Huei-Jeng Chen

Journal de Théorie des Nombres de Bordeaux

Huei-Jeng CHENOn shuffle of double Eisenstein series in positive characteristicTome 29, no 3 (2017), p. 815-825.<http://jtnb.cedram.org/item?id=JTNB_2017__29_3_815_0>© Société Arithmétique de Bordeaux, 2017, tous droits réservés.L’accès aux articles de la revue « Journal de Théorie des Nom-bres de Bordeaux » (http://jtnb.cedram.org/), implique l’accordavec les conditions générales d’utilisation (http://jtnb.cedram.org/legal/). Toute reproduction en tout ou partie de cet article sousquelque forme que ce soit pour tout usage autre que l’utilisation àfin strictement personnelle du copiste est constitutive d’une infrac-tion pénale. Toute copie ou impression de ce fichier doit contenir laprésente mention de copyright.cedramArticle mis en ligne dans le cadre duCentre de diffusion des revues académiques de mathématiqueshttp://www.cedram.org/Journal de Théorie des Nombresde Bordeaux 29 (2017), 815–825On shuffle of double Eisenstein series in positivecharacteristicpar Huei-Jeng CHENRésumé. L’étude du présent article s’inspire du résultat deGangl, Kaneko et Zagier sur connexion entre les valeurs zêtadoubles et les formes modulaires. Nous introduisons la séried’Eisenstein double Er,s en caractéristique positive avec les va-leurs zêta doubles ζA(r, s) comme terme constant et calculons lest-expansions de la série d’Eisenstein double. De plus, on en dé-duit les relations de shuffle de deux séries d’Eisenstein qui corres-pondent aux relations de shuffle des valeurs zêta doubles dans [4].Abstract. The study of the present paper is inspired by Gangl,Kaneko and Zagier’s result of the connection with double zeta val-ues and modular forms. We introduce double Eisenstein series Er,sin positive characteristic with double zeta values ζA(r, s) as theirconstant term and compute the t-expansions of the double Eisen-stein series. Moreover, we derive the shuffle relations of doubleEisenstein series which match the shuffle relations of double zetavalues in [4].1. IntroductionThe multiple zeta values (abbreviated as MZV’s) ζ(s1, . . . , sr), wheres1, . . . , sr are positive integers with s1 ≥ 2, are generalizations of zeta val-ues. These numbers were first studied by Euler for the case of r = 2. Hederived certain interesting relations between double zeta values, such asthe sum formula∑k−1n=2 ζ(n, k − n) = ζ(k). The MZV’s have many connec-tions with various arithmetic and geometric points of view, we refer thereader to [1, 12, 13]. Gangl–Kaneko–Zagier [5] introduced and studied dou-ble Eisenstein series Gr,s with double zeta value ζ(r, s) as its constant termfor r ≥ 3 and s ≥ 2. They showed that certain relations between double zetavalues can be “lifted” to relations between corresponding double Eisensteinseries.Thakur [9] introduced the multizeta values ζA(s1, . . . , sr) (abbreviatedas MZV’s too) in positive characteristic and [11] showed the existence ofManuscrit reçu le 23 août 2016, révisé le 21 octobre 2016, accepté le 2 décembre 2016.2010 Mathematics Subject Classification. 11J91, 11M36.Mots-clefs. Double zeta values, Eisenstein series, t-expansions, shuffle relations.816 Huei-Jeng Chenshuffle relations for multizeta values. These shuffle relations can be derivedfrom explicit formulas (see [4]). Unlike the classical double shuffle relation,there is no other expression of the product of two single (Carlitz) zetavalues than the shuffle relation mentioned above at date. In the recentwork of Chang [3], he established an effective criterion for computing thedimension of the space generated by double zeta values and a fundamentalperiod of the Carlitz module raised to the weight power in terms of specialpoints in the tensor power of the Carlitz module. One views his work as analgebraic/arithmetic point of view.In the present note the author also introduces double Eisenstein seriesEr,s in positive characteristic with double zeta values ζA(r, s) as their con-stant term. Moreover, the shuffle relations of double zeta values in [4] matchthe shuffle relations of the corresponding double Eisenstein series. We men-tion that the approach is from the idea of Gangl–Kaneko–Zagier, but theproofs are entire different. We use Goss polynomials to compute the t-expansions (an analogy to Fourier expansions) of the double Eisenstein se-ries, and use partial fractional decomposition method to show the desiredrelation.Let DZk denote the Fq(θ)-vector space spanned by double zeta values ofweight k (In the classical case we denote by DZk the Q-vector space spannedby double zeta values of weight k). It is a difficult problem in computing thedimension of DZk. In the classical case, Gangl–Kaneko–Zagier [5] showedthat the structure of the DZk are well connected to the space of modularformsMk of weight k for the full modular group Γ1 = PSL(2,Z) by means ofdouble Eisensetin series. With additional work, they bounded the dimensionof DZk in terms of the dimension of the space of cusp forms Sk of weight kfor Γ1. In a subsequent paper we shall explore the space DZk in connectionwith a suitable space of weight k Drinfeld modular forms for GL2(Fq[θ]).2. Preliminaries2.1. Notation. We adopt the notations below in the following sections.Fq := a finite field with q = pm elements.K := Fq(θ), the rational function field in the variable θ.∞ := the zero of 1/θ, the infinite place of K.| |∞ := the nonarchimedean absolute value onK corresponding to∞.K∞ := Fq((1/θ)), the completion of K with respect to | · |∞.C∞ := the completion of a fixed algebraic closure ofK∞ with respectto | · |∞.A := Fq[θ], the ring of polynomials in the variable θ over Fq.A+ := the set of monic polynomials in A.Ad := the set of polynomials in A of degree d.On shuffle of double Eisenstein series in positive characteristic 817Ad+ := Ad ∩A+, the set of monic polynomials in A of degree d.[n] := θqn − θ.Dn :=n−1∏i=0θqn − θqi = [n][n− 1]q · · · [1]qn−1 .Ln :=n∏i=1θqi − θ = [n][n− 1] · · · [1].ln := (−1)nLn.2.2. Shuffle relations of double zeta values. For any tuple(s1, . . . , sr) ∈ Nr,ζA(s1, . . . , sr) :=∑ai∈A+degθ a1>···>degθ ar1as11 . . . asrr∈ K∞.We mention that they are not only non-vanishing [10] but also transcen-dental over K [2]. Here r is called the depth and w := s1 + · · ·+ sr is calledthe weight of the presentation ζA(s1, . . . , sr). Recall that for the case r = 2,we have the following shuffle relation of double zeta values.Theorem 2.1 ([4, Theorem 3.1]). For any r, s ∈ N, we haveζA(r)ζA(s) = ζA(r, s) + ζA(s, r) + ζA(r + s)+∑i+j=r+sq−1|j[(−1)r−1(j − 1r − 1)+ (−1)s−1(j − 1s− 1)]ζA(i, j).3. Eisenstein series and Double Eisenstein series3.1. Double Eisenstein series. Let Ω := C∞ − K∞ = {z ∈ C∞ | z /∈K∞} be the Drinfeld upper half-plane, which has a natural structure asa connected admissible open subspace of the rigid analytic space P1(C∞)(see [7]).Definition 3.1. Letm, a ∈ A and z ∈ Ω. We write “mz+a  0” ifm ∈ A+or m = 0, a ∈ A+. Suppose that mz + a  0 and nz + b  0. We write“mz + a  nz + b” if one of the following conditions holds:(1) m ∈ A+, n = 0;(2) m = n = 0, a, b ∈ A+ with deg a > deg b;(3) m,n ∈ A+ with degm > degn.Definition 3.2. For r, s, k ∈ N, let Ek be the Eisenstein series of weight kdefined on z ∈ Ω :Ek(z) :=∑mz+a∈Az+Amz+a01(mz + a)k .818 Huei-Jeng ChenThe double Eisenstein series Er,s is defined byEr,s(z) :=∑mz+a,nz+b∈Az+Amz+anz+b01(mz + a)r(nz + b)s .We note that since Ek(z) and Er,s(z) converge uniformly on an admissiblecover of Ω, they are both rigid analytic functions on Ω (cf. [8]).Remark 3.3. The modular Eisenstein series in [8] are defined byEk(z) =∑mz+a∈Az+A1(mz + a)k .It can be checked that Ek(z) = 0 if q−1 - k and Ek(z) = −Ek(z) if q−1 | k.In the latter case Ek(z) is a Drinfeld modular form of weight k and type m(m is a class in Z/(q − 1)Z) for GL2(A).3.2. t-expansions. Let (−θ)1q−1 be a fixed choice of (q− 1)-st root of −θand π̃ := (−θ)qq−1∏∞i=1(1− θθqi )−1 be the fundamental period of the Carlitzmodule. Let t(z) be the rigid analytic function on Ω given byt(z) =∑b∈π̃A1(π̃z + b) .We have t(z + a) = t(z) for any a ∈ A. The function t(z) serves as auniformizing parameter “at the infinity”, so that every function f(z) onΩ satisfying the property f(z + a) = f(z) for any a ∈ A can be writtenas the form f(z) = f̃(t(z)) with respect to t. This is analogous to q(z) =exp(2π√−1z) in the classical case.Let expπ̃A(z) := z∏λ∈π̃A,λ 6=0(1− zλ) =∑i≥0 αizqi be the lattice functionon C∞ with respect to the lattice π̃A. We refer to [6] for more details ofexpπ̃A(z).Proposition 3.4 ([7, Proposition 6.5, Proposition 6.6]). There exists asequence of polynomials Gk(X) ∈ K[X] for k ∈ N satisfying the followingproperties:(1) For z ∈ Ω,π̃kGk(t(z)) =∑b∈A1(z + b)k .(2) Gk(X) = X(Gk−1 + α1Gk−q + · · ·+ αiGk−qi), where i = [logq k].(3) Gk(0) = 0 for all k.Definition 3.5. We call Gk(x) the k-th Goss polynomial.On the domain Ω, Ek(z) and Er,s(z) are invariant under the translationsz 7→ z + b for b ∈ A. We can use Proposition 3.4 to derive the expansionsof Eisenstein and double Eisenstein series with respect to t.On shuffle of double Eisenstein series in positive characteristic 819Proposition 3.6. The t-expansions of Ek(z) and Er,s(z) are given byEk(z) = ζA(k) +∑m∈A+π̃kGk(t(mz))andEr,s(z) = ζA(r, s) +∑m∈A+π̃rζA(s)Gr(t(mz))+∑m,n∈A+degm>degn≥0π̃r+sGr(t(mz))Gs(t(nz)).Proof. By Proposition 3.4(1) we have∑m∈A+,n∈A1(mz + n)k =∑m∈A+π̃kGk(t(mz)).It follows thatEk(z) =∑m=0,n∈A+1nk+∑m∈A+,n∈A1(mz + n)k= ζA(k) +∑m∈A+π̃kGk(t(mz)).Er,s(z) =∑m=n=0,a,b∈A+deg a>deg b1arbs+∑m∈A+,a∈A;n=0,b∈A+1(mz + a)rbs+∑m,n∈A+,a,b∈Adegm>degn1(mz + a)r(nz + b)s= ζA(r, s) +∑m∈A+π̃rζA(s)Gr(t(mz))+∑m,n∈A+degm>degnπ̃r+sGr(t(mz))Gs(t(nz)). 4. Shuffle of double Eisenstein seriesIn this section we will derive the shuffle relation of a product of twoEisenstein series.820 Huei-Jeng ChenTheorem 4.1. On the domain Ω, we haveEr(z)Es(z) = Er,s(z) + Es,r(z) + Er+s(z)+∑i+j=r+sq−1|j[(−1)r−1(j − 1r − 1)+ (−1)s−1(j − 1s− 1)]Ei,j(z).Proof. By Proposition 3.6 we haveEr(z)Es(z) = ζA(r)ζA(s) + ζA(r)∑n∈A+π̃sGs(t(nz))+ ζA(s)∑m∈A+π̃rGr(t(mz)) +∑m,n∈A+π̃r+sGr(t(mz))Gs(t(nz)).Let Ci,jr,s := (−1)r−1(j−1r−1)+ (−1)s−1(j−1s−1). Then we can expand Er,s(z) +Es,r(z) + Er+s(z) +∑i+j=r+sq−1|jCi,jr,sEi,j(z) as follows.Er,s(z) + Es,r(z) + Er+s(z) +∑i+j=r+sq−1|jCi,jr,sEi,j(z) = ∆1 + ∆2 + ∆3 + ∆4,where∆1 = ζA(r, s) + ζA(s, r) + ζA(r + s) +∑i+j=r+sq−1|jCi,jr,sζA(i, j),∆2 = ζA(s)∑m∈A+π̃rGr(t(mz)) + ζA(r)∑n∈A+π̃sGs(t(nz)),∆3 =∑m,n∈A+degn>degm≥0π̃r+sGr(t(nz))Gs(t(mz))+∑m,n∈A+degn>degm≥0π̃r+sGr(t(mz))Gs(t(nz)),∆4 =∑m∈A+π̃r+sGr+s(t(mz))+∑i+j=r+sq−1|jCi,jr,sζA(j)∑m∈A+π̃iGi(t(mz))+∑i+j=r+sq−1|jCi,jr,s∑m,n∈A+degn>degmπ̃r+sGi(t(mz))Gj(t(nz)).On shuffle of double Eisenstein series in positive characteristic 821By Theorem 2.1 we have ζA(r)ζA(s) = ∆1. So it suffices to show that∞∑l=0∑m,n∈Al+π̃r+sGr(t(mz))Gs(t(nz)) = ∆4.We will prove it by partial fraction decomposition. For any two distinctelements a, b in an integral domain R (we adopt R for A or A[z] here) wehave1arbs=∑i+j=r+s1(a− b)j((−1)s(j−1s−1)ai+(−1)j−r(j−1r−1)bi).Consider∞∑l=0∑m,n∈Al+π̃r+sGr(t(mz))Gs(t(nz))=∞∑l=0∑m∈Al+π̃r+sGr(t(mz))Gs(t(mz))+∞∑l=0∑m,n∈Al+m 6=nπ̃r+sGr(t(mz))Gs(t(nz)).We rewrite the first part of the sum as∑m∈A+π̃r+sGr(t(mz))Gs(t(mz))=∑m∈A+a,b∈A1(mz + a)r(mz + b)s=∑m∈A+a=b∈A1(mz + a)r(mz + b)s +∑m∈A+a6=b∈A1(mz + a)r(mz + b)s=∑m∈A+π̃r+sGr+s(t(mz))+∑m∈A+a6=b∈A∑i+j=r+s1(a− b)j((−1)s(j−1s−1)(mz + a)i +(−1)j−r(j−1r−1)(mz + b)i)=∑m∈A+π̃r+sGr+s(t(mz))+∑i+j=r+s∑m∈A+a,f∈A,f 6=01f j((−1)s(j−1s−1)(mz + a)i +(−1)j−r(j−1r−1)(mz + f + a)i).822 Huei-Jeng ChenFor a fixed pair (i, j), we have that∑m∈A+a,f∈A,f 6=01f j(−1)s(j−1s−1)(mz + a)i =∑η∈F∗q∑m∈A+a∈A,f∈A+1(ηf)j(−1)s(j−1s−1)(mz + a)i=0, if q − 1 - j;−∑m∈A+a∈A,f∈A+1f j(−1)s(j−1s−1)(mz + a)i , if q − 1|j.Similarly,∑m∈A+a,f∈A,f 6=01f j(−1)j−r(j−1r−1)(mz + f + a)i =∑m∈A+b,f∈A,f 6=01f j(−1)j−r(j−1r−1)(mz + b)i=0, if q − 1 - j;−∑m∈A+b∈A,f∈A+1f j(−1)j−r(j−1r−1)(mz + b)i , if q − 1|j.This implies that∑m∈A+π̃r+sGr+s(t(mz))+∑i+j=r+s∑m∈A+a,f∈A,f 6=01f j((−1)s(j−1s−1)(mz + a)i +(−1)j−r(j−1r−1)(mz + f + a)i)=∑m∈A+π̃r+sGr+s(t(mz))+∑i+j=r+sq−1|j∑m∈A+b∈A,f∈A+1f j(−1)s−1(j−1s−1)+ (−1)j−r−1(j−1r−1))(mz + a)i=∑m∈A+π̃r+sGr+s(t(mz)) +∑i+j=r+sq−1|jCi,jr,sζA(j)∑m∈A+π̃iGi(t(mz)).On shuffle of double Eisenstein series in positive characteristic 823By using a similar argument, the second sum can be expressed as∞∑l=1∑m,n∈Al+,m 6=nπ̃r+sGr(t(mz))Gs(t(nz))=∞∑l=1∑m,n∈Al+m6=n,a,b∈A1(mz + a)r(nz + b)s=∞∑l=1∑m,n∈Al+m6=n,a,b∈A∑i+j=r+s1((m−n)z+(a−b))j((−1)s(j−1s−1)(mz + a)i +(−1)j−r(j−1r−1)(nz + b)i)=∞∑l=1∑i+j=r+s∑m∈Al+,n′∈A<ln′ 6=0,a,b′∈A(−1)s(j−1s−1)(n′z + b′)j(mz + a)i+∞∑l=1∑i+j=r+s∑n∈Al+,m′∈A<lm′ 6=0,a′,b∈A(−1)j−r(j−1r−1)(m′z + a′)j(nz + b)i=∞∑l=1∑i+j=r+s∑f∈F∗q1f j∑m,n′∈A+,a,b′∈Adegn′<degm=l(−1)s(j−1s−1)(n′z + b′)j(mz + a)i+∞∑l=1∑i+j=r+s∑f∈F∗q1f j∑m′,n∈A+,a′,b∈Adegm′<degn=l(−1)j−r(j−1r−1)(m′z + a′)j(nz + b)i=∞∑l=1∑i+j=r+sq−1|j∑m,n′∈A+,a,b′∈Adegn′<degm=lCi,jr,s1(n′z + b′)j(mz + a)i=∞∑l=1∑i+j=r+sq−1|jCi,jr,s∑m,n′∈A+degn′<degm=lπ̃r+sGi(t(mz))Gj(t(n′z)).Combining the first sum with the second sum, we have∞∑l=0∑m,n∈A+degm=degn=lπ̃r+sGr(t(mz))Gs(t(nz))824 Huei-Jeng Chen=∞∑l=0∑m∈A+degm=lπ̃r+sGr(t(mz))Gs(t(mz))+∞∑l=0∑m,n∈A+degm=degn=l,m6=nπ̃r+sGr(t(mz))Gs(t(nz))=∑m∈A+π̃r+sGr+s(t(mz)) +∑i+j=r+sq−1|jCi,jr,sζA(j)∑m∈A+π̃iGi(t(mz))+∞∑l=1∑i+j=r+sq−1|jCi,jr,s∑m,n′∈A+degn′<degm=lπ̃r+sGi(t(mz))Gj(t(n′z))= ∆4. Remark 4.2. From Proposition 3.4(3), we see that the constant terms inthe t-expansions of Er(z)Es(z) andEr,s(z) + Es,r(z) + Er+s(z)+∑i+j=r+sq−1|j[(−1)r−1(j − 1r − 1)+ (−1)s−1(j − 1s− 1)]Ei,j(z)are ζA(r)ζA(s) andζA(r, s) + ζA(s, r) + ζA(r + s)+∑i+j=r+sq−1|j[(−1)r−1(j − 1r − 1)+ (−1)s−1(j − 1s− 1)]ζA(i, j)respectively. It follows that the relation in Theorem 4.1 can be viewed as a“lifting” of the shuffle relation in Theorem 2.1.References[1] F. Brown, “Mixed Tate motives over Z”, Ann. Math. 175 (2012), no. 2, p. 949-976.[2] C.-Y. Chang, “Linear independence of monomials of multizeta values in positive character-istic”, Compos. Math. 150 (2014), no. 11, p. 1789-1808.[3] ———, “Linear relations among double zeta values in positive characteristic”, Camb. J.Math. 4 (2016), no. 3, p. 289-331.[4] H.-J. Chen, “On shuffle of double zeta values over Fq [t]”, J. Number Theory 148 (2015),p. 153-163.[5] H. Gangl, M. Kaneko & D. Zagier, “Double zeta values and modular forms”, in Auto-morphic forms and zeta functions (Tokyo, 2004), World Scientific, 2006, p. 71-106.[6] E.-U. Gekeler, “On the coefficients of Drinfeld modular forms”, Invent. Math. 93 (1988),no. 3, p. 667-700.[7] D. Goss, “The algebraist’s upper half-plane”, Bull. Am. Math. Soc. (1980), no. 2, p. 391-415.On shuffle of double Eisenstein series in positive characteristic 825[8] ———, “π-adic Eisenstein series for function fields”, Compos. Math. 41 (1980), p. 3-38.[9] D. S. Thakur, Function field Arithmetic, World Scientific, 2004, xv+388 pages.[10] ———, “Power sums with applications to multizeta and zeta zero distribution for Fq [t]”,Finite Fields Appl. 15 (2009), no. 4, p. 534-552.[11] ———, “Shuffle relations for function field multizeta values”, Int. Math. Res. Not. 2010(2010), no. 11, p. 1973-1980.[12] D. Zagier, “Values of zeta functions and their applications”, in First European Congressof Mathematics Vol. II (Paris, 1992), Progress in Mathematics, vol. 120, Birkhäuser, 1994,p. 497-512.[13] J. Zhao, Multiple zeta functions, multiple polylogarithms and their special values, Series onNumber Theory and Its Applications, vol. 12, World Scientific, 2016, xxi+595 pages.Huei-Jeng ChenTaida Institute for Mathematical SciencesNo.1, Sec.4, Roosevelt RoadTaipei, TaiwanE-mail: hjchen1204@ntu.edu.tw

On shuffle of double Eisenstein series in positive characteristic

Chen, Huei-Jeng

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On shuffle of double Eisenstein series in positive characteristic

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