne easily checks from lemma 10 that dJ = − J ∧ J . It follows that the formal elementΞ = [ J ] + [ J | J ] + [ J | J | J ] + . . . lies in V ( X F n ) ⊗ Q S . The topological dual of S may beidentified with the tensor coalgebra T c ( U ), where U is the Q -vector space spanned bythe alphabet { x , x , y , . . . , y n − } dual to { x , x , y , . . . , y n − } . Thus we can viewΞ ∈ Hom( T ( U ) , V ( X F n )) . On the other hand, consider the element J (1) = [ ν n ] x + [ ω (0) n ] x + [ η ] y + . . . + [ η n − ] y n − ∈ H ( X F n ) ⊗ U ∨ . It defines an isomorphism j : U ∼ = H ( X F n ), and hence j : T c ( U ) ∼ = T c ( H ( X F n )).Composing j − with Ξ gives a mapgr ℓ ( V ( X F n )) ∼ = T c ( H ( X F n )) −→ V ( X F n )which splits the quotient map. Note that it follows from the proof that this splittinghas integral coefficients. (cid:3) It follows that V ( X F n ) is canonically isomorphic to the tensor coalgebra spannedby [ ω ], [ ν ], [ η i ], equipped with the shuffle product. The Hodge and weight filtrationsare induced from the corresponding filtrations on X n . More precisely we have(5.3) [ x | . . . | x n ] ∈ F r V ( X F n ) if (cid:12)(cid:12) { i : x i = ν } (cid:12)(cid:12) ≤ n − r . The weight comes from a grading w : gr ℓ V ( X F n ) → N which is defined by(5.4) w ([ x | . . . | x n ]) = n + (cid:12)(cid:12) { i : x i ∈ { η j }} (cid:12)(cid:12) , and is obtained by giving ω (0) and ν weight 1, and the η i ’s weight 2. ULTIPLE ELLIPTIC POLYLOGARITHMS 17 Example: the bar construction on E × . In the case of a single puncture:gr ℓ · V ( X ) ∼ = T ( Q [ ω (0) ] ⊕ Q [ ν ]) . The formal power series connection J reduces in this case to J = ν x + Ω( ξ, − ad( x )) x ∈ X hh x , x ii = ν x + x ω (0) − [ x , x ] ω (1) + [ x , [ x , x ]] ω (2) + . . . and gives an explicit way to lift any word in the letters [ ω (0) ] , [ ν ] to V ( X ). Corollary 24. The weight and the length filtrations on V ( X ) coincide. Examples 25. The elements of V ( X ) of length at most one are 1 , [ ω (0) ] , [ ν ]. In length ≤ ω (0) | ω (0) ], [ ω (0) | ν ] + [ ω (1) ], [ ν | ω (0) ] − [ ω (1) ], and [ ν | ν ] . Structure of V ( X n ) . One of the main results of [3] implies: Theorem 26. There is an isomorphism of algebras (5.5) V ( X n ) ∼ = n O i =1 V ( X F i ) . The length and weight filtrations on V ( X n ) coincide.Proof. The bar Gauss-Manin connection ∇ B : V ( X F n ) → X n − ⊗ Q V ( X F n ) , which isdefined in [3], is nilpotent with respect to the weight grading. It implies the existenceof a map V ( X F n ) → V ( X n ) and an isomorphism V ( X n ) ∼ = V ( X n − ) ⊗ Q V ( X F n ) ofalgebras, from which the statement follows by induction (see [3] for the proofs). (cid:3) Note that (5 . 5) does not respect the Hopf algebra, or differential structures on X n .It is however a complete algebraic description of all iterated integrals on E ( n ) , and inparticular, enables one to write down a basis for them. Remark . Theorem 26 is proved in [3] by first showing that the bar-de Rham coho-mology of X n is trivial. This is equivalent to the exactness of the sequence:0 −→ Q −→ X n ⊗ Q V ( X n ) −→ X n ⊗ Q V ( X n ) −→ . . . −→ X nn ⊗ Q V ( X n ) −→ . One way to think of the refined weight grading on X n is as follows. It defines aweight grading on T c ( X n ) with the property that the inclusion V ( X n ) ֒ → T c ( X n ) iscompatible with the weight filtration on V ( X n ). ∗ ∗ ∗ Averaging unipotent functions Introduction. The main idea for constructing multivalued functions on an ellip-tic curve is to use the Jacobi uniformization E = C × /q Z and average a function on C × with respect to multiplication by q . Consider the exampleof the multivalued function Li ( z ) = − log(1 − z ). Let q ∈ C × such that | q | < z ∈ C × such that 1 / ∈ q R z . The spiral q R z can be lifted to a universal covering spaceof C \{ , } , and the function Li ( z ) has a well-defined analytic continuation alongit. Near the origin, the function Li ( z ) vanishes, but at the point z = ∞ it has alogarithmic singularity, so the naive average diverges. One way to ensure convergenceis to consider the generating series E ( z ; u ) = X m ∈ Z u m Li ( q m z )where u − is chosen small enough to dampen the logarithmic singularity at infinity,but not so small as to wreck the convergence at the origin. For m ≪ 0, the asymptoticis u m log( q m z ), which is bounded if u > 1. For m ≫ u m q m z , which is bounded if u < | q | − . Thus for 1 < u < | q | − the series E ( z ; u )converges absolutely, and is almost periodic with respect to multiplication by q .From this one can easily show that E ( z ; u ) has a simple pole at u = 1. Now onemust view E ( z ; u ) as a function of ξ and α , where u = e ( α ) and z = e ( ξ ). Thus thepole at u = 1 contributes a pole at α = 0 which can be removed to obtain E reg ( ξ ; α ) = E ( ξ ; α ) − α . This function now admits a Taylor expansion at the point α = 0. The procedure forconstructing multivalued functions on the elliptic curve E × is to take the coefficientsof α i , i ≥ f ( t , t ) on M , ( C ) = { ( t , t ) : t , t = 0 , , t = t } , withsingularities along the removed hyperplanes. We wish to average the function(6.1) X m ,m ∈ Z u m u m f ( q m t , q m t )The first problem that we encounter is that the function f ( t , t ) is simply not well-defined as t , t → ∞ since its singularities t i = ∞ , t = t do not cross normally atthat point, and so the limit depends on the direction in which it is approached. Thestandard solution is to blow-up the points (0 , ∞ , ∞ ), as below: t = 1 t = ∞ t = 1 t = ∞ ULTIPLE ELLIPTIC POLYLOGARITHMS 19 Note that we do not need to blow up the point t = t = 1, which is also a non-normal crossing point, because there are only finitely many lattice points { ( q Z t , q Z t ) } in its neighbourhood. The domain of summation naturally decomposes into the sixsectors pictured above, each of which is homeomorphic to a square [0 , × [0 , f ( t , t ) in the neighbourhood of each sector, one findsnecessary and sufficient conditions on u , u to ensure the absolute convergence of(6 . u , u plane are in one-to-one correspondencewith the boundary divisors in the figure, and depend on the local asymptotic behaviourof f . One can then remove poles in the α i plane (where u i = e ( α i )) and compute Taylorexpansions to extract multivalued functions on E (2) with unipotent monodromy.The plan of the second, analytic, part of this paper is as follows.(1) First we construct an explicit partial compactification of M ,n which is adaptedto this averaging procedure.(2) By studying the analytic properties of the multiple polylogarithms (2 . 3) we findnecessary and sufficient conditions on the dual variables u i to ensure absoluteconvergence of the averaging function: X m ,...,m r ∈ Z u m . . . u m r r I n ,...,n r ( q m t , . . . , q m r t r )(3) From its differential equation, we compute the pole structure in the u i coordi-nates. The multiple elliptic polylogarithms can be defined as the coefficientsin its regularized Taylor expansion at α = . . . = α r = 0, where u i = e ( α i ).Note that, since the singularities in the space of α i parameters are not normalcrossing, the regularization must be performed with some care. We do notaddress the question of explicit regularization in the present paper.The entire procedure from (1) to (3) will work more generally for any functionssatisfying some growth conditions on certain toric varieties. Section § § 7, with examples given in § 9. In § 8, which is independent from the restof the paper, we show how to compute the asymptotics of the Debye polylogarithmsexplicitly at infinity using a certain coproduct. Finally, in § 10 we prove that all iteratedintegrals on a punctured elliptic curve can be obtained by this averaging procedure.6.2. Preliminaries. Let q = e ( τ ), where τ is in the upper-half plane, and let t i = e ( ξ i ), for i = 1 , . . . , r , where ξ , . . . , ξ r are in the domain U defined by (3 . § M ,r +3 ( C ): σ : R n −→ f M ,r +3 ( C )( s , . . . , s r ) ( q s t , . . . , q s r t r ) , where we view M ,r +3 ( C ) ⊂ C r in simplicial coordinates. Suppose that we have amultivalued function f ( t , . . . , t r ) on M ,r +3 ( C ), with a fixed branch in the neighbour-hood of some ( t , . . . , t r ) such that t i t − j / ∈ q R for i = j and t i / ∈ q R . Then f ( t , . . . , t r )admits a unique analytic continuation to the image of σ ( R n ), and in particular, thevalues f ( q n t , . . . , q n r t r ) are well-defined for all ( n , . . . , n r ) ∈ Z r .We apply this to the multiple polylogarithm functions(6.2) I m ,...,m n ( t , . . . , t n ) = X a ,...,a n ≥ t a . . . t a n n a m ( a + a ) m . . . ( a + . . . + a n ) m n , which give rise to multivalued unipotent functions on M ,r +3 ( C ) ⊂ C r , and vanishalong the divisors t i = 0. The power series expansion above defines a canonical branchin the neighbourhood of the origin.6.3. Compactification of the hypercube. Let S = { , . . . , n } and let us write P S for Hom( S, P ) ∼ = ( P ) n . Let (cid:3) n ⊂ P S denote the real hypercube [0 , ∞ ] n . For anydisjoint pair of subsets I, J ⊂ S , let F JI = \ i ∈ I { t i = 0 } ∩ \ j ∈ J { t j = ∞} ⊆ P S denote the corresponding coordinate linear subspace. The sets F JI ∩ (cid:3) n give the stan-dard stratification of the hypercube by its faces.Working first in simplicial coordinates, consider the divisor X = [ ≤ i 2. Following the standard practice for blowing up linear subspaces,we blow up the set of faces in F of smallest dimension, followed by the strict transformsof faces F I where | I | = n − 1, and so on, in increasing order of dimension, until thestrict transforms of all elements in F have been blown up. Now repeat the sameprocedure with F ∞ , and denote the corresponding space by P S , with(6.3) π : P S −→ P S It does not depend on the chosen order of blowing-up.Let us denote the strict transform of any face F I (resp. F J ) by D I (resp. D J ),for all | I | , | J | ≥ 2. Let D i (resp. D j ) denote the strict transform of the divisor F ∅ i (resp. F j ∅ ) which corresponds to a facet of the original hypercube, and let us denoteby D = S | K |≥ D K ∪ D K , the union of all the above. The strict transform of (cid:3) n is acertain polytope C n , whose facets are in bijection with the irreducible components of D , which number 2 × (2 n − X ′ ⊂ P S denote the strict transform of X . Proposition 28. The divisor D ∪ X ′ ⊂ P S is locally normal crossing near D . The proof will be given by computing explicit normal coordinates in every localneighbourhood of D , using a decomposition into sectors.6.4. Sector decomposition. Let us view M ,n ( R ) in simplicial coordinates as thecomplement of divisors of the form t i = t j and t i = 1 in ( R × ) n . Then (cid:3) n ∩ M ,n +3 ( R )admits a decomposition into ( n + 1)! connected components:∆ π = { < t π (1) < . . . < t π ( n +1) < ∞} , where π is a permutation of (1 , . . . , n + 1), the t i are simplicial coordinates on eachcomponent of ( P ) n , and where t n +1 = 1. The permutation π should be viewed as adihedral ordering of the n + 3 marked points 0 , , ∞ , t , . . . , t n on P ( R ). To every such π we associate local ‘sector’ coordinates on M ,n +3 ( C ) as follows:(6.4) s π = t π (1) t π (2) , . . . , s πn = t π ( n ) t π ( n +1) . ULTIPLE ELLIPTIC POLYLOGARITHMS 21 The coordinates s πi give a homeomorphism from ∆ π to the unit cube (0 , n . When π is the trivial permutation, the coordinates s πi are the same coordinates x i used todefine the multiple polylogarithms in § π , we define the open affine scheme U π = Spec Z [ s π , . . . , s πn , { ( Y i ≤ k ≤ j s πk − − } ≤ i ≤ j ≤ n ] . Note that the U π are all canonically isomorphic, and (0 , n ⊂ U π ( R ). Lemma 29. For every π , U π defines an affine chart on M ,n +3 : M ,n +3 ⊂ U π ⊂ M ,n +3 Proof. Consider the set of forgetful maps (or ‘cross-ratios’) f T : M ,n +3 → M , ∼ = P ,where T is a subset of any 4 of the n + 3 marked points. Then M ,n +3 ⊂ M ,n +3 is theopen subscheme where all f T ’s are non-zero. It suffices to check that U π is isomorphicto the open subscheme of M ,n +3 where some of the f T ’s are non-zero. For this, onecan write each s πk and Q i ≤ k ≤ j s πk − s πi . We omit the details. (cid:3) Definition 30. Let U n = S π U π ⊂ M ,n +3 be the scheme obtained by gluing allcharts U π together. Viewing S n +1 as the stabilizer of 0 , ∞ in Aut( M ,n +3 ) ∼ = S n +3 ,we have U n = [ π ∈ S n +1 π ( U id ) , where U id corresponds to the trivial permutation.The smooth scheme U n ⊂ M ,n +3 is equipped with a set of normal crossing divisorsdefined in each chart by the vanishing of the s πk . Example 31. Let n = 2. The coordinate square (0 , ∞ ) × (0 , ∞ ) ⊂ M , ( R ) is coveredby six sectors ∆ π as shown below after blowing up F = (0 , F = ( ∞ , ∞ ). D t = 1 D D D t = 1 D D ∆ π Consider the sector denoted ∆ π , where π = (1 , t , t ), which corresponds to thedihedral ordering 0 < < t < t < ∞ on the marked points of M , ( R ). Its sectorcoordinates are s π = t − , s π = t /t . The divisor s π = 0 corresponds to the partition { , , t }|{ t , ∞} on M , , and the divisor s π = 0 corresponds to { , }|{ t , t , ∞} .In the general case, we have: Lemma 32. The divisor s πk = 0 corresponds on U π ⊂ M ,n +3 to the partition ( § { , t π (1) , . . . , t π ( k ) } (cid:12)(cid:12) { t π ( k +1) , . . . , t π ( n +1) , ∞} . Therefore the scheme U n is the complement in M ,n +3 of the set of divisors A corre-sponding to partitions in which the marked points and ∞ lie in the same component.The simplicial coordinates t , . . . , t n give a canonical map U n → ( P ) n , which iden-tifies U n with the blow-up P S of ( P ) n defined in § P S \ X ′ ∼ = M ,n +3 \ A ∼ = U n . This identifies the following divisors on P S \ X ′ , M ,n +3 \ A , and U n , respectively: D I ↔ { , t k : k ∈ I } (cid:12)(cid:12) {∞ , , t k : k / ∈ I } ↔ s π | I | = 0 on every U π st π ( I ) = ID J ↔ { , , t k : k / ∈ J } (cid:12)(cid:12) {∞ , t k : k ∈ J } ↔ s πn +1 −| J | = 0 on every U π st π ( J ) = J We deduce that D I ∩ D I ′ = ∅ if and only if I ⊆ I ′ or I ′ ⊆ I (and likewise for D J , D J ′ )and D I ∩ D J = ∅ if and only if I ∩ J = ∅ .Proof. Straightforward. One must only verify that the sector coordinates s πk are pre-cisely the local coordinates that one obtains when one blows up ( P ) n along divisors F I and F J in order of increasing dimension. (cid:3) In conclusion, we have three descriptions of the space U n : first, as a certain blow-upof ( P ) n along the boundary of the hypercube; second, as the gluing together of affineschemes U π ; and third, as the complement in M ,n +3 of a certain family of divisors. Corollary 33. Let D ⊂ U n be an irreducible divisor defined locally by the vanishingof an s πk . Then D ∼ = U k − × U n − k , where U is a point. It follows that the polytope C n (which was defined to be the strict transform π − ( (cid:3) n ) ) has the following productstructure on its facets: C n ∩ D I ∼ = C | I |− × C n −| I | and similarly for C n ∩ D J . Absolute convergence of multivalued series. Let(6.5) S = { ( q, t , . . . , t n ) : 0 < | q | < | t | < . . . < | t n | < , t i t − j / ∈ q R , t i / ∈ q R } , and let f be a unipotent function on ( C × ) n \{ t i = t j } , i.e., a multivalued function on M ,n +3 ( C ) ⊂ M ,n +3 ( C ) with everywhere local unipotent monodromy around bound-ary divisors (definition 2). Consider a sum(6.6) F ( t , . . . , t n ; q ) = X m ,...,m n ∈ Z u m . . . u m n n f ( q m t , . . . , q m n t n ) , By § f ( q m t , . . . , q m n t n ) are well-defined if we fix a branch of f nearsome point ( t , . . . , t n ), where ( q, t , . . . , t n ) ∈ S . We give sufficient conditions on theauxilliary variables u i to ensure the absolute convergence of such a series, by boundingthe terms of F in different sectors. Definition 34. For 0 < ε ≪ 1, let U επ ⊂ U π ( C ) denote the open set of points { ( s π , . . . , s πn ) ∈ U π ( C ) : | s πi | < i, and | s πi | < ε for some 1 ≤ i ≤ n } , and let U ε = S π ∈ S n +1 U επ .Let K be a compact subset of S defined in (6 . Lemma 35. For ( q, t , . . . , t n ) in K , there are only finitely many m = ( m , . . . , m n ) such that ( q m t , . . . , q m n t n ) lies in the complement of U ε , and f ( q m t , . . . , q m n t n ) isuniformly bounded for such m . ULTIPLE ELLIPTIC POLYLOGARITHMS 23 Proof. The first part follows since the complement of U ε in M ,n +3 is compact, anddoes not contain the total transform of any divisors t i = 0 , ∞ . The definition of S ensures that q m i t i = q m j t j and q m i t i = 1 for all m i , m j , i = j , and since these are thepossible singularities of f ( q m t , . . . , q m n t n ), it is uniformly bounded on K . (cid:3) All the remaining terms of (6 . 6) lie in some sector U επ . Let us fix one such per-mutation π and work in local sector coordinates s π , . . . , s πn . Then U επ can be furtherdecomposed into smaller pieces as follows. For any non-empty A ⊆ { , . . . , n } , let N πA = { ( s π , . . . , s πn ) : | s πi | < ε for i ∈ A, > | s πi | ≥ ε for i / ∈ A } . We clearly have: U επ = [ ∅6 = A ⊆{ ,...,n } N πA .N π N π N π ε ε Proposition 36. There is a constant C ∈ R such that for all ( s π , . . . , s πn ) ∈ N πA , | f ( s π , . . . , s πn ) | ≤ C (cid:0) Y i ∈ A κ i ( s πi ) (cid:1) f A (( s πj ) j / ∈ A ) , where f A ( s πj ) j / ∈ A is a unipotent function on T i ∈ A { s πi = 0 } , and κ i ( s ) = | s | M i log w | s | , where f vanishes along s πi = 0 to order M i ≥ , and w is some integer ≥ .Proof. This follows immediately from the local expansion of a unipotent multivaluedfunction in the neighbourhood of a normal crossing divisor (definition (2 . (cid:3) Recall from definition (6 . 4) that s πi = t π ( i ) t − π ( i +1) . Thus the action of the summationindex ( m , . . . , m n ) ∈ Z n on the sector coordinate s πi is given by s πi q p πi s πi , where(6.7) p πi = m π ( i ) − m π ( i +1) . By lemma 32, the divisor s πk = 0 corresponds to a divisor D I or D J . Let us define(6.8) v πk = Y i ∈ I u i or v πk = Y j ∈ J u − j accordingly, where u i are the parameters in (6 . v π ) p π . . . ( v πn ) p πn = u m . . . u m n n . Thus, for the terms of (6 . 6) which lie in the sector U π , we can write u m . . . u m n n f ( q m t , . . . , q m n t n ) = ( v π ) p π . . . ( v πn ) p πn f ( q p π s π , . . . , q p πn s πn )in local coordinates. Dropping cluttersome π ’s from the notation, we have: Corollary 37. For all ( q, t , . . . , t n ) ∈ K , there exists a constant C such that (cid:12)(cid:12) v p . . . v p n n f ( q p s π , . . . , q p n s πn ) (cid:12)(cid:12) ≤ C Y i ∈ A | v i q M i | p i | p i | w for all ( p , . . . , p n ) ∈ Z n such that ( q p s π , . . . , q p n s πn ) ∈ N πA .Proof. Consider the bound in proposition 36. The function f A on the right-hand sidecan be uniformly bounded by a version of lemma 35, applied to T i ∈ A { s πi = 0 } , whichis isomorphic to a product of U k ’s. Since log w | q p i s πi | = ( p i log | q | + log | s πi | ) w ≤ C i | p i | w for some constant C i , for all i ∈ A , we deduce the bound stated above. (cid:3) Theorem 38. Suppose that the chosen branch of f vanishes along all divisors D I oftype with multiplicity | I | . Then (6 . converges absolutely on compacta of the polydisc < u , . . . , u n < | q | − Proof. Consider first a divisor D I of type 0. Then, in the previous corollary, the divisor D I = { s πk = 0 } in some chart U π will correspond in the right-hand side to terms of theform | v πk q | I | | p p w , where p is large and positive. The assumptions on u i imply that | v πk q | I | | p p w = (cid:16) Y i ∈ I | u i q | (cid:17) p p w < , and therefore | v πk q | I | | p p w tends to zero exponentially fast in p . Now consider a divisor D J of type ∞ . It corresponds to terms of the form | v πk q M k | p p w , where p is large andpositive and M k , w ≥ 0. But by the assumptions on u i , we have | v πk | = (cid:16) Y j ∈ J u − j (cid:17) < , and so once again, | v πk q M k | p p w tends to zero exponentially fast in p . (cid:3) Structure of the poles. We first make some general remarks about the polestructure as follows from the proof of theorem 38. In § Corollary 39. Let f satisfy the conditions of theorem . For every I = ∅ , let w I de-note the order of the logarithmic singularity of f along D I . Every codimension h face ofthe polytope C n corresponds to a flag I ( I ( . . . ( I h , where I , . . . , I h ⊆ { , . . . , n } ,and is contained in the intersection E = D I ∩ . . . ∩ D I h . Let s , . . . , s h , s h +1 , . . . , s n denote local normal coordinates in which D I k is given by s k = 0 for ≤ k ≤ h . Thenin the neighbourhood of E , the function f has an expansion of the form f = X i ≤ w I ,...,i h ≤ w Ih f i ,...,i h ( s h +1 , . . . , s n ) log i ( s ) . . . log i h ( s h ) , where ( s h +1 , . . . , s n ) are coordinates on E . After averaging, each term in the sumwhich is indexed by i , . . . , i h contributes singularities to (6 . of the form ( Y k ∈ I u k − − j . . . ( Y k ∈ I h u k − − j h , with j ≤ i + 1 , . . . , j h ≤ i h + 1 . In particular, the term f ,..., ( s h +1 , . . . , s n ) which isconstant in s , . . . , s h , contributes a simple pole of the form ( Y k ∈ I u k − − . . . ( Y k ∈ I h u k − − . ULTIPLE ELLIPTIC POLYLOGARITHMS 25 Proof. Following the method of proof of the previous theorem, one sees that the state-ment reduces to a local computation in the one-dimensional situation. In this case, itis clear that the averaging procedure applied to log i z , for i ≥ 0, gives X m ≥ u m log i q m z = X m ≥ u m ( m log q + log z ) i , which has a pole at u = 1 of order at most i + 1. (cid:3) Remark . Corollary 39 gives an upper bound on the singularities which occur: itcan happen that summing over one sector gives rise to spurious poles in the u i ’s, whichcancel on taking the total contribution over all sectors.The upshot of the previous corollary is that if we know the differential equationssatisfied by f , then we can deduce the pole structure of F completely from thesedifferential equations, up to constants of integration (see § n ! maximal flags I ( I ( . . . ( I n − ( I n . Example 41. In the case n = 2, we have poles along u i = 1 coming from logarithmicsingularites along D i , for i = 1 , 2, and along u u = 1 coming from D . The typicalcontribution from D , for example, is of the form w X i =0 R i ( ξ ; u )( u − i +1 where R i ( ξ ; u ) is the result of averaging a function of t . The two maximal flags { } ⊂ { , } and { } ⊂ { , } which correspond to the corners D ∩ D and D ∩ D ,give constant contributions of the form c , ( u − u u − 1) and c , ( u − u u − c , (resp. c , ) is the regularized limit of f at D ∩ D (resp. D ∩ D ).For an explicit computation of such a pole structure, see § Elliptic multiple polylogarithms In this section, we apply the results of § Theorem 42. The series obtained by averaging the classical multiple polylogarithm E n ,...,n r ( ξ , . . . , ξ r ; u , . . . , u r ) = X m ,...,m r ∈ Z u m . . . u m r r I n ,...,n r ( q m t , . . . , q m r t r ) converges for < u , . . . , u r < | q | − , and ( q, t , . . . , t r ) ∈ S . It defines a (generatingseries) of functions on E ( r ) with poles given by products of consecutive u i ’s only: u i = | q | − for ≤ i ≤ r , and Y i ≤ k ≤ j u k = 1 for all ≤ i ≤ j ≤ r . In order to extract a convergent Taylor expansion in the variables α i , where u i = e ( α i ), it suffices to know the exact asymptotic behaviour of I n ,...,n r ( t , . . . , t r ) atinfinity. This is carried out for the Debye polylogarithms in § Analytic continuation of polylogarithms. The function I n ,...,n r ( t , . . . , t r )has a convergent Taylor expansion at the origin, and so defines the germ of a multival-ued analytic function on M ,n +3 ( C ) ⊂ C n . As is well-known, it is unipotent by (2 . t i = 0.It therefore extends by analytic continuation to a multivalued function on the blow-up U n \ X ( C ), and can have at most logarithmic divergences along the boundary com-ponents D J and D I of X . It turns out that for some of these components D , there isno logarithmic divergence and we can speak of the continuation I m ,...,m n ( t , . . . , t n ) (cid:12)(cid:12) D of I m ,...,m n ( t , . . . , t n ) to D . This is not well-defined, since the function is multivalued.However, if D also meets the strict transform of a divisor t i = 0, for some i , there is acanonical branch which vanishes at { t i = 0 } ∩ D , and defined in a neighbourhood of { t i = 0 } . The following lemma is the key to the absolute convergence of (7 . Lemma 43. Let D I be of type . Then I m ,...,m n ( t , . . . , t n ) (cid:12)(cid:12) D I vanishes to order | I | .Proof. The sum (6 . 2) converges in the neighbourhood of the origin and locally definesa holomorphic function which vanishes along the divisors t i = 0. It therefore vanisheson any exceptional divisor D I lying above the origin to order | I | . (cid:3) Setting f ( t , . . . , t n ) = I m ,...,m n ( t , . . . , t n ) proves the first part of theorem 42. Lemma 44. Let J ⊂ { , . . . , n } be non-empty, and / ∈ J . Let J c = { , . . . , n }\ J andwrite J c = { i , . . . , i k } where i < . . . < i k . Then I m ,...,m n ( t , . . . , t n ) (cid:12)(cid:12) D J = ( − | J | I m ′ ,...,m ′ k ( t i , . . . , t i k ) , where m ′ = m i + . . . + m i − , m ′ = m i + . . . + m i − ,. . . , m ′ k = m i k + . . . + m n .Proof. This follows immediately from the iterated integral representation (2 . (cid:3) Corollary 45. Every multiple polylogarithm I m ,...,m n ( t , . . . , t n ) is the analytic con-tinuation to some exceptional divisor of a multiple logarithm I ,..., ( t , . . . , t N ) . Thus we can restrict ourselves to considering only multiple logarithms if we wish.Another way to interpret lemma 44 is to notice that the terms I (cid:12)(cid:12) D J are in one-to-onecorrespondence with the terms in the so-called stuffle product formula.7.2. Elliptic Multiple Polylogarithms. The functions obtained by averaging mul-tiple polylogarithms satisfy differential equations which are easily deduced from (2 . Lemma 46. The function F ( ξ ; u ) is the averaged weighted generating series for zz − : F ( ξ ; u ) = − πi X n ∈ Z q n z − q n z u n . Proof. By decomposing the domain of summation into various parts we obtain: X n ∈ Z q n z − q n z u n = X n< − q − n z − − q − n z − u n − X n< u n + z − z + X n> q n z − q n z u n = X n> X m> q mn ( − z − m u − n + z m u n ) + z − z + 11 − u which is the definition of the Eisenstein-Kronecker series − (2 iπ ) − F ( ξ ; u ). (cid:3) Lemma 47. The averaged weighted generating series for d Li ( z ) is: X n ∈ Z d Li ( zq n ) u n = F ( ξ ; u ) dξ . Likewise, the result of averaging z − s d Li ( z ) is e ( − ξs ) F ( ξ ; u ) dξ . ULTIPLE ELLIPTIC POLYLOGARITHMS 27 Proof. Follows from the previous lemma using the fact that dξ = iπ dzz . (cid:3) Let us define the (unregularized) multiple elliptic polylogarithm to be: E n ,...,n r ( ξ , . . . , ξ r ; α , . . . , α r ) = X m ,...,m r ∈ Z u m . . . u m r r I n ,...,n r ( q m t , . . . , q m r t r ) . where u i = e ( α i ) for 1 ≤ i ≤ r . Theorem 48. The total derivative dE ,..., ( ξ , . . . , ξ n ; α , . . . , α n ) equals = n X k =1 dE ( ξ k − ξ k +1 ; α k ) E ,..., ( ξ , . . . , b ξ k , . . . , ξ n ; α , . . . , α k + α k +1 , . . . , α n ) − n X k =2 dE ( ξ k − ξ k − ; α k ) E ,..., ( ξ , . . . , b ξ k , . . . , ξ n ; α , . . . , α k − + α k , . . . , α n ) where ξ n +1 = 0 , α n +1 = 0 , and dE ( ξ ; α ) = F ( ξ ; α ) dξ .Proof. Since it converges uniformly, we can differentiate term by term in the definitionof E ,..., . The differential equation then follows from the corresponding differentialequation (2 . 5) for I ,..., ( t , . . . , t n ). The key observation is that a term such as X m ,...,m n ∈ Z dI (cid:16) q m k t k q m k +1 t k +1 (cid:17) I ,..., ( q m t , . . . , \ q m k t k , . . . , q m n t n ) u m . . . u m n n can be rewritten in the form X m ,...,m n ∈ Z dI (cid:16) q m k − m k +1 t k t k +1 (cid:17) u m k − m k +1 k × I ,..., ( q m t , . . . , \ q m k t k , . . . , q m n t n ) u m . . . ( u k u k +1 ) m k +1 . . . u m n n and the region of summation decomposes into a product after a triangular change ofbasis of the summation variables ( m , . . . , m n ) ( m , . . . , m k − m k +1 , . . . , m n ). (cid:3) Elliptic Debye polylogarithms. Recall the definition of the classical Debyepolylogarithms (definition 1). Let us write α = ( α , . . . , α r ) and likewise for β . Definition 49. The generating series of elliptic Debye polylogarithms is: E r ( ξ , . . , ξ r ; α, β ) = X m ,. . ,m r ∈ Z e ( m α + · · · + m r α r )Λ r ( q m t , . . , q m r r t r ; β , . . , β r )The absolute convergence of the series is guaranteed by theorem 38.One of the main reasons for considering such a generating series is because of amysterious modularity property relating the parameters α and β (see [13] when r = 1). Proposition 50. Let r ≥ . The differential d E r ( ξ , . . , ξ r ; α, β ) is equal to = n X k =1 d E ( ξ k − ξ k +1 ; α k , β k ) E r − ( ξ , . . , b ξ k , . . , ξ r ; α , . . , α k + α k +1 , . . , α n , β , . . ) − n X k =2 d E ( ξ k − ξ k − ; α k , β k ) E r − ( ξ , . . , b ξ k , . . , ξ r ; α , . . , α k − + α k , . . , α n , β , . . ) where ξ n +1 = α n +1 = β n +1 = 0 , and in the right-hand side, the arguments in the β ’sare of the same form as those for the α ’s. In the case r = 1 , we have (7.1) d E ( ξ ; α ; β ) = e ( − βξ ) F ( ξ ; α − τ β ) dξ . The proof follows immediately from theorem 48 . The structure of the poles of elliptic polylogarithms. Let us write(7.2) γ i = α i − τ β i for 1 ≤ i ≤ r . Proposition 51. The Debye elliptic polylogarithms (definition 49) have at most simplepoles along the divisors which have consecutive indices only: X i ≤ j ≤ k γ i = 0 and X i ≤ j ≤ k α j = 0 . The multiple elliptic polylogarithm E m ,. . ,m n ( ξ , . . , ξ n ; α , . . , α n ) has poles along divi-sors of the form P i ≤ j ≤ k α j = 0 of order at most m + . . . + m n + 1 .Proof. By induction. Suppose that E r has simple poles along P i ≤ j ≤ k α j = 0, and P i ≤ j ≤ k γ j = 0 with consecutive indices only. This is automatically true for r ≤ d E r +1 only has simple poles along P i ≤ j ≤ k α j = 0 and P i ≤ j ≤ k γ j = 0. Thus the sameconclusion also holds for E r +1 , except that the constants of integration might giverise to supplementary poles. To see that such constants of integration must be zero,let I be a set of non-consecutive indices. The divisor D I meets a divisor of the form t j = 0, for j / ∈ I , along which the function Λ( t , . . . , t n ; s , . . . , s n ) vanishes. It followsfrom the discussion above that Λ has no divergence in the neighbourhood of D I ∩{ t j = 0 } , and hence no pole in either P i ∈ I α i = 0 or P i ∈ I γ i = 0. This proves theresult for the generating series E r . The corresponding statement for its coefficients E m ,. . ,m r ( ξ , . . , ξ r ; α , . . , α r ) follows on taking a series expansion in the β i . (cid:3) This method for computing the pole structure is illustrated below ( § Asymptotics of Debye polylogarithms. In the previous section we showed that the polar contributions in the averaging pro-cess come from the asymptotic expansion of polylogarithms at infinity. This expansioncan be computed explicitly in terms of a combinatorially defined coproduct.8.1. The coproduct for Debye polylogarithms. The Debye multiple polyloga-rithms are defined by iterated integrals, and so by the general theory [6] admit acoproduct which is dual to the composition of paths. We describe it explicitly below. Definition 52. Let n ≥ 1. Let I = { , . . . , n } be an ordered set of indices and let β , . . . , β n be formal variables satisfying P ni =1 β i = 0. Define a string in I to be aconsecutive subsequence S = ( i , i , . . . , i l ) of length 2 ≤ l < n , which is either inincreasing or decreasing order and such that i = n . Let β S = β i + β i + . . . + β i l − . For any such string S = ( i , . . . , i l ), let A S denote the symbol(8.1) A S = ( t i : t i : . . . : t i l ; β i , β i , . . . , β i l − , − β S ) . Let H n denote the commutative ring over Z generated by all symbols A S as S rangesover the set of strings in 1 , , . . . , n . The length of a string defines a grading on H n .The Debye polylogarithm defines a map from H n to generating series of multivaluedfunctions. If a string S is given by (8 . 1) then we have(8.2) Λ( A S ) = Λ l − (cid:16) t i t i ℓ , t i t i ℓ , . . . , t i l − t i ℓ ; β i , β i , . . . , β i l − (cid:17) . Denote the last element of a string by ℓ ( i , i , . . . , i l ) = i l , and define the sign ε ( S )of S to be 1 if S is in increasing order, or ( − l − if it is in decreasing order. ULTIPLE ELLIPTIC POLYLOGARITHMS 29 Definition 53. A finite collection S = { S α } of strings is admissible if the stringsintersect at most in their last indices, i.e., if S α , S β are in S and α = β then either(1) S α ∩ S β = ∅ or (2) S α ∩ S β = { ℓ } , where ℓ = ℓ ( S α ) = ℓ ( S β ) . Given an admissible set of strings S = { S α } , define the set of remaining indices R S = (cid:0) I \ [ S α (cid:1) ∪ [ ℓ ( S α )with the ordering induced from I , and define the corresponding quotient sequence Q S = ( t j : t j : . . . : t j m ; ˜ β j , ˜ β j , . . . , ˜ β j m ) , where ( j , j , . . . , j m ) = R S and ˜ β j = β j + P α,ℓ ( S α )= j β S α . Definition 54. Define a map ∆ ′ : H n −→ H n ⊗ H n by(8.3) ∆ ′ A J = X S = { S ,...,S k } ε ( S ) A S . . . ε ( S k ) A S k ⊗ Q S where J ⊆ { , . . . , n } , and the sum is over all non-empty admissible collections ofstrings S in J such that Q S has at least two elements.Consider the map which sends A S of (8 . 1) to t β i i . . . t β il i l , with β i l = − β S , andextend by multiplicativity. Then each term in (8 . 3) maps to ± t β . . . t β n n . Example 55. Writing β ij for β { i,j } = β i + β j , and so on, formula (8 . 3) gives:∆ ′ ( t : t : t ; β , β , β ) = ( t : t ; β , − β ) ⊗ ( t : t ; β , β )(8.4) − ( t : t ; β , − β ) ⊗ ( t : t ; β , β )+ ( t : t ; β , − β ) ⊗ ( t : t ; β , β ) . In general, a typical term in ∆ ′ ( t : . . . : t ; β , . . . ; β ) is( t : t : t ; β , β , − β )( t : t ; β , − β )( t : t ; β , − β ) ⊗ ( t : t ; β , β ) Proposition 56. Let ∆ : H n → H n ⊗ H n be ∆ = 1 ⊗ id + id ⊗ ′ . Then H n ,equipped with ∆ , is a commutative graded Hopf algebra.Proof. We omit the proof. In fact it suffices to show that the 1-part of the coproductcoincides with the differential for the Debye polylogarithms (lemma 57 below). (cid:3) Let ∆ ( m +1) : H n → H ⊗ m +1 n denote the m -fold iteration of ∆. Let ∆ ( m +1) ⋆,...,⋆, ,⋆,...,⋆ denote its component whose corresponding tensor factor contains only strings of lengthtwo. Thus ∆ ,⋆ extracts all ordered pairs of neighbouring indices except ( n, n − Lemma 57. The differential equation for Λ can be rewritten as d Λ = µ ◦ ( d Λ ⊗ Λ) ◦ ∆ ,⋆ , where µ denotes the multiplication map.Proof. Follows from the definition of Λ together with the differential equation (2 . (cid:3) Example 58. For any i = j , let t [ i,j ] = ( t i : . . . : t j ) denote the tuple of consecutiveelements. Contributions to ∆ ,⋆ are of the following kinds (omitting indices β i ):( t i − : t i ) ⊗ ( t : . . . : b t i − : . . . : t n ) 1 < i ≤ n (8.5) ( t i : t i − ) ⊗ ( t : . . . : b t i : . . . : t n ) 1 < i < n (8.6) Contributions to ∆ ⋆, are of the following kinds: t [1 ,n − ⊗ ( t n − : t n )(8.7) t [2 ,n ] ⊗ ( t : t n )(8.8) t [ n − , ⊗ ( t : t n )(8.9) t [ i, t [ i +1 ,n ] ⊗ ( t : t n ) 1 < i < n − t [1 ,k ] t [ i,k ] t [ i +1 ,n ] ⊗ ( t k : t n ) 1 < k < i < n − Asymptotic of the Debye polylogarithms. Let J be a subset of { , , . . . , n } .We study the asymptotics of Λ( A ,...,n ), as defined by (8 . 2) when t j for j ∈ J simulta-neously tend to infinity: i.e., for some finite values t , . . . , t n , we set:(8.12) t j = T t j , for j ∈ J , t k = t k for k / ∈ J , and let T → ∞ . Caveat 59. The Debye polylogarithms are multivalued, and so their asymptotics areonly well-defined up to monodromy. For divisors of the form D I , where I ( { , . . . , n } ,and i = I , there is a canonical branch in the neighbourhood of t i = 0 , where it vanishes(see § D { ,...,n } must one make some choice. In the followingtheorem, this ambiguity is contained in the constant C in equation (8 . . We call a string S = ( i , i , . . . , i l ) essential if i l / ∈ J and i , . . . , i l − ∈ J , and regular if: either all indices belong to J , or none of its indices belongs to J . SetΛ reg ( A S ) = (cid:26) Λ( A S ) if S is regular , , and likewise define Φ( A S ) to be 0 if S is non-essential andΦ( A S ) = t − β i i t − β i i . . . t − β in i n β i β i ,i . . . β i ,i ,...,i n − if S is essential . Theorem 60. With the assumptions (8 . above, for any < ε << we have (8.13) Λ( t : t : . . . : t n ; β , β , . . . , β n ) = µ ◦ (Φ ⊗ Λ reg ⊗ C ) ◦ ∆ (3) + O ( T ε − ) for some functions C ( t : t : . . . : t n ; β , β , . . . , β n ) = C ( β , β , . . . , β n ) which areconstant in the t ’s, and where µ denotes the triple product.Proof. Induction on the depth n . For n = 2 this theorem reduces to the well-knownasymptotics of the classical Debye polylogarithm. For strings of length two d Λ( t : t ; β , β ) = t − β t − β d Li ( t /t )where β + β = 0. It follows from this that for S = (1 , A S = ( t : t ; β , β ), d Λ( A S ) = d Φ( A S ) + O ( T ε − ) if S is essential ,d Λ reg ( A S ) if S is regular ,O ( T ε − ) otherwise . This follows from the fact that Λ diverges at most logarithmically at infinity, andlog( T ) a O ( T ε − ) = O ( T ε − ). Hence, by lemma 57 it follows that asymptotically(8.14) d Λ ∼ µ ◦ ( d Φ ⊗ Λ) ◦ ∆ ,⋆ + µ ◦ ( d Λ reg ⊗ Λ) ◦ ∆ ,⋆ , where a ∼ b means that a − b = O ( T ε − ). For the induction step, we first check thatthe differential of the difference between both sides of (8 . 13) vanishes. By inductionhypothesis, we replace Λ in (8 . 14) by (8 . d Λ ∼ µ ◦ ( d Φ ⊗ Φ ⊗ Λ reg ⊗ C + d Λ reg ⊗ Φ ⊗ Λ reg ⊗ C ) ◦ ∆ (4)1 ,⋆,⋆,⋆ . ULTIPLE ELLIPTIC POLYLOGARITHMS 31 Now compute the differential of the right-hand side of (8 . µ ◦ ( d Φ ⊗ Λ reg ⊗ C ) ◦ ∆ (3) + µ ◦ (Φ ⊗ d Λ reg ⊗ Λ reg ⊗ C ) ◦ ∆ (4) ⋆, ,⋆,⋆ . where in the second term we used lemma 57 applied to d Λ reg . In order to show that(8 . 16) and the right-hand side of (8 . 15) coincide, it suffices to use the coassociativityof the coproduct and to show that the following expression vanishes:Ω = d Φ + µ ◦ (Φ ⊗ d Λ reg ) ◦ ∆ ⋆, − µ ◦ ( d Φ ⊗ Φ + d Λ reg ⊗ Φ) ◦ ∆ ,⋆ as the difference of (8 . 16) and (8 . 15) is µ ◦ (Ω ⊗ Λ reg ⊗ C ) ◦ ∆ (3) . We will prove that(8.17) d Φ = µ ◦ (cid:2) − (Φ ⊗ d Λ reg ) ◦ ∆ ⋆, + ( d Φ ⊗ Φ) ◦ ∆ ,⋆ + ( d Λ reg ⊗ Φ) ◦ ∆ ,⋆ (cid:3) applied to ξ , where ξ = ( t : t : . . . : t n ; β , β , . . . , β n ). Case when ξ is essential . Then J = { , . . . , n − } . It follows from example 58that the only quotient sequences arising from ∆ ,⋆ are of the form ( t i : t n ) for some i < n , and are therefore not regular. Thus the first term in the right-hand side of(8 . 17) vanishes. The only contributions to the second summand come from the string( t n − , t n ); only the strings ( t i − , t i ) and ( t i , t i − ), for 1 < i < n , contribute to the lastsummand. For such a string a , let ξ/a denote its quotient. From the definitions:( d Λ reg ⊗ Φ)( a ⊗ ξ/a ) = β , ,...,i − d Li( t i − t − i ) Φ( ξ ) if a = ( t i − , t i )( d Λ reg ⊗ Φ)( a ⊗ ξ/a ) = β , ,...,i − d Li( t i t − i − ) Φ( ξ ) if a = ( t i , t i − )Using the fact that d Li( t i − t − i ) − d Li( t i t − i − ) = d log( t i ) − d log( t i − ) a straightforwardcalculation shows that both sides of (8 . 17) agree on ξ . Case when ξ is non-essential . Either n ∈ J or some i < n is not in J . Supposefirst that n ∈ J . The first term of (8 . 17) vanishes as either the argument of Φ is notessential, or the argument of Λ reg is not regular. The second and third summandsvanish since the arguments of Φ are non-essential. Hereafter, we assume n / ∈ J .Now suppose that J c contains at least 3 elements J c ⊇ { i, j, n } . Then the entireright-hand side of (8 . 17) vanishes, since every argument of Φ is always non-essential.It only remains to check the equality of (8 . 17) when J c consists of two elements { k, n } for some k < n . Consider the second and third terms on the right-hand sideof (8 . . 5) and (8 . 6) are essential only for the strings( t k : t k +1 ) and ( t k : t k − ). These are non-essential so the second factor d Φ ⊗ Φ vanishesfor all possible values of k . The third factor d Λ reg ⊗ Φ is non-trivial only when k = n − t n − : t n ). In fact, in the case k = n − 1, we have contributions fromΦ ⊗ d Λ reg (8 . 7) and d Λ reg ⊗ Φ applied to (8 . i = n − 1. They cancel.Thus in all remaining cases k < n only the first term Φ ⊗ d Λ reg of (8 . 17) can benon-zero. If k = 1 then we get terms in the first summand corresponding to (8 . . . n − X i =1 ( − i − a − i, b − i +1 ,n − = 0where a [ i,j ] = β i ( β i + β i − ) . . . ( β i + . . . + β j ), if i ≥ j and is equal to 1 otherwise, and b [ i,j ] = β i ( β i + β i +1 ) . . . ( β i + . . . + β j ), if i ≤ j and is equal to 1 otherwise. The generalcase 1 < k < n − X 18) and (8 . 19) are easily checked by taking the residues along thedivisors β i + . . . + β n − = 0 and induction. In conclusion, we have proved that Ω vanishes, and hence, by induction hypothesis,the differential of the difference between both sides of (8 . 13) is O ( T ε − ). Thus thedifference between both sides is a constant plus O ( T ε − ), which proves the theorem. (cid:3) Asymptotics in depths 1 and 2. In depth 1 we have,Λ( t ; β ) ∼ β − t − β + C ( β ) as t → ∞ , where C ( β ) = 2 iπ (1 − e ( β )) − (see lemma 62 below).Let β = β + β , t = t − = t t − . In depth two, the coproduct (8 . 4) yieldsΛ( t , t ; β , β ) ∼ t − β β Λ( t ; β ) + Λ( t ; β ) C ( β ) + C as t → ∞∼ t − β β (cid:2) Λ( t ; β ) − t β Λ( t ; β ) (cid:3) + C as t → ∞∼ t − β t − β β β + (cid:2) Λ( t ; β ) − Λ( t ; β ) (cid:3) C ( β )(8.20) + t − β β C ( β ) + C as t , t → ∞ where C , C , C are constant power series in β , β to be determined. The constant C is clearly zero as can be seen by letting t → . C (let t → C can be computed as follows.8.3.1. Limit at D ∩ D . From the second line of (8 . t →∞ lim t →∞ Λ( t , t ; β , β ) = 0 , since C = 0. Now let t , t → ∞ and then let t /t → ∞ . The third line gives aconstant contribution C ( β , β ) − C ( β ) C ( β ). It follows that(8.22) C ( β , β ) = C ( β ) C ( β ) . Limit at D ∩ D . From the first line of (8 . t →∞ lim t →∞ Λ( t , t ; β , β ) = C ( β ) C ( β ) . Now let t , t → ∞ and then let t /t → ∞ . The third line gives the constantcontribution C ( β ) C ( β )+ C ( β , β ), which yields a second equation for C ( β , β ).Note however, that the two limit computations are for different branches (see caveat59), and differ by the monodromy of the third line of (8 . 20) around the point t = t on D . The monodromy of Λ( t ; β ) (resp. Λ( t − ; β )) is πi (resp. − πi ) around a positiveupper semi-circle from 1 − to 1 + . Therefore, by equating the two different formulae for C ( β , β ) gives rise to an associator, or pentagon, equation:(8.24) (cid:0) C ( β ) + C ( β ) − iπ (cid:1) C ( β ) = C ( β ) C ( β )which is indeed satisfied by C ( β ) = πi − e ( β ) .8.4. Rationality of the constants. The argument above generalizes: Proposition 61. Let v ∈ U n be a vertex of U n , i.e., v is an intersection of bound-ary divisors D I of dimension . Then there is a branch of the Debye polylogarithm Λ( t , . . . , t n ; β , . . . , β n ) in a neighbourhood of v which is locally of the form X I =( i ,...,i n ) f I ( s , . . . , s n ) log i s . . . log i n s n where f I (0 , . . . , ∈ Q [ πi ] , and s , . . . , s n are local sector coordinates at v = (0 , . . . , . ULTIPLE ELLIPTIC POLYLOGARITHMS 33 Proof. It suffices to show that the constant coefficients lie in Q [ πi ]. But this followsfrom a standard associator argument: the 1-skeleton of the polytope C n ⊂ U n ( R ) isconnected, and the restriction of Λ to a one-dimensional stratum is a depth one Debyepolyogarithm, whose limiting values at infinity have the desired property, by (9 . C n , we deduce that theconstants at v are expressible as sums and products of the coefficients of C ( β ). (cid:3) Examples in depths 1,2 Depth 1: the classical elliptic polylogarithms. Let q = e ( τ ) with Im ( τ ) > z = e ( ξ ) with ξ in the fundamental domain D ( § L ( z ; β ) = X n ≥ Li n ( z ) β n − , which we wish to average over the spiral (0 , ∞ ) ∼ = q R z in the universal covering spaceof M , ( C ). The calculations are simplified if one considers the Debye generating seriesΛ( z ; β ) = z − β L ( z ; β ). Since d Li n ( z ) = z − Li n − ( z ) dz for n ≥ 2, we have:(9.1) d Λ( z ; β ) = z − β d Li ( z ) . Note that Λ( z ; β ) vanishes at z = 0, and so by theorem 38 the series(9.2) E ( z ; u, β ) = X n ∈ Z u n Λ( q n z ; β )converges absolutely for 1 < u < | q | − , and may have poles at u = 1, which are givenby the asymptotics of Λ( z ; β ) at z = ∞ . Since d Li( z ) is asymptotically − d log( z ) atinfinity, we deduce from (9 . 1) that there is some constant C ( β ) such that:(9.3) Λ( z ; β ) ∼ β − z − β + C ( β ) . Lemma 62. The constant at infinity is given by (9.4) C ( β ) = − β − + iπ + X n ≥ ζ (2 n ) β n − = 2 iπ − e ( β ) . Proof. The following functional equation follows from (9 . 1) and differentiating:(9.5) Λ( z ; β ) + Λ( z − ; − β ) = β − z − β + C ( β )Evaluating at z = 1 gives the expression for C ( β ), since Li n (1) = ζ ( n ), for n ≥ (cid:3) The corresponding constants in all higher dimensions are explicitly computable from C ( β ). It follows from (9 . 3) that the singular part of E ( z ; u, β ) comes from:1 β X n< u n (cid:0) ( q n z ) − β + C ( β ) (cid:1) = z − β β ( q − β u − 1) + C ( β ) u − e ( − βξ ) β ( e ( γ ) − 1) + C ( β ) e ( α ) − . where u = e ( α ), and γ = α − βτ . The second expression defines a Taylor series in β with coefficients in Q [ u, (1 − u ) − , log q, iπ ]. Thus the singular part of E ( ξ ; α, β ) is(9.6) E sing ( ξ ; α ; β ) = e ( − ξβ ) βγ + C ( β ) α . In conclusion, the regularized generating series for the classical elliptic polylog is:(9.7) E reg ( ξ ; α ; β ) = X n ∈ Z e ( αn )Λ( e ( ξ + nτ ) , β ) − e ( − ξβ ) βγ − C ( β ) α , which admits a Taylor expansion in α, β at the origin. Thus we write E reg ( ξ ; α ; β ) = X m,n ≥ Λ Em,n ( ξ ; τ ) α m β n , where Λ Em,n ( ξ ; τ ) are the classical elliptic polylogarithms of [13], and equal to thefunctions denoted ( − n Λ m,n ( ξ ; τ ) in loc. cit. , Definition 2.1. Remark . In order to retrieve the explicit formula of [13], Definition 2.1, one can writeΛ Em,n ( ξ ; τ ) as an average of certain modified (and regularized) Debye polylogarithms.For this, one simply replaces the term α − in (9 . 6) by the expression(9.8) 1 α = P ( α ) − X m> e ( mα )where P is a power series whose coefficients are related to Bernoulli numbers. Replacingthe term in γ − by a similar expression to (9 . 8) leads to the required result.9.2. Depth 2: the double elliptic polylogarithms. Consider the generating seriesof depth two Debye multiple polylogarithms:Λ( t , t ; β , β ) = t − β t − β X m ,m ≥ I m ,m ( t , t ) β m − β m − . The generating series of elliptic multiple polylogarithms is:(9.9) E ( ξ , ξ ; u , u , β , β ) = X m ,m ∈ Z u m u m Λ( q m t , q m t ; β , β )which converges absolutely for 1 < u , u < | q | − by theorem 38, and has poles along u = 1 , u = 1 , u u = 1 corresponding to logarithmic singularities of Λ( t , t ) along D , D , D . Let γ i = α i − τ β i , where e ( α i ) = u i , and q = e ( τ ). Lemma 64. The singular part of E ( ξ , ξ ; α , α , β , β ) is E sing2 = E sing(1)2 + E sing(2)2 , where E sing( i )2 comes from singularities along divisors of codimension i . We have E sing(2)2 = e − β ξ − β ξ β β γ γ + e − β ξ C ( β ) β γ α − e − β ξ C ( β ) β γ α + e − β ξ C ( β ) β γ α + C , α ( α + α ) where C , = C ( β ) C ( β ) and C ( β ) is the power series defined by (9 . , and E sing(1)2 = R β γ + R β γ + A α + A α , where A = E reg ( ξ ; α , β ) C ( β ) (cid:12)(cid:12) α =0 A = (cid:0) E reg ( ξ ; α , β ) − E reg ( ξ ; − α , β ) (cid:1) C ( β ) (cid:12)(cid:12) α =0 R = e − β ξ E reg ( ξ ; α ; β ) (cid:12)(cid:12) γ =0 R = e − β ξ E reg ( ξ ; α , β ) − e − β ξ E reg ( ξ ; α , β ) (cid:12)(cid:12) γ =0 . Here, e a = e ( a ) , α = α + α , β = β + β , and ξ = − ξ = ξ − ξ .Proof. We can compute the singularities of E from the differential equation d E ( ξ , ξ ; α , α , β , β ) = d E ( ξ − ξ ; α , β ) E ( ξ ; α + α , β + β )(9.10) − d E ( ξ − ξ ; α , β ) E ( ξ ; α + α , β + β )+ d E ( ξ ; α , β ) E ( ξ ; α , β ) ULTIPLE ELLIPTIC POLYLOGARITHMS 35 The fact that d E ( ξ , ξ , α , α ; β , β ) = 0 reduces to the Fay identity. Substitutingthe singular parts E sing (given by (9 . . 10) yields the pole structure of E .Using ( f g ) sing = ( f g sing + f sing g ) − f sing g sing , the differential equation for E (7 . 1) andthe additivity of the exponential function, we have E sing2 = E sing(1)2 − E sing(2)2 , where d E sing(2) = d (cid:16) e − ξ β − ξ β β β γ γ + e − ξ β C ( β ) β γ α − e − ξ β C ( β ) β γ α + e − β ξ C ( β ) β γ α (cid:17) d E sing(1)2 = h d E ( ξ ; α , β ) − d E ( ξ ; α , β ) i C ( β ) α + d E ( ξ , α , β ) C ( β ) α + κ | γ + γ =0 β γ + κ | γ =0 β γ + κ | γ =0 β γ where γ = γ + γ and κ = (cid:0) F ( ξ , γ ) + F ( ξ , γ ) (cid:1) e − β ξ − β ξ dξ κ = e − β ξ − β ξ F ( ξ , γ ) dξ − β e − β ξ E ( ξ ; α ; β ) dξ κ = β e − β ξ E ( ξ ; α , β ) dξ − β e − β ξ E ( ξ ; α ; β ) dξ It follows from the expansion ( iii ) of Proposition-Definition 4 plus the fact that E ( ξ, τ )is an odd function of ξ that κ | γ + γ = 0, and therefore does not contribute. There isan obvious of d E sing(2)2 . By integrating, we deduce that: E sing(1)2 = R β γ + R β γ + A α + A α + α , since d E reg and d E are equal up to higher order poles. It remains to add the constantsof integration. Since these give at most simple poles in the α ’s and correspond to thelimiting values in the corners, they contribute C , α ( α + α ) + C , α ( α + α ) , where C , and C , are the constant part of the asymptotic of the Debye doublepolylogarithm near D ∩ D and D ∩ D given by (8 . 21) and (8 . (cid:3) As in the depth 1 case, we therefore define the depth 2 multiple elliptic polyloga-rithms to be the coefficients in the Taylor expansion: E − E sing2 = X m i ,n j ≥ Λ E ( m ,m ) , ( n ,n ) ( ξ , ξ ; τ ) α m α m β n β n , where E is given by (9 . E sing2 by the previous lemma.9.3. Singular part computed from the coproduct. Another way to arrive atlemma 64 is from the computation of the asymptotic of the depth 2 Debye polyloga-rithms given in example 8 . 3. In general, we have: Corollary 65. The singular structure of the elliptic Debye polylogarithm E n is ob-tained by averaging the asymptotic of the ordinary Debye polylogarithms. In particular,it is explicitly computable from the coproduct (8 . and the constant terms C . In fact, the asymptotic of the Debye polylogarithms in the neighbourhood of bound-ary divisors of all codimensions can be computed from the coproduct in two differentways. The first, via theorem 60, is to compute the asymptotic in the neighbourhoodof codimension 1 divisors, and by induction apply the theorem to the arguments ofΛ to obtain the asymptotic in all codimensions. The other, is directly from formula(8 . 13) which immediately gives the asymptotic in all codimensions, provided that thedefinition of ‘essential’, ‘regular’, and the constants C are modified accordingly. Iterated integrals on E × We compute the integrable words corresponding to the elliptic Debye polylogarithmsviewed as functions of one variable, and compare with the bar construction. From thiswe deduce that all iterated integrals on E × are obtained by averaging.10.1. Projective coordinates and degeneration. Let G m = P \{ , ∞} , let n ≥ ⊂ G n +1 m for the union of all the diagonals. There is an isomorphism(10.1) ( G n +1 m \ ∆) / G m ∼ −→ M ,n +3 . If we write homogeneous coordinates on the left-hand side as ( t : . . . : t n +1 ), then theisomorphism is given by ( t : . . . : t n +1 ) ( t t − n +1 , . . . , t n t − n +1 ). Let β , . . . , β n +1 beformal parameters satisfying β + . . . + β n +1 = 0. Recall that we set:(10.2) Λ( t : . . . : t n +1 ; β , . . . , β n +1 ) = Λ( t t − n +1 , . . . , t n t − n +1 ; β , . . . , β n )Forgetting the marked point t n +1 gives rise to a fibration M ,n +3 → M ,n +2 (10.3) ( t : . . . : t n +1 ) ( t : . . . : t n )whose fiber over the point ( t : . . . : t n ) of M ,n is isomorphic to G m \{ t , . . . , t n } . Thefunctions (10 . Lemma 66. For constant t , . . . , t n (i.e., dt i = 0 for i ≤ n ), we have d Λ( t : . . . : t n +1 ; β , . . . , β n +1 ) = d Λ( t n : t n +1 ; β n , − β n ) × (10.4) Λ( t : . . . : t n − : t n +1 ; β , . . . , β n − , β n + β n +1 ) For ≤ i < j ≤ n , let M ij denote analytic continuation along a small loop around t i = t j . Then the functions (10 . are single-valued around t i = t j for i, j ≥ : ( M ij − id ) Λ( t : . . . : t n +1 ; β , . . . , β n +1 ) = 0 if i, j ≤ n . Proof. The differential equation (10 . 4) follows from the differential equation for Λ. Toprove the singlevaluedness, note that M ij commutes with ∂/∂t n +1 for i, j ≤ n . From(10 . 4) the result follows by induction plus the fact that Λ( t : . . . : t n +1 ) vanishes as t n +1 tends to ∞ . Alternatively, via (2 . . 2) of the function I m ,...,m n ( t , . . . , t n ) at the origin shows that it has trivial monodromy around t i = t j for i < j ≤ n . Thus the same is true of Λ( t : . . . : t n +1 ) by definition. (cid:3) The elliptic analogue of (10 . 1) is as follows. Letting ∆ ⊂ E n +1 denote the union ofall diagonals, and using the notation ( ξ : . . . : ξ n +1 ) for coordinates on E n / E , we have:( E n +1 \ ∆) / E ∼ −→ E ( n ) (10.5) ( ξ : . . . : ξ n +1 ) ( ξ − ξ n +1 , . . . , ξ n − ξ n +1 ) . Again, forgetting the marked point ξ n +1 gives rise to a fibration E ( n ) → E ( n − (10.6) ( ξ : . . . : ξ n +1 ) ( ξ : . . . : ξ n )whose fiber over the point ( ξ : . . . : ξ n ) is isomorphic to E\{ ξ , . . . , ξ n } . Definition 67. Define the elliptic Debye hyperlogarithm to be the generating series: G n ( ξ ; ξ , . . . , ξ n , α , . . . , α n , β , . . . , β n ) = E n ( ξ − ξ, . . . , ξ n − ξ ; α , . . . , α n , β , . . . , β n )viewed as a multivalued function of the single variable ξ ∈ E\{ ξ , . . . , ξ n } . ULTIPLE ELLIPTIC POLYLOGARITHMS 37 It follows from equation (10 . 4) that, for constant ξ , . . . , ξ n (i.e., dξ i = 0),(10.7) dG n ( ξ ; ξ , . . . , ξ n ; α , . . . , α n , β , . . . , β n ) = d E ( ξ n − ξ ; α n , β n ) × G n − ( ξ ; ξ , . . . , ξ n − ; α , . . . , α n − , β , . . . , β n − ) Remark . There is no obvious way to determine the constant of integration in (10 . E ( n ) is for itto vanish along a tangential base point at 1 on the Tate curve at infinity, which is notthe case for the averaged functions E n . Thus, the comparison between the averagedfunctions E n and such iterated integrals must take into account the constants.In order to circumvent this issue, let ̺ ∈ E\{ ξ , . . . , ξ n } be any point. Consider the n + 1 square matrix M ij with 1’s along the diagonal, 0’s below the diagonal, and M ij = G j − i ( ξ ; ξ i , . . . , ξ j − ; α i , . . . , α j − , β i , . . . , β j − ) for 1 ≤ i < j ≤ n + 1 . Denote this matrix by M ξ , viewed as a function of ξ . The differential equation (10 . dM ξ = M ξ Ω for some square matrix Ω of1-forms. It follows that M − ̺ M ξ satisfies the same equation. Therefore, we define G ̺i ( ξ ; ξ , . . . , ξ i ; α , . . . , α i , β , . . . , β i ) = ( M − ̺ M ξ ) ,i +1 for 0 ≤ i ≤ n These functions satisfy the differential equation (10 . 7) and vanish at ξ = ̺ if i ≥ Reminders on iterated integrals. Given a smooth manifold M over R , asmooth path γ : [0 , → M , and smooth one-forms ω , . . . , ω n on M , the iteratedintegral of ω , . . . , ω n along γ is defined to be 1 if n = 0, and for n ≥ Z γ ω . . . ω n = Z ≤ t n ≤ ... ≤ t ≤ γ ∗ ( ω )( t ) . . . γ ∗ ( ω n )( t n ) . Let A be the C ∞ de Rham complex on M , and let V ( A ) denote the zeroth cohomologyof the reduced bar complex of A . Choose a basepoint ̺ ∈ M , and let I M denote thedifferential R -algebra of multivalued holomorphic functions on M with global unipotentmonodromy. A theorem due to Chen [6] states that the map V ( A ) → V ( M ) given by X I =( i ,...,i n ) c I [ ω i | . . . | ω i n ] X I c I Z γ ω i . . . ω i n (10.8)is an isomorphism, where γ is any path from ̺ to z , and the iterated integrals are viewedas functions of the endpoint z . In particular, they only depend on the homotopy classof γ relative to its endpoints. The differential with respect to z is(10.9) ∂∂z X I c I Z γ ω i . . . ω i n = X I c I ω i ∧ Z γ ω i . . . ω i n . By successive differentiation, and using formula (10 . V ( A ) which corresponds via (10 . 8) to any given function in I ( M ). We shallapply this in the following situation. Suppose that X ֒ → A is a connected Q -model for A , so we have an isomorphism V ( X ) ⊗ Q R ∼ = V ( A ). Denote the image of the map V ( X )in I ( M ) by I ( M ) Q . It defines a Q -structure on the algebra I ( M ). If F ∈ I ( M ) Q , itsbar element in V ( X ) will be a unique element of T ( X ) by (5 . 2) (since X is connected).In the sequel, M will be a single elliptic curve with several punctures. We have a fam-ily of functions F ⊂ I ( M ), which are the functions obtained by averaging, and wantto show that F = I ( M ) Q . For this we shall write the elements of F as elements of V ( X ) ⊗ Q R by computing their differential equations, and check that: firstly they liein V ( X ), and secondly, using our explicit description of V ( X ), that they span V ( X ). Integrable words corresponding to the elliptic polylogarithms.Definition 69. Define the shuffle exponential to be the formal power series: e x ( αν ) = X n ≥ α n n ! ν x n = X n ≥ α n [ ν | . . . | ν ] | {z } n ∈ T ( Q [ ν ])[[ α ]] . The leading term in the series ( n = 0) is the empty word. Note that if w , . . . , w n aresymbols and x = e x ( αw ) then we have(10.10) x x [ w | w | . . . | w n ] = [ x | w | x | w | . . . | x | w n | x ]as an equality of power series in α with coefficients in T ( Q w ⊕ Q w ⊕ . . . ⊕ Q w n ). Lemma 70. Let ̺, ξ ∈ E , and let α, β be formal parameters, and γ = α − τ β . Thenwe have the following equality of generating series of multivalued functions: (10.11) e ( β̺ + γr ̺ ) (cid:0) E ( ξ ; α, β ) − E ( ̺ ; α, β ) (cid:1) = Z ξ̺ [Ω( ξ ; γ ) | e x ( − β ′ ω (0) − γν )] where we write ̺ = s ̺ + r ̺ τ (recall that ξ = s + r τ ).Proof. Recall that ω (0) = dξ and ν = 2 πidr . Let β ′ = 2 πiβ . It therefore followsimmediately from definition 69 and the shuffle product for iterated integrals that e ( − βξ − γr + β̺ + γr ̺ ) = Z ξ̺ e x ( − β ′ ω (0) − γν ) . We have d E ( ξ ; α, β ) = e ( − βξ ) F ( ξ ; γ ) dξ and Ω( ξ ; γ ) = e ( γr ) F ( ξ ; γ ) dξ . Hence d E ( ξ ; α, β ) = e ( − βξ − γr ) Ω( ξ ; γ ) . Combining these two facts, we see that, by (10 . d (LHS of (10.11)) = Ω( ξ ; γ ) Z ξ̺ e x ( − β ′ ω (0) − γν ) , so the differentials of both sides of (10 . 11) agree, and both vanish at ξ = ̺ . (cid:3) It is straightforward to verify that the coefficients in the right-hand side of (10 . V ( X ) ⊗ Q C . For n ≥ 1, let us define(10.12) H n ( ξ ; α ; β ) = e ( β ,...,n ̺ + γ ,...,n r ̺ ) G ̺n ( ξ ; ξ , . . . , ξ n ; α , . . . , α n , β , . . . , β n )where we recall that β ,...,n = β + . . . + β n , and likewise for γ . By construction, thecoefficients of H are combinations of elliptic multiple polylogs.Recall that X n is our rational model for the de Rham complex of E ( n ) , and a a : X n → X F n is the restriction to the fiber. For any a , . . . , a k ∈ X n , let us write[ a | . . . | a k ] = [ a | . . . | a k ] ∈ X ⊗ kF n , and extend this definition in the obvious way for formal power series in T ( X n ). Proposition 71. The generating series of functions H n is the iterated integral: (10.13) H n ( ξ ; α, β ) = Z ξ̺ W n ( ξ ) , on the fiber E\{ ξ , . . . , ξ n } of (10 . , where W ( ξ ) = [Ω( ξ − ξ ; γ ) | e x ( − β ′ ω (0) − γ ν )] , and W n is defined inductively for n ≥ by W n ( ξ ) = [Ω( ξ n − ξ ; γ n ) | e x ( − β ′ n ω (0) − γ n ν ) x W n − ( ξ )] . Formula (10 . also remains valid in the case when the marked points ξ i , ≤ i ≤ n ,are not necessarily distinct. ULTIPLE ELLIPTIC POLYLOGARITHMS 39 Proof. The case n = 1 is essentially equation (10 . n > 1, we have by (10 . dH n ( ξ ) = e ( β n ̺ + γ n r ̺ ) d E ( ξ n − ξ ; α n , β n ) H n − ( ξ )and furthermore, H n ( ̺ ) = 0. The proof of the proposition in the generic case, i.e.,when all ξ i are distinct follows by induction just as lemma 70. Finally, it follows fromlemma 66 that H n ( ξ ; α, β ) has no singularities along ξ i = ξ j for 1 ≤ i < j ≤ n , and soequation (10 . 13) remains true after degeneration of the arguments ξ i . (cid:3) Note that both sides of (10 . 13) have simple poles in the variables γ , . . . , γ n . Here-after, extracting the coefficients of a generating series such as either side of (10 . 13) willmean multiplying by γ . . . γ n and taking the Taylor expansion in α i , β i .10.4. Comparison theorem. Let Σ = { σ , . . . , σ m } be distinct points on E , where σ = 0. Fix a basepoint ̺ ∈ E\ Σ. Define F ̺ ( E\ Σ) to be the Q -algebra of multivaluedfunctions spanned by the function r − r ̺ , and the coefficients of the functions(10.14) H n ( ξ ; ξ , . . . , ξ n ; α, , for all n ≥ , where ξ , . . . , ξ n ∈ Σ . For every σ ∈ Σ, let us write ω ( i ) σ for the coefficients ofΩ( ξ − σ ; α ) = X n ≥ ω ( i ) σ α i − , and set η σ = ω (1) σ − ω (1) σ , for σ = 0. Recall that our model X F n for the puncturedelliptic curve E\ Σ is generated by ν and the ω ( i ) σ for i ≥ , σ ∈ Σ (lemma 10). Theorem 72. The map R ξ̺ : V ( X F n ) → F ̺ ( E\ Σ) is an isomorphism.Proof. By (10 . W n ( ξ ) of proposition 71 can also be written:(10.15) [Ω( ξ n − ξ ; γ n ) | P n | Ω( ξ n − − ξ ; γ n,n − ) | P n − | . . . | Ω( ξ − ξ ; γ n,..., ) | P ]where P i = e x ( − β ′ n,...,i ω (0) − γ n,...,i ν ) for 1 ≤ i ≤ n . The functions (10 . 14) correspondto the constant terms in (10 . 15) with respect to β i , namely the iterated integrals:(10.16) Z ξ̺ [Ω( ξ n − ξ ; α n ) | e x ( − α n ν ) | . . . | Ω( ξ − ξ ; α n,..., ) | e x ( − α n,..., ν )] . One easily checks that (10 . 16) is integrable, but this also follows from equation (10 . ξ , and not the path of inte-gration chosen. Thus the coefficients of (10 . 16) with respect to α lie in V ( X F n ).It suffices to show that the iterated integral of every element of V ( X F n ) arises inthis way. For this, choose any numbers ε , . . . , ε n ∈ { , } . By the multilinearity ofbar elements, the iterated integral from ̺ to ξ of any integrable word of the form(10.17) [Ω( ξ n − ξ ; α n ) − ε n Ω( ξ ; α n ) | e x ( − α n ν ) | . . .. . . | Ω( ξ − ξ ; α n,..., ) − ε Ω( ξ ; α n,..., ) | e x ( − α n,..., ν )]also lies in F ̺ ( E\ Σ). Now let π ℓ : V ( X F n ) → gr ℓ V ( X F n ) be the map which projectsonto the associated graded for the length filtration, and extended to power series inthe obvious way. It kills all Massey products of weight ≥ 2. In particular, π ℓ (cid:0) Ω( σ − ξ ; α ) − Ω( ξ ; α ) (cid:1) = − η σ π ℓ (cid:0) Ω( σ − ξ ; α ) (cid:1) = ω (0) α − Applying π ℓ to (10 . 17) (multiplied by α . . . α n to clear the poles in α i ) gives a gener-ating series in α , . . . , α n whose coefficients are all words of the form(10.18) m k ν i k . . . m ν i m ν i where i , . . . , i k are any non-negative integers, and m i = (cid:26) η ξ i if ε i = 1 ω (0) if ε i = 0It is easy to verify that every word in { ν, ω (0) , η σ , . . . , η σ m } × is a linear combinationof shuffle products of ν . . . ν with elements (10 . ℓ V ( X F n ) ∼ = T ( Q ν ⊕ Q ω (0) ⊕ Q η σ ⊕ . . . ⊕ Q η σ m ) , that the iterated integral of every element in V ( X F n ) appears a linear combination ofproducts of the function r − r ̺ = Z ξ̺ ν with coefficients of (10 . (cid:3) In particular, every iterated integral on E × can be obtained in this way. Our model V ( X ) defines a Q -structure on the de Rham fundamental groupoid of E × , hence: Corollary 73. The periods of the prounipotent fundamental groupoid ̺ Π ξ ( E × ) for anyinitial point ̺ ∈ E × and endpoint ξ ∈ E × , lie in the Q -algebra generated by r − r ̺ , andthe coefficients of (10 . with respect to the α i ’s.Remark . The previous corollary concerns a general complex elliptic curve. If E × happens to be defined over Q , then the Q -structure on the de Rham fundamentalgroupoid induced from the Q -structure on E × is is not the same as the one we consid-ered above. These two Q -structures could in principle be compared using [14], § Generalizations. One can extend theorem 72 to the case where ̺ is a tangen-tial basepoint at one of the points σ ∈ Σ. As a result, a higher-dimensional versionof theorem 72 can also be deduced from theorem 26, which states that the iteratedintegrals on the configuration space E ( n ) are products of iterated integrals on the fibersof the map E ( n ) → E ( n − , which is the one-dimensional case treated above. 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