Linear independence of indefinite iterated Eisenstein integrals Nils Matthes Abstract 6 1 We prove linear independence of indefinite iterated Eisenstein integrals over the 0 fraction field of the ring of formal power series Z[[q]]. Our proof relies on a general 2 criterium for linear independence of iterated integrals, which has been established n by Deneufchâtel, Duchamp, Minh and Solomon. As a corollary, we obtain C-linear a independence of indefinite iterated Eisenstein integrals,which has applications to the J study of elliptic multiple zeta values, as defined by Enriquez. 1 2 1 Introduction ] T N Given a collection ω ,...,ω of smooth one-forms on a smooth manifold M, and a 1 r . smooth path γ :[0,1]→M, one defines their iterated integral as h t a ω ...ω = γ∗(ω )...γ∗(ω ), (1.1) m 1 r 1 r Zγ Z0≤t1≤...≤tr≤1 [ where γ∗(ω ) = f (t )dt denotes the pull back of ω along γ. In the case of a single i i i i i 1 differential one-form ω, (1.1) is simply the path integral of ω along γ. v A classical application of iterated integrals is the construction of solutions to cer- 3 4 tain systems of linear differential equations via the method of Picard iteration (cf. 7 e.g.[14]). However,iteratedintegralsalsoappearin number theory,prominentexam- 5 plesbeingmultiplepolylogarithms andmultiplezetavalues,whichareiteratedintegrals 0 on P1\{0,1,∞} (see for example the lecture notes [8] for an introduction from the . 1 point of view of iterated integrals). It is known that the multiple polylogarithms are 0 linearly independent over C [13]. Using rather different techniques, this result has 6 beengeneralized[5],withCreplacedbyanarbitraryfieldoffunctions satisfyingsome 1 extra conditions. : v On the other hand, another family of iterated integrals arising in number theory i X areiteratedintegralsofmodularforms. Theirstudy hasbeeninitiatedbyManin[11], and was later extended in [3, 7, 9]. Known in the literature under the names iterated r a Eichler integrals [3] or iterated Shimura integrals [11], these are iterated integrals on the upper half-plane, which generalize the classicalEichler integrals [10], and are also closely related to L-functions of modular forms [11, 3]. Iterated integrals of modular forms also appear in the study of elliptic multiple zeta values [4, 6, 2, 12], the latter being a natural genus one analogue of the classical multiple zeta values. In [2], a procedure for decomposing elliptic multiple zeta values into certainC-linearcombinationsof(indefinite) iteratedintegralsofEisensteinseries (called iterated Eisenstein integrals for short)1 is described. The uniqueness of this decomposition, important both for the mathematical theory as well as for applica- tions to physics [1], depended on the C-linear independence of the iterated Eisenstein integrals in question. 1All modular forms appearing in this paper are modular forms for the group SL2(Z). 1 In this paper, we prove linear independence of iterated Eisenstein integrals, first overthefractionfieldFrac(Z[[q]])oftheringofformalpowerseriesinonevariablewith integer coefficients, where q is viewed as a coordinate on the open unit disk. By the main result of [5], it is enough to prove that Frac(Z[[q]]) does not contain primitives of Eisenstein series, which in turn follows from a computation of their denominators. Having established linear independence over Frac(Z[[q]]), the linear independence of iterated Eisenstein integrals over Q follows immediately, since Q ⊂ Frac(Z[[q]]). Finally,byextendingscalarsfromQtoC,weobtainthedesiredC-linearindependence of iterated Eisenstein integrals. Acknowledgments. Very many thanks to Pierre Lochak for bringing the paper [5] to my attention, as well as for helpful discussions and remarks. This paper is part of the author’s doctoral thesis at Universität Hamburg, and I would like to thank my advisor Ulf Kühn for helpful remarks. 2 Iterated Eisenstein integrals 2.1.Eisenstein series. For k ≥ 1 denote by G the Hecke-normalized Eisenstein 2k series (cf. e.g. [17]), which is the function on the upper half-plane H = {z ∈ C | Im(z)>0}, defined by the convergentseries B G (q)=− 2k + σ (n)qn ∈Q⊕qZ[[q]], q =e2πiτ, (2.1) 2k 4k 2k−1 n≥1 X where B denotes the 2k-th Bernoulli number, and σ (n)= d2k−1. We also 2k 2k−1 d|n set G ≡−1. 0 P The function G is holomorphic, and, for k ≥2, it is a modular form for SL (Z). 2k 2 Write G∞ for the constant term in its q-expansion, and likewise G0 (q) for G (q)− 2k 2k 2k G∞. Note that for k ≥1, we have 2k B G∞ =− 2k, G0 (q)= σ (n)qn. (2.2) 2k 4k 2k 2k−1 n≥1 X 2.2.Regularization of iterated integrals. We would now like to define iterated Eisenstein integrals i∞ G (q )dτ ...G (q )dτ (2.3) 2k1 1 1 2kn n n Zτ as functions depending onsome startpoint τ ∈H,where the integrationis performed along some path from τ to the cusp i∞2. Unfortunately, in this case the usual def- inition of iterated integrals (1.1) produces divergent integrals, already in the case of single Eisenstein integrals, i.e. for n = 1. In order to overcome this problem, we describe a regularization scheme for such iterated integrals, introduced by Brown in [3]. For the rest of this subsection, we follow [3]. Let W = C[[q]]<1 be the C-algebra of formal power series, which converge on the open q-disk D = {q ∈ C||q| < 1}, and denote by D∗ := D\{0} the punctured disk. Via the universal covering map exp:H→D∗, τ 7→e2πiτ, (2.4) wecanconsiderW asaC-subalgebraoftheC-algebraO(H)ofholomorphicfunctions on the upper half-plane. 2The value of the iterated integral does not depend on the choice of path, since the Eisenstein series are holomorphic functions on a one-dimensional complex manifold. 2 WriteW =W0⊕W∞ withW0 =qC[[q]]andW∞ =C. Forapowerseriesf ∈W, define f0 to be its image in W0 under the natural projection, and define f∞ ∈ W∞ likewise. Denote by Tc(W) the tensor coalgebra on the C-vector space W, which comes equipped with a shuffle product . We will use bar notation for elements of Tc(W), and define a map R:Tc(W)→(cid:1)Tc(W) by the formula n R[f |...|f ]= (−1)n−i[f |...|f ] [f∞|...|f∞ ]. (2.5) 1 n 1 i (cid:1) n i+1 i=0 X We can now make the Definition 2.1. Given f ,...,f ∈ W ⊂ O(H) as above, their regularized iterated 1 n integral is defined as n i∞ 0 I(f ,...,f ;τ):= R[f |...|f ] [f∞ |...|f∞] , (2.6) 1 n 1 i dτ i+1 n dτ i=0Zτ Zτ X where b b [f |...|f ] := f (τ )dτ ...f (τ )dτ . (2.7) 1 n dτ 1 1 1 n n n Za Za Proposition 2.2. For all f ,...,f ∈ W, I(f ,...,f ;τ) is well-defined, i.e. (2.6) 1 n 1 n is finite, and we have ∂ I(f ,...,f ;τ)=−f (τ )I(f ,...,f ;τ ). (2.8) ∂τ τ=τ0 1 n 1 0 2 n 0 (cid:12) Proof: [3], Lemma 4(cid:12).5 and Proposition 4.7 i). (cid:12) Thesecondpartoftheprecedingpropositionistheanalogueforregularizediterated integralsofthe differentialequationsatisfiedbyordinaryiteratedintegrals([8], p.40). It will be crucial in the proof of linear independence of iterated Eisenstein integrals. 2.3.Iterated integrals on the q-disk. We have seen that I(f ,...,f ;τ) is a holo- 1 n morphic function on the upper half-plane. Using the change of coordinates (2.4), we can rewrite I(f ,...,f ;τ) as a regularizediterated integral on the punctured q-disk 1 n 1 n 0 1 I(f1,...,fn;τ)= (2πi)n i=0Zq R[f1|...|fi]dqq Zq [(f∞)i+1|...|(f∞)n]dqq. (2.9) X The virtue of representation (2.9) is that one sees that I(f ,...,f ;τ)∈W[log(q)], log(q):=2πiτ, (2.10) 1 n and thereforeevery linearidentity betweenthe I(f ,...,f ;τ) reduces, by comparing 1 n coefficients,toalinearsystemofequations. Also,notethatifallf ∈W :=Q[[q]]∩W, i Q then (2πi)nI(f ,...,f ;τ)∈W [log(q)]. 1 n Q Definition 2.3. For k ,...,k ≥ 0, we define the (indefinite, Hecke-normalized) 1 n iterated Eisenstein integral to be G(2k ,...,2k ;q)=(2πi)nI(G ,...,G ;τ)∈W [log(q)]. (2.11) 1 n 2k1 2kn Q Note that by Proposition 2.2, 1 ∂ ∂ G(2k ,...,2k ;q)=q G(2k ,...,2k ;q) 2πi∂τ τ=τ0 1 n ∂q q=q0 1 n (cid:12) (cid:12) (cid:12) =−G(cid:12) (q )G(2k ,...,2k ;q ). (2.12) (cid:12) 2(cid:12)k1 0 2 n 0 3 Example 2.4. In length one, we have (cf. [3], Example 4.10) B σ (n) G(2k;q)= 2k log(q)− 2k−1 qn. (2.13) 4k n n≥1 X Later on, we will also need the integral over the non-constant part G0 of the 2k Eisenstein series G . We denote this by 2k 0 dq σ (n) G0(2k;q):= G0 (q ) 1 =− 2k−1 qn. (2.14) 2k 1 q n Zq 1 n≥1 X 3 Proof of linear independence Having defined iterated Eisenstein integrals in the last section, we now turn to the proofoftheirlinearindependence. Thelargerpartofthissectionisdevotedtoproving linear independence over Frac(Z[[q]]), the fraction field of the ring of formal power series with integer coefficients. In order to achieve this, we use the following general linear independence result for iterated integrals, which is (a special case of) the main result of [5] (Theorem 2.1). Let X be analphabet (not necessarilyfinite), anddenote by X∗ the free monoid on X. Theorem 3.1 (Deneufchâtel, Duchamp, Minh, Solomon). Let (A,d) be a differential algebra over a field k of characteristic zero, whose ring of constants ker(d) is precisely equal to k. Let C be a differential subfield of A (i.e. a subfield such that dC ⊂ C). Suppose that S ∈AhhXii is a solution to the differential equation dS =M ·S, (3.1) where M = u x ∈ ChhXii is a homogeneous series of degree 1, with initial x∈X x condition S =1, where S denotes the coefficient of the empty word in the series S. 1 1 P The following are equivalent: (i) The family of coefficients (Sw)w∈X∗ of S is linearly independent over C. (ii) The family {u } is linearly independent over k, and we have x x∈X dC∩Span ({u } )={0}. (3.2) k x x∈X We are now in a position to prove our main result. Theorem 3.2. The family of iterated Eisenstein integrals (2.11) is linearly indepen- dent over Frac(Z[[q]]). Proof: We will apply Theorem 3.1 with the following parameters: • k = Q, A = Q((q))[log(q)] with differential d = q ∂ , and C = Frac(Z[[q]]) (the ∂q latter is a differential field by the quotient rule for derivatives) • X ={a } , u =−G (q), hence 2k k≥0 a2k 2k M(τ)=− G (q)a . 2k 2k k≥0 X With these conventions, it follows from (2.12) that the formal series i∞ i∞ 1+ [M(τ )] + [M(τ )|M(τ )] +...∈O(H)hhXii, (3.3) 1 dτ 1 2 dτ Zτ Zτ 4 where the iterated integrals are regularized as in Section 2.2, is a solution to the differential equation dS = M ·S, with S = 1. Consequently, the coefficient of the 1 word w = a ...a in S is equal to G(2k ,...,2k ;q). Moreover, since Q-linear 2k1 2kn 1 n independence of the Eisenstein series is well-known (cf. e.g. [16], VII.3.2), it remains to verify (3.2) in our situation. To this end, assume that there exist α ∈ Q, all but finitely many of which are 2k equal to zero, such that α G (q)∈dC. (3.4) 2k 2k k≥0 X Clearingdenominators,wemayassumethatα ∈Z. Furthermore,fromthedefinition 2k of d=q ∂ , one sees that the image dC of the differential operator d does not contain ∂q any constant except for zero. Therefore, the constant term α G∞ of (3.4) k≥0 2k 2k vanishes; in other words P α G (q)= α G0 (q)∈qQ[[q]]. (3.5) 2k 2k 2k 2k k≥0 k≥1 X X NowthedifferentialdisinvertibleonqQ[[q]],andinvertingdisthesameasintegrating. Hence (3.4) is equivalent to α G0(2k;q)∈C. (3.6) 2k k≥1 X But this is absurd, unless all the α vanish, as we shall see now. Indeed, if f ∈ 2k C = Frac(Z[[q]]), then there exists m ∈ Z\{0} such that f ∈ Z[m−1]((q)). This follows from the well-known inversion formula for power series. On the other hand, the coefficient of qp in G0(2k;q), for p a prime number, is given by σ (p) p2k−1+1 1 2k−1 = ≡ mod Z (3.7) p p p (cf. (2.14)). Thus, we must have 1 α ∈Z[m−1], (3.8) p 2k k≥1 X for every prime number p. But then the integer α is divisible by infinitely k≥1 2k many primes (namely, at least all the primes which don’t divide m), which implies P α =0. k≥1 2k Now assume that k is the smallest positive, even integer with the property that 1 Pα 6=0. Consider the coefficient of qpk1 in G0(2k;q), which is equal to k1 σ2k−1(pk1) = 1 k1 pj(2k−1) ≡ pk11 mod Z if 2k >k1 (3.9) pk1 pk1 j=0 (pk11 + p1 mod Z if 2k =k1 X (cf. (2.14)). By(3.6), αk1+ 1 α ∈Z[m−1],andbywhatwehaveseenbefore, p pk1 k≥1 2k α =0. Hence k≥1 2k P α k1 ∈Z[m−1], (3.10) P p for every prime number p, which again implies α = 0, in contradiction to our k1 assumptionα 6=0. Therefore,in (3.6),we musthaveα =0 forallk ≥1 and(3.2) k1 2k is verified. 5 Having established linear independence of iterated Eisenstein integrals over the field Frac(Z[[q]]), linear independence over C follows almost immediately. Corollary3.3. ThefamilyofiteratedEisensteinintegralsG(2k ,...,2k ;q)forn≥0 1 n and all k ≥0 is linearly independent over the complex numbers. i Proof: LetG ,...,G beiteratedEisensteinintegrals,andassumewehavearelation 1 n n α G =0 (3.11) i i i=1 X with α ∈ C. Since Q ⊂ Frac(Z[[q]]), it follows from Theorem 3.2 that the matrix of i the coefficients of the G , considered as series in log(q)kql for k,l ≥ 0, has maximal i rank n. Therefore α =0 for i=1,...,k. i Remark 3.4. By the shuffle product formula, the C-vector space spanned by the iterated Eisenstein integrals is a C-algebra. Corollary 3.3 implies that it is a free shuffle algebra, and thus a polynomial algebra by [15]. References [1] J. Broedel, C. R. Mafra, N. Matthes, and O. Schlotterer. Elliptic multiple zeta values and one-loop superstring amplitudes. Journal of High Energy Physics, 7:112, July 2015. [2] J. Broedel, N. Matthes, and O. Schlotterer. Relations between elliptic multiple zeta values and a special derivation algebra. 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Translated from the French, Graduate Texts in Mathematics, No. 7. [17] D. Zagier. Periods of modular forms and Jacobi theta functions. Invent. Math., 104(3):449–465,1991. E-mail: [email protected] Fachbereich Mathematik (AZ) Universität Hamburg Bundesstrasse 55 D-20146 Hamburg 7