Mathematische Annalen

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1 Math. Ann. (203) 357:09 8 DOI 0.007/s Mathematische Annalen Double zeta values, double Eisenstein series, and modular forms of level 2 Masanobu Kaneo Koji Tasaa Received: 2 December 20 / Revised: December 202 / Published online: 3 April 203 Springer-Verlag Berlin Heidelberg 203 Abstract We study the double shuffle relations satisfied by the double zeta values of level 2, and introduce the double Eisenstein series of level 2 which satisfy the double shuffle relations. We connect the double Eisenstein series to modular forms of level 2. Mathematics Subject Classification (2000) M32 F Introduction In [7], Gangl, Zagier and the first author studied in detail the double shuffle relations satisfied by the double zeta values ζ(r, s) = m>n>0 m r n s (r 2, s ), () and revealed in particular various connections between the space of double zeta values and the space of modular forms as well as their period polynomials on the full modular group PSL 2 (Z). They also defined the double Eisenstein series and deduced the double shuffle relations for them, and in [9] we illustrated a way to connect the double Eisenstein series to the period polynomials of modular forms (of level ). This wor is partially supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (S) , (B) , and Grant-in-Aid for JSPS Fellows (No ). M. Kaneo (B) K. Tasaa Kyushu University, 744, Motooa, Nishi-u, Fuuoa , Japan maneo@math.yushu-u.ac.jp K. Tasaa -tasaa@math.yushu-u.ac.jp 23

2 092 M. Kaneo, K. Tasaa In the present paper, we consider the double shuffle relations of level 2 and study the formal double zeta space, whose generators are the formal symbols corresponding to the double zeta values of level 2 (Euler sums) and the defining relations are the double shuffle relations. One of the relations we obtain in the formal double zeta space (Theorem ) has an interesting application to the problem of representations of integers as sums of squares, and this will be given in the subsequent paper by the second author [2]. We then proceed to define the double Eisenstein series of level 2 and show that they also satisfy the double shuffle relations (Theorem 3), and have connections lie in the case of level to double zeta values, modular forms, and period polynomials, of level 2 (Theorem 5 and Corollary ). 2 The double zeta values of level 2 The double zeta values of level 2 we are referring to are the following four types of real numbers given for integers r 2 and s : ζ ee (r, s) = ζ oe (r, s) = m>n>0 m, n: even m>n>0 m: odd, n: even m r n s, ζeo (r, s) = m>n>0 m: even, n: odd m r n s, ζoo (r, s) = m>n>0 m, n: odd m r n s. m r n s, These numbers can be written as simple linear combinations of the original multiple zeta values () and the numbers often referred to as Euler sums defined by ζ(r, s) = m>n>0 ( ) n m r,ζ(r, s) = ns m>n>0 and vice versa. Explicitly, we have the relations ( ) m m r,ζ(r, s) = ns m>n>0 ( ) m+n m r n s, ζ ee (r, s) ζ(r, s) ζ oe (r, s) ζ eo (r, s) = ζ(r, s) 4 ζ(r, s). (2) ζ oo (r, s) ζ(r, s) (The matrix on the right is invertible.) Note that, from the obvious relations and 23 ζ(r, s) = ζ ee (r, s) + ζ eo (r, s) + ζ oe (r, s) + ζ oo (r, s) ζ(r, s) = 2 r+s ζ ee (r, s),

3 Double zeta values, double Eisenstein series, and modular forms of level we have the relation (2 r+s )ζ ee (r, s) = ζ eo (r, s) + ζ oe (r, s) + ζ oo (r, s). We shall hereafter only consider ζ eo (r, s), ζ oe (r, s), and ζ oo (r, s). Moreover define ζ e () = n>0, even n and ζ o () = n>0, odd n. Then in the standard manner we can show the following double shuffle relations. Proposition For positive integers r, s 2, we have ζ o (r)ζ e (s) = ζ oe (r, s) + ζ eo (s, r) = (( ) i ζ oe (i, j) + r i+ j=r+s i 2, j i+ j=r+s i 2, j ( ) ) i ζ oo (i, j), s ζ o (r)ζ o (s) = ζ oo (r, s) + ζ oo (s, r) + ζ o (r + s) = (( ) ( )) i i + ζ eo (i, j). r s Proof The first equality in each sequence of identities is obtained as usual from the manipulation of the defining series. For the second, we use the following integral representations of each zeta value and the shuffle product of integrals: ζ o () = ζ e () = ζ eo (r, s) = ζ oe (r, s) = >t >t 2 > >t >0 >t >t 2 > >t >0 >t >t 2 > >t r+s >0 dt r+ dt r+s t r+ dt dt 2 dt t t 2 t dt dt 2 dt t t 2 t dt dt 2 dt r t t 2 t r dt r+s t r+s tr+s 2, >t >t 2 > >t r+s >0 dt r+ t r+ dt r+s t r+s dt dt 2 dt r t t 2 t r tr+sdt r+s tr+s 2, dt t 2, t dt t 2, dt r t 2 r dt r t 2 r 23

4 094 M. Kaneo, K. Tasaa ζ oo (r, s) = >t >t 2 > >t r+s >0 dt r+ t r+ dt r+s t r+s dt dt 2 dt r t t 2 t r dt r+s tr+s 2. t r dt r t 2 r The first two are easy to deduce, and to see the rest for double zetas, we use the expression (2) of each double zeta value in terms of Euler sums and the standard integral representations ζ(r, s) = ζ(r, s) = ζ(r, s) = ζ(r, s) = >t >t 2 > >t r+s >0 >t >t 2 > >t r+s >0 >t >t 2 > >t r+s >0 >t >t 2 > >t r+s >0 dt t dt t dt t dt t dt r t r dt r dt r+ dt r+s t r t r+ t r+s dt r+s t r+s, dt r ( dt r ) dt r+ dt r+s ( dt r+s), t r + t r t r+ t r+s + t r+s dt r dt r dt r+ dt r+s ( dt r+s), t r t r t r+ t r+s + t r+s dt r ( dt r ) dt r+ dt r+s dt r+s. t r + t r t r+ t r+s t r+s Noting the identities t 2 = ( 2 t ( ) ), + t t t 2 = ( 2 t + ( ) ), + t we obtain the desired integral expressions and hence the proposition by shuffle products of integrals. Now we introduce the level 2 version of the formal double zeta space studied in [7] as follows. Let > 2 and DZ be the Q-vector space spanned by formal symbols Zr,s eo, Z r,s oe, Z r,s oo, Poe r,s, Poo r,s (r, s, r + s = ), and Z o with the set of relations P oe r,s = Z oe r,s + Z eo s,r = i+ j= i, j P oo r,s = Z oo r,s + Z oo s,r + Z o = for r, s, r + s =, so that 23 (( ) ( ) ) i i Zi, oe j r + Zi, oo j, (3) s i+ j= i, j (( ) i + r ( )) i Zi, eo j (4) s {Q-linear combinations of Z eo r,s DZ =, Z r,s oe, Z r,s oo, Poe r,s, Poo r,s, Z o}. Q-linear span of relations (3), (4)

5 Double zeta values, double Eisenstein series, and modular forms of level Since the elements Pr,s oe given by DZ = where the defining relations (5) and (6)are and Poo r,s are written in Z s, we can also regard the space as {Q-linear combinations of Z eo r,s, Z oe r,s, Z oo r,s, Z o } Q-linear span of relations (5), (6) Z oe r,s + Z eo s,r = i+ j= i, j Z oo r,s + Z oo s,r + Z o = (( ) ( ) ) i i Zi, oe j r + Zi, oo j, (5) s i+ j= i, j (( ) i + r ( )) i Zi, eo j s. (6) Note that the relations (3) and (4) (as well as (5) and (6)) correspond to those in Proposition when r, s 2, under the correspondences Zr,s eo ζ eo (r, s), Zr,s oe ζ oe (r, s), Zr,s oo ζ oo (r, s), Z o ζ o (), Pr,s oe ζ o (r)ζ e (s), Pr,s oo ζ o (r)ζ o (s), because in that case the binomial coefficients for i = on the right vanishes. For our later applications it is convenient to allow the divergent Z, eo, Poe, etc., and in fact the double shuffle relations in Proposition can be extended for r = ors = by using a suitable regularization procedure for ζ(, s) etc. developed in [2] (the case of m = 2 in their notation). Specifically, by setting ζ o () := 2 (T + log 2), ζ e () := (T log 2) (7) 2 and, for s 2 ζ eo (, s) = 2 ζ o (s)t 2 (log 2)ζ o (s) ζ oe (s, ), ζ oe (, s) = 2 ζ e (s)t + 2 (log 2)ζ e (s) ζ eo (s, ), ζ oo (, s) = 2 ζ o (s)t + 2 (log 2)ζ o (s) ζ oo (s, ) ζ o (s + ) where T is a formal variable, the equations in Proposition are valid for all r, s except (r, s) = (, ). 23

6 096 M. Kaneo, K. Tasaa Theorem Suppose is even and 4. InDZ, we have ) 2 r=2 r:even Z oo r, r = 4 Z o. 2) Each Pr, r oe with r even can be written as a Q-linear combination of Poo i, j (i, j : even, i + j = ) and Z o Proof Consider the generating functions Z eo (X, Y ) = Zr,s eo Xr Y s, Z oe (X, Y ) = r+s= Z oo (X, Y ) = Zr,s oo Xr Y s. r+s= r+s= Z oe r,s Xr Y s, Here and in the following, the sum r+s= always means r+s=, r,s. The double shuffle relations (5) and (6) are equivalent to the relations Z oe Z oo (X, Y )+Z eo (Y, X)=Z oe (X + Y, Y )+Z oo (X + Y, X), (8) (X, Y )+Z oo (Y, X)+ Z o X Y X Y =Z eo (X + Y, Y )+Z eo (X + Y, X). Substituting X =, Y = 0in(8) and X =, Y = in(9), we respectively obtain Z, oe + Z, eo = Z, oe + Zr, r oo, (0) 2 r= ( ) r Zr, r oo + Z o We divide () by 2 and add (0) to obtain and hence ) of Theorem. 2 2 Z o = 2 r=2 r: even To prove 2), we need the following lemma. r= (9) = 2Z eo,. () Z oo r, r Lemma Let 4 be an even integer and a i, j, b i, j, c i, j be rational numbers. Then the following two statements are equivalent. 23

7 Double zeta values, double Eisenstein series, and modular forms of level ) The relation a i, j Zi, eo j + i+ j= i+ j= b i, j Z oe i, j + i+ j= c i, j Z oo i, j 0 (mod QZ o ) holds in DZ (as before i+ j= means i+ j=, i, j ). 2) There exist some homogeneous polynomials F, G Q[X, Y ] of degree 2 such that F(Y, X ) + F(X 2, Y 2 ) F(X 2, X 2 + Y 2 ) F(X 3 + Y 3, X 3 ) + G(X 3, Y 3 ) + G(Y 3, X 3 ) G(X, X + Y ) G(X + Y, X ) = ( ) 2 a i, j X i Y j + ( ) 2 b i, j X2 i Y j 2 i i i+ j= i+ j= + ( ) 2 c i, j X3 i Y j 3. i i+ j= Proof This is an analogue of Proposition 2.2 in [7]. Tae F(X, Y ) = ( 2) r X r Y s (and G = 0) and compute the coefficients of F(Y, X ) + F(X 2, Y 2 ) F(X 2, X 2 + Y 2 ) F(X 3 + Y 3, X 3 ) using binomial theorem. Then the relation in ) is exactly (not only mod QZ o but as an exact equality) the relation (5). Similarly, by taing G(X, Y ) = ( 2 r ) X r Y s (and F = 0) and computing the coefficients of G(X 3, Y 3 ) + G(Y 3, X 3 ) G(X, X + Y ) G(X + Y, X ), we see that the relation in ) is the relation (6) modulo QZ o. Since any relation of the form in ) in DZ should come from a linear combination of (5) and (6) modulo QZ o, and any homogeneous polynomial is a linear combination of monomials, we obtain the lemma. Using the lemma, we are going to produce enough relations of the form α r,s Pr,s oe β r,s Pr,s oo (mod QZ o ) (2) r+s= r,s: even r+s= r,s: even such that we can solve these in Pr,s oe. In view of the relations P oe r,s = Z oe r,s + Z eo s,r, Poo r,s Z oo r,s + Z oo s,r (mod QZ o ) (3) and the lemma, we obtain the relation of the form (2) if we can tae F and G in 2) of Lemma so that the coefficients satisfy (i) a i, j = b j,i, (ii) c i, j = c j,i, (iii) a i, j = b i, j = c i, j = 0 for all odd i, j. We now wor for convenience with inhomogeneous polynomials. Recall the usual correspondences f (x) = F(x, ) and F(X, Y ) = Y 2 f (X/Y ), and the action of 23

8 098 M. Kaneo, K. Tasaa the group Ɣ = PGL 2 (Z) on the space of polynomials of degree at most 2by(we are assuming is even) ( ) ab f (x) = (cx + d) 2 f 2 cd We extend this action to the group ring Z[Ɣ] by linearity. Set T = ( ), S = 0 ( ) 0,ε= 0 ( ) ax + b. (4) cx + d ( ) 0,δ= 0 ( ) 0. 0 Then the left-hand side of the equation in 2) of Lemma can be written in inhomogeneous form as ( ) f δ g (TST+TSε) (x )+ ( f ) ( TST) (x2 ) ( f ) TSε g (+δ) (x3 ). (5) (We write instead of 2.) Lemma 2 Suppose the polynomial f (x) (ofdegreeatmost 2) satisfies f TSTε = f and put g = 2 f T ε. Then the expression (5) gives the coefficients (in Lemma -2) satisfying the above three conditions (i), (ii), (iii). Proof Inserting g = 2 f T ε into (5) and using the assumption f TSTε = f, which is equivalent to f TS = f T ε since (T ε) 2 =, and also using the identities TSTST = S, T εt = ε, εs = δ, δε = εδ = S in Ɣ, we can write (5) as ( f δ( ε) ) (x ) + ( f ( ε) ) (x 2 ) ( f T ( ε) ) (x 3 ). (6) Now the condition (iii) (the polynomial is even) is clear from this (being illed by +ε), and the conditions (i) and (ii) are respectively the consequences of the equations f δ( ε)δ = f ( ε), f T ( ε)δ = f T δ f TS= f T εs f T ε = f T ( ε). Noting TSTε = ( 0 ) and hence ( ) ( ) x x ( x + ) = x, and ( x + ) x + x + 2 = x 2, we see that the polynomials x r (x 2) 2 r for r = 0, 2,..., 2 (even) satisfy the condition f TSTε = f in Lemma 2. With this choice of f (for r = 0, 2,..., 4) and g in Lemma 2, we compute the coefficients in Lemma by noting (3), (6) and by using 23

9 Double zeta values, double Eisenstein series, and modular forms of level x r (x 2) 2 r ( ε) = x r (x 2) 2 r x r (x + 2) 2 r 2 r ( ) 2 r = 2 r i x r+i i = i= i:odd 2 ( 2 = r ( 2 r i r i=r+2 i:even ) 2 to obtain a relation of the form 2 i=r+2 i:even ( i r i=r+2 i:even ) 2 i x i (r + i i ) ( 2 i )( i r ) 2 i x i, ) 2 i Pi, i oe linear combination of Poo even,even (mod QZ o ). When we put r = 4,...,2, 0, we can solve these congruences successively in each Pi, i oe for i = 2, 4,...,2 (because the system is triangular). This completes the proof of Theorem. 3 The double Eisenstein series of level 2 3. Definition and the double shuffle relations We introduce the double Eisenstein series of level 2 and first show that they satisfy the double shuffle relations. Let ev (resp. od) be the set of even (resp. odd) integers and τ a variable in the upper half-plane. Define the three double Eisenstein series G eo r,s Gr,s eo (τ) := (2πi) r s λ>μ>0 λ ev τ+ev μ ev τ+od (mτ + n) r (m τ + n ) s, Gr,s oe (τ) := (2πi) r s λ>μ>0 λ ev τ+od μ ev τ+ev λ r μ s = (2πi) r s λ r μ s, (τ), Goe r,s mτ+n>m τ+n >0 m ev,n ev m ev,n od Goo r,s (τ) := (2πi) r s (τ), and Goo r,s λ>μ>0 λ ev τ+od μ ev τ+od λ r μ s. (τ) by Here, the positivity mτ +n > 0 of a lattice point means either m > 0orm = 0, n > 0, and mτ + n > m τ + n means (m m )τ + (n n )>0. We assume r 3 and s 2 for the absolute convergence. (7) 23

10 00 M. Kaneo, K. Tasaa All the series in (7) is easily seen to be invariant under the translation τ τ +, and hence have Fourier expansions. The Fourier series developments can be deduced in a quite similar manner to the full modular case [7]. In particular, our double zeta values of level 2 appear as constant terms. Theorem 2 Let r 3 and s 2 be integers and set = r + s. We have the following q-series expansions (q = e 2πiτ ). G eo r,s (τ) = ζ eo (r, s) + g eo r,s (q) + {( ( ) ) p ( ) s + δ p,s s G oe r,s (τ) = ζ oe (r, s) + g oe r,s (q) + { ( ) p ( ) s s G oo r,s (τ) = ζ oo (r, s) + g oo r,s (q) + p+h= p> ( } p ζ o (p)gh ) ζ e (q) + ( )p+r o (p)g oh r (q), p+h= p> ( } p ζ o (p)gh o (q)+δ p,s ζ e (p)gh ) ζ o (q)+( )p+r o (p)g eh r (q), p+h= p> {( ( ) ( p p ( ) s + ( ) p+r s r } )) ζ e (p)g oh (q) + δ p,s ζ o (p)g oh (q), where δ p,s is Kronecer s delta, ζ (r, s) = (2πi) r s ζ (r, s) and ζ () = (2πi) ζ ()( =e or o), and the g s are the following q-series: g eo r,s g oe r,s r+s ( ) (q) = 2 r+s (r )!(s )! ( ) (q) = 2 r+s (r )!(s )! g oo r,s (q) = r+s ( ) r+s 2 r+s (r )!(s )! m>m >0 u,v>0 m>m >0 u,v>0 m>m >0 u,v>0 ( ) v u r v s q um+vm, ( ) u u r v s q um+vm, ( ) u+v u r v s q um+vm, and g e r (q) = 23 ( )r 2 r (r )! u,m>0 u r q um, g o r (q) = ( )r 2 r (r )! ( ) u u r q um. u,m>0

11 Double zeta values, double Eisenstein series, and modular forms of level 2 0 Proof Put, for positive integers r and s, ϕ e r (q) = ( )r 2 r (r )! u r q u/2,ϕr o (q) = u>0 ( )r 2 r (r )! ( ) u u r q u/2. Then the series gr (q), g r,s (q) in the theorem can be written by using ϕ r (q) as gr e (q) = ϕr e (q2m ), gr o (q) = ϕr o (q2m ), m>0 m>0 gr,s eo (q) = ϕr e (q2m )ϕs o (q2m ), gr,s oe (q) = g oo r,s (q) = m>m >0 m>m >0 ϕ o r (q2m )ϕ o s (q2m ), g ee r,s (q) = m>m >0 m>m >0 u>0 ϕ o r (q2m )ϕ e s (q2m ), ϕ e r (q2m )ϕ e s (q2m ). The computation of the Fourier series can be carried out in a completely similar fashion as done in [7], dividing the sum of the defining series into four terms, according as m = m = 0, m = m > 0, m > m = 0, m > m > 0. For instance, in the case of G eo r,s (τ), we compute G eo r,s (τ) = m=m =0 n>n >0 m,m,n ev n od + m=m >0 n>n m,m,n ev n od + using the partial fraction decomposition (τ + n) r (τ + n ) s =( )s and the formulas n Z n Z m>m =0 n >0 m,m,n ev n od + r ( ) s + i i i=0 m>m >0 m,m,n ev n od s ( ) r + j + ( ) j j j=0 (τ + 2n) l = ( 2πi)l 2 l (l )! (τ + 2n + ) l = ( 2πi)l 2 l (l )! (τ + n) r i (2πi) r s (mτ + n) r (m τ + n ) s, (τ + n ) s j (n n ) s+i (n n ) r+ j u l q u/2 = (2πi) l ϕl e (q) (l 2), u>0 ( ) u u l q u/2 = (2πi) l ϕl o (q) (l 2) u>0 23

12 02 M. Kaneo, K. Tasaa (consequences of the standard Lipschitz formula, and when l = weuse lim lim N N n= N N N n= N τ + 2n = πi 2 + ( 2πi) 2 u>0 τ + 2n + = πi 2 + (2πi)ϕo (q) q u/2 = πi 2 + (2πi)ϕe (q), instead). We leave the details to the reader. We remar that each series in Theorem 2 is in R + qq[[q]] + R[[q]], and the terms in R[[q]] ( imaginary part ) only come from the terms having ζ (p) with odd p as coefficients (note that ζ (p) is in Q or R according to the parity of p). Now we extend the definition of the double Eisenstein series for any (nonconverging) r, s (except r = s = ), by using q-series. For this, we separately define the imaginary part and the combinatorial part. First we define the imaginary parts as I eo r,s (q) = I oe r,s (q) = I oo r,s (q) = p+h= p:odd p+h= p:odd {( ( ) ) p ( ) s + δ p,s ζ o (p)gh e s (q) ( } p +( ) ) ζ p+r o (p)g oh r (q), { ( ) p ( ) s ζ o (p)gh o s (q) + δ p,s ζ e (p)gh o (q) ( } p +( ) ) ζ p+r o (p)g eh r (q), {( ( ) p ( ) s s p+h= p:odd +δ p,s ζ o (p)g o h (q) }. + ( ) p+r ( p r )) ζ e (p)gh o (q) The sum is over p, h with p odd. Note that the regularized values ζ o () and ζ e () are defined by (7) and thus for any positive integers r, s, these series are elements of R[T ][[q]]. Secondly, we define the part in qq[[q]] which is referred to as the combinatorial double Eisenstein series. Put β eo r,s (q) = 23 p+h=r+s {( ( ) ( } p p ( ) )+δ s p,s β o p s ge h )β (q)+( )p+r op r goh (q),

13 Double zeta values, double Eisenstein series, and modular forms of level 2 03 β oe r,s (q) = β oo r,s (q) = where p+h=r+s { ( ) p ( ) s β o p s go h (q) + δ p,sβ e p go h (q) } )β op geh (q), ( p +( ) p+r r p+h=r+s {( ( ) s ( p s ) ( )) } p +( ) p+r β ep r goh (q)+δ p,sβ op goh (q), β e r = B r 2 r+ r!,βo r = ( 2 r )B r 2 r! (B r = the Bernoulli number), and as before the condition p + h = r + s includes p, h. Let g e r (q) := m>0 mϕ e r+ (q2m ), g o r (q) := m>0 mϕ o r+ (q2m ) (r 0), (8) and for integers r, s let where εr,s eo (q) =δ r,2gs o (q) δ r,gs o (q) + δ s,(gr e (q) + g r e (q)) + δ r,δ s, α, εr,s oe (q) =δ r,2gs e (q) δ r,gs e (q) + δ s,(gr o (q) + g r o (q)) + δ r,δ s, α 2, εr,s oo (q) =δ r,2gs o (q) δ r,gs o (q) + δ s,(gr o (q) + g r o (q)) + δ r,δ s, α 3, α = g0 o (q) 2 g 0 e (q), α 2 = α,α 3 = 4g2 o (q) + 2 g 0 e (q). (9) Note that each εr,s, is 0 when r 3 and s 2, and in the other boundary cases, the definition is adopted (quite similarly as in [7]) so that the extended double Eisenstein series below satisfy the double shuffle relations. The combinatorial double Eisenstein series are then defined, for positive integers r, s, by C eo r,s C eo r,s C oo r,s (q) =geo r,s (q) + βeo r,s (q) + 4 εeo r,s (q), (q) =goe r,s (q) + βoe r,s (q) + 4 εoe r,s (q), (q) =goo r,s (q) + βoo r,s (q) + 4 εoo r,s (q). Lastly, the constant term of the double Eisenstein series is given by the (regularized) double zeta values. 23

14 04 M. Kaneo, K. Tasaa Definition For any integers r, s with (r, s) = (, ), we define G eo G oe G oo r,s (q) := ζ eo (r, s) + Cr,s eo r,s (q) := ζ oe (r, s) + Cr,s oe r,s (q) := ζ oo (r, s) + Cr,s oo (q) + I eo r,s (q), (q) + I oe r,s (q), (q) + I oo r,s (q). To state the double shuffle relations in the forms (3) and (4) {( for these ) series, we ab need usual Eisenstein series for the congruence subgroup Ɣ 0 (2) = SL cd 2 (Z) } c 0 mod 2. For each integer 3, let the series G (i ) (τ) and G (0) (τ) be defined by G (i ) (τ) := λ>0 λ ev τ+od λ = mτ+n>0 m: even, n: odd (mτ + n), and G (0) (τ) := λ>0 λ od τ+z λ = mτ+n>0 m: odd (mτ + n). When 4 is even, the functions G (i ) (τ) and G (0) (τ) are the Eisenstein series for Ɣ 0 (2) associated to cusps i and 0 respectively, and as such they are modular of weight with respect to Ɣ 0 (2). The Fourier series of G (i ) (τ) and G (0) (τ) are given as follows. Let G (τ) be the Eisenstein series of weight for SL 2 (Z): G (τ) := Zτ+Z mτ+n>0 (σ (n) = d n ( 2πi) = ζ() + (mτ + n) ( )! σ (n)q n, n d ). (20) (Note that this gives a non-zero function even when is odd.) With this we have G (i ) (τ) = G (2τ) 2 G (τ) = ζ o () + ( 2πi) 2 ( )! G (0) (τ) = G (τ) G (2τ) = ( 2πi) ( )! n d n n/d:odd ( ( ) d d ) q n, n d d n qn. (2) We define the q-series G (q), G (i ) (q), and G (0) (q) for any by the (convergent) q-series on the right-hand sides of (20) and (2), with the regularized values (7) and ζ() = T (= ζ e () + ζ o ()). Finally we set 23

15 Double zeta values, double Eisenstein series, and modular forms of level 2 05 G o (q) = (2πi) G (i ) (q) = ζ o () + g o (q), G e (q) = 2 (2πi) G (q) = ζ e () + g e (q). Theorem 3 For any integers r, s with (r, s) = (, ), we have Gr o (q)ge s (q) + 4 (δ r,2gs e (q) + δ s,2gr o (q)) = Goe r,s (q) + Geo s,r (q) = (( ) ( ) ) i i Gi, oe j r (q) + Gi, oo j s (q), i+ j=r+s Gr o (q)go s (q) + 4 (δ r,2g o s (q) + δ s,2gr o (q)) = Goo r,s (q) + Goo s,r (q) + Go r+s (q) = (( ) ( )) i i + Gi, eo j r s (q). i+ j=r+s The proof of the theorem will be postponed to Sect Double Eisenstein series and period polynomials In this subsection, we describe a mysterious connection between our double Eisenstein series and the period polynomials associated to cusp forms on Ɣ 0 (2). Thisindof connection was first observed in the full modular case [9], which will be recalled briefly in the appendix for the convenience of the reader because the reference [9] circulated only among participants of the conference. Let us recall the theory of period polynomials for Ɣ 0 (2) given in [6] and [8]. We follow the formulation of [0]. Recall the group Ɣ 0 (2) is generated by two elements (see e.g. [, Theorem 4.3])) T = ( ), M = 0 ( ). 2 Let be a positive even integer and V be the space of polynomials with rational coefficients of degree at most 2: V := {P(X) Q[X] deg( f ) 2}. The group Ɣ 0 (2) acts on V as in (4) and this action extends to that of the group ring Z[Ɣ 0 (2)] as usual. Consider the subspace W of V defined by W := { P V P 2 ( T )( + M) = 0 }. 23

16 06 M. Kaneo, K. Tasaa For a cusp form f S (2) := {the space of cusp forms onɣ 0 (2)}, the period polynomial r f (X) is given by r f (X) := i 0 f (τ)(x τ) n dτ. It is implicitly shown in the proof of Proposition 3 in [8] that r f (X) W C. Now we consider the even and odd parts of polynomials separately. Put ε = ( 0 0 ). By the identity ε( T )( + M) = ( T )( + M)T ε (every matrix identity is regarded projectively, i.e., as that in Ɣ 0 (2)/ {±}), we see that if P W then P ( ± ε) W, and so we have the direct sum decomposition W = W + W, where W + (resp. W ) is the even (resp. odd) part of W : W ± := { P V P ε =±P and P ( T )( + M) = 0 }. We also denote by r ± f (X) the even and odd part of r f (X), r ± f (X) := 2 r f (X) ( ± ε), and by r ± the map r ± : S (2) f r ± f (X) W ± C. For the space W + of even polynomials, we have two obvious elements and X 2. This is clear for because ( T ) = 0. For X 2, we note the identity ( T )( + M) = ( TM)( + M) (because M 2 = ) and TM = ( 0 2) and thus X 2 TM = X 2. Hence, we have the decomposition 23 W + = Q Q X 2 W +,0,

17 Double zeta values, double Eisenstein series, and modular forms of level 2 07 where W +,0 4 } := {P(X) V P(X) = a i X i, P ( T )( + M) = 0. i=2 even Let r +,0 be the map S (2) W +,0 obtained by the composition of r + and the natural projection W + W +,0. From the wors of Fuuhara and Yang [6] and Imamoḡlu and Kohnen [8], we obtain the following Theorem 4 For even, the two maps r +,0 : S (2) W +,0 C and r : S (2) W C are isomorphisms of vector spaces. Proof We now from [6] and [8] that both maps are injective. So all we have to show is the dimensions of the target spaces are equal to the dimension of S (2), which is equal to [/4]. We only calculate the dimension of W +,0, since the other is similar and only the former is relevant to the subsequent story involving the double Eisenstein series. Put T = ( 0 ). Obviously P = 0 is equivalent to P T = 0. For an even polynomial P(X) = 2 i 4 i:even a i X i with no constant term and no X 2 term, we compute the condition P ( T )( + M)T = 0forP being in W +,0.By ( T )( + M)T = T TT t T + MT (TMT = t T = ( )) 0 and TT = ( ) 0, MT = ( ) 0, the condition becomes ( ) ( X + ) (P 2 X X + ( X (X + ) (P 2 X + ( P ) P )) X + ( )) = 0, X + 23

18 08 M. Kaneo, K. Tasaa which is written as 2 i 4 i:even ( a i (X i ) ( X + ) 2 i (X + ) 2 i) = 0. Using binomial theorem, we can write this as 2 j 3 j:odd 2 i 4 i:even (( ) ( i j i 2 j )) a 2 i X j = 0. Therefore, the space W +,0 is the set of polynomials P(X) = 2 i 4 i:even a i X i whose (rational) coefficients satisfy a set of linear relations 2 i 4 i:even (( ) ( i j i 2 j )) a 2 i = 0 ( j =, 3,..., 3). (22) Clearly the equations for j and 2 j have just opposite sign, and so we have to loo only at the equations for j < /2, the total number of equations being [( 2)/4]. But then for i < j the coefficient of a 2 i is zero, and the coefficient matrix is upper triangular with non-zero diagonals, or more explicitly, the equation can be solved to express a 4, a 6,...,a 2[(+2)/4] by the other coefficients. We thus see that the dimension of W +,0 is /2 2 ([( + 2)/4] ) =[/4], as desired. Remar We have not succeeded in characterizing the (codimension 2) image of S (2) by r + in W + C. It is expected that such a characterization as in [0] should exist. Amazingly enough, the coefficient matrix in (22) appears exactly when we loo at the imaginary part of the double Eisenstein series of level 2, which we are going to explain. We loo at the imaginary part of Gr, r oo (τ) (as q-series) for r even. Let π : C[[q]] R[[q]] be the natural projection to imaginary part (note that by imaginary part we mean the term in R[[q]], not the coefficient of ). As Theorem 2 and Definition (for r = 2) shows, imaginary parts come from the terms with ζ e (p)gh o (τ) for odd p, and we have in matrix form 23

19 Double zeta values, double Eisenstein series, and modular forms of level 2 09 G oo 2, 2 (q) ζ e ( 3)g G oo 4, 4 π (q) 3 o(q). = Q ζ e ( 5)g5 o (q)., (23) G oo 2,2 (q) ζ e (3)g 3 o (q) where Q is the (/2 ) (/2 2) matrix given by Q = (( ) ( 2 j 2 j 2i 2i )). i /2 j /2 2 This is exactly the coefficient matrix of (22)! Let DE be the Q-vector space generated by Gr, r oo (r = 2, 4,..., 2). Theorem 5 Let 4 be a positive even integer. ) dim DE = 2, so that the series Gr, r oo (τ) (r even) are linearly independent over Q. 2) The space DE contains Q (2πi) G (i ) (τ) S Q (2), where SQ (2) is the space of cusp forms on Ɣ 0 (2) having rational Fourier coefficients. Proof We first prove 2). By Theorem 3,weseethemap Zr,s eo oe Geo s,r (q), Zr,s oo Goe r,s (q), Zr,s Goo r,s (q), Z o Go (q), P oe r,s Go r (q)ge s (q) + 4 (δ r,2g e s (q) + δ s,2g o r (q)), P oo r,s Go r (q)go s (q) + 4 (δ r,2g o s (q) + δ s,2g o r (q)) gives a realization of the space DZ in DE. Then, the first part of Theorem ensures that the space DE contains G o (q) = (2πi) G (i ) (τ), and by Theorem 3 we see that Gr o(q)ge s (q) and Go r (q)go s (q) (r +s =, r, s > 2) are contained in DE. Because of the relation (2πi) G r (0) (τ)g (i ) s (τ) = (2 r )Gr e (q)go s (q) Go r (q)go s (q) (q = e2πiτ ) and the fact shown by Imamoglu and Kohnen in [8] that these cusp forms G r (0) (τ) G s (i ) (τ) generate the space S (2), we obtain the assertion 2). For ), first we note by definition the inequality dim DE 2. 23

20 0 M. Kaneo, K. Tasaa Since elements in Q (2πi) G (i ) (τ) S Q (2) have no imaginary parts, they sit in the ernel of the projection π from DE to R[[q]], thus dim er π + dim S (2) = [ ]. 4 As for the dimension of the image of π, we see that it is equal to the ran of the matrix Q because the series g3 o(q), go 5 (q),...,go 3 (q) are linearly independent over C. This can be seen as follows. For an odd prime p, the coefficient of q p in gr o (q) is + p r times a constant independent of p. Hence by picing distinct odd prime numbers p 3, p 5,...,p 5 and looing at the coefficients of q, q p 3, q p 5,...,q p 5 in g3 o(q), go 5 (q),...,go 3 (q), we see the desired linear independence because the coefficient matrix is essentially the (non-vanishing) Vandermonde determinant. We thus have [ ] + 2 dim im π = ranq = 4 and therefore dim DE [ ] + 4 [ ] = 2. Therefore we conclude dim DE = 2 and also er π = Q (2πi) G (i ) (τ) S Q (2). Corollary For an even integer > 2, we have dim ζ oo (2r, 2r) r /2 Q 2 dim S (2). Proof By taing the constant term of the q-series, we obtain the surjective map μ : DE ζ oo (2r, 2r) r /2 Q. By the theorem, the ernel of μ contains the space S Q (2) and hence we obtain the corollary. 23

21 Double zeta values, double Eisenstein series, and modular forms of level 2 Remar 2 The above corollary says that among the /2 numbers ζ oo (even, even) there are at least dim S (2) linear relations. It seems that the /2 numbers ζ oo (odd, odd) are linearly independent over Q, and the total space ζ oo (r, r) 2 r Q is spanned by ζ oo (odd, odd) and ζ(). (Recall the sum formula in Theorem, so that ζ() is contained in the above space.) The conjectural dimension of this space is thus /2. We also conjecture that the space of usual double zeta values of even weight is contained in the space spanned by ζ oo (r, r) except ζ oo (, ): ζ(r, r) 2 r Q ζ oo (r, r) 2 r 2 Q, and that the space ζ oo (2r, 2r) r /2 Q is contained in the usual double zeta space: ζ oo (2r, 2r) r /2 Q ζ(r, r) 2 r Q. When is odd, we can prove that every ζ oo (r, r) except ζ oo (, ) is a linear combination of ζ(r, r): ζ oo (r, r) 2 r 2 Q ζ(r, r) 2 r Q. To prove this we use the identity of Y. Komori, K. Matsumoto, and H. Tsumura ([]) ( + ( ) r ) ζ 2 (r, s) + ( + ( ) s) ζ 2 (s, r) 3 )) = ζ(i)ζ( i) ζ(), i=0 even (( ) i 2 +i+ + r ( i s valid when r, s 2 and r + s = : odd, where ζ 2 (r, s) = (m + 2n) r m s = ζ oo (r, s) + ζ ee (r, s). m,n For ζ oo (, ), it seems we need (log 2)ζ( ) other than usual double zeta values, but we have not proved this. 3.3 Proof of Theorem 3 As in [7], we prove Theorem 3 by dividing it into three parts: the constant term, the imaginary part, and the combinatorial part. The double shuffle relation of the constant term is nothing but that of double zeta values, namely Proposition and its regularization. As for the imaginary part, the assertion is as follows. 23

22 2 M. Kaneo, K. Tasaa Lemma 3 For each integer > 2, we define generating functions I eo (X, Y ), (X, Y ), I oo (X, Y ) by I oe I eo I oo Then we have p+h= p:odd (X, Y ) := r+s= (X, Y ) := r+s= I eo r,s Xr Y s, I oe (X, Y ) := I oo r,s Xr Y s. ( X h Y p + X p Y h ) ζ o (p)g o h (q)=i oo = I eo eo (X +Y, X)+I (X +Y, Y ), (X h Y p ζ e (p)gh o (q)+ X p Y h ζ ) o (p)g e h (q) p+h= p:odd = I oe oo (X + Y, X)+I (X + Y, Y ), r+s= I oe r,s Xr Y s, oo (X, Y )+I (Y, X) =I oe where gh o(q) and g h e (q) are the series defined in (8). Proof By definition, each generating function can be given as I eo (X, Y ) = ( X h Y p X h (X Y ) p ) ζ o (p)gh e (q) p+h= p:odd + Y h (X Y ) p ζ o (p)gh o (q), p+h= p:odd I oe (X, Y ) = X h Y p ζ e (p)gh o (q) p+h= p:odd + p+h= p:odd I oo (X, Y ) = p+h= p:odd + p+h= p:odd p+h= p:odd Y h (Y X) p ζ o (p)g e h (q), eo (X, Y )+I (Y, X) X h (X Y ) p ζ o (p)g o h (q) ( Y h (Y X) p X h (Y X) p ) ζ e (p)g o h (q) X h Y p ζ o (p)g o h (q). The lemma follows from these by a simple calculation using binomial theorem and we omit the details. 23

23 Double zeta values, double Eisenstein series, and modular forms of level 2 3 For the computation of the combinatorial part, we prepare the generating functions as follows. β(x) := β p X p = ( 2 X ) e X, p>0 β e (X) := ( ) β e p X p = 2 4 X, p>0 e X 2 β o (X) := β o p X p =, 4 p>0 e X 2 + g e (X) := g e p (τ)x p = 2 p>0 u>0 g o (X) := g o p (τ)x p = ( ) u e ux 2 p>0 u>0 ( g p e (τ)x p = e ux 2 2X g e (X) := p>0 g o (X) := p>0 g o p (τ)x p = 2X e ux q u 2 q u, u>0 ( u>0 2 q u q u, q u ( q u ) 2 4ge 2 (τ) ), ( ) u e ux q u 2 ( q u ) 2 4go 2 (τ) ), g eo (X, Y ) := r,s g oe (X, Y ) := r,s g oo (X, Y ) := r,s g eo r,s (τ)xr Y s = 4 g oe r,s (τ)xr Y s = 4 g oo r,s (τ)xr Y s = 4 u,v>0 u,v>0 ( ) v e ux+vy 2 ( ) u e ux+vy 2 ( ) u+v e ux+vy u,v>0 2 q u q u q u q u q u q u+v q u+v, q u+v q u+v, q u+v q u q u+v, β eo (X, Y ) := r,s β oe (X, Y ) := r,s β oo (X, Y ) := r,s β eo r,s (τ)xr Y s = β o (Y )g e (X) β o (X Y )(g e (X) g o (Y )), β oe r,s (τ)xr Y s = β e (Y )g o (X) β o (X Y )(g o (X) g e (Y )), β oo r,s (τ)xr Y s = β o (Y )g o (X) β e (X Y )(g o (X) g o (Y )), 23

24 4 M. Kaneo, K. Tasaa ε eo (X, Y ) := r,s ε eo r,s (τ)xr Y s = Xg o (Y ) Y g o (Y ) g o 0 (τ) + Xge (X) + g e 0 (τ) + ge (X) + α, ε oe (X, Y ) := εr,s oe (τ)xr Y s r,s = Xg e (Y ) Y g e (Y ) g e 0 (τ) + Xgo (X) + g o 0 (τ) + go (X) + α 2, ε oo (X, Y ) := εr,s oo (τ)xr Y s = Xg o (Y ) Y g o (Y )+ Xg o (X)+g o (X)+α 3, r,s where α,α 2 and α 3 are defined as in (9). Let C eo (X, Y ) := Cr,s eo Xr Y s, r,s C oe (X, Y ) := Cr,s oe Xr Y s, r,s C oo (X, Y ) := Cr,s oo Xr Y s. Then by definition we have r,s C eo (X, Y ) = g eo (X, Y ) + β eo (X, Y ) + 4 εeo (X, Y ), C oe (X, Y ) = g oe (X, Y ) + β oe (X, Y ) + 4 εoe (X, Y ), C oo (X, Y ) = g oo (X, Y ) + β oo (X, Y ) + 4 εoo (X, Y ). The double shuffle relations of the combinatorial double Eisenstein series are stated as Lemma 4 Put Q oe (X, Y ) := g o (X)g e (Y ) + β o (X)g e (Y )+β e (Y )g o (X) + 4 (Xg e (Y )+Y g o (X)), Q oo (X, Y ) := g o (X)g o (Y ) + β o (X)g o (Y ) + β o (Y )g o (X) + 4 (Xgo (Y )+Y g o (X)), C o (X) := g o (X) α 3 2 X. Then we have Q oe (X, Y ) = C oe (X, Y ) + C eo (Y, X) = C oe (X + Y, Y ) + C oo (X + Y, X), Q oo (X, Y ) = C oo (X, Y ) + C oo (Y, X) + C o (X) C o (Y ) X Y = C eo (X + Y, X) + C eo (X + Y, Y ). 23

25 Double zeta values, double Eisenstein series, and modular forms of level 2 5 Proof Computations are parallel to those in [7], though tedious, and we omit the details. The two lemmas and Proposition complete the proof of Theorem 3. Appendix: The double Eisenstein series and the period polynomials in the case of SL 2 (Z) In this appendix we briefly recall the relation described in [9] between the double Eisenstein series and modular forms for SL 2 (Z). The double Eisenstein series for SL 2 (Z) was first defined and studied in [7]: G r,s (τ) := (2πi) r s λ>μ>0 λ,μ Z τ+z (mτ + n) r (m τ + n ) s. Its Fourier series is given there as G r,s (τ) = ζ(r, s) + g r,s (q) + λ r μ s = (2πi) r s p+h= p> ( ( ) ( p p ( ) s + ( ) p+r s r where ζ(r, s) = (2πi) r s ζ(r, s), ζ(p) = (2πi) p ζ(p), and g r,s (q) = g h (q) = ( ) r+s (r )!(s )! ( )h (h )! u,m>0 m>n>0 u,v>0 u h q um. mτ+n>m τ+n >0 m,n,m,n Z ) ) + δ p,s ζ(p)g h (q), u r v s q um+vn, By extending the definition in the case of non-absolute convergence using q-series, we showed that the double Eisenstein series satisfy the double shuffle relations (in the form described in [7]), that the space of double Eisenstein series contains the space of modular forms on SL 2 (Z), and made a connection to the period polynomial by looing at the imaginary parts of the q-expansions of G r,s (τ). Specifically, the imaginary parts are given, lie (23), by G 2, 2 (τ) ζ ( 3)g 3 (q) G 3, 3 (τ) π. G 2,2 (τ) = ζ( 5)g 5 (q) Q()., ζ(3)g 3 (q) 23

26 6 M. Kaneo, K. Tasaa where Q () is the ( 3) (/2 2) matrix given by ( ( ) ( Q () 2 j = ( ) i ( ) i 2 j i 2 i ) + δ 2 i,2 j ) Rather surprisingly, this contains exactly Q as a minor. For example, Q () 2 = , Q 2 = i 3 j / Precisely, the i-th row of Q is the 2i -st row of Q (). The right ernel of Q () corresponds to the even period polynomials (without constant term) of weight for SL 2 (Z), an example being t (, 3, 3, ) in the right ernel of Q () 2 and the corresponding period polynomial X 8 3X 6 + 3X 4 X 2 of weight 2. As in Corollary, by looing at the constant term of the double Eisenstein series and by using the connection to the period polynomial just mentioned, we obtain the upper bound of the dimension of the space of double zeta values: dim ζ(r, r) 2 r Q 2 dim S (). Also, elements in the left ernel of Q () produce expressions of modular forms in terms of double Eisenstein series. By comparing the Fourier coefficients, we obtain certain formulas for Fourier coefficients of modular forms. Let us loo at some examples in weight 2. As the simplest example, tae (0, 0, 0, 0,, 0, 0, 0, 0) in the left ernel of Q () 2. This corresponds to the relation G 6,6 (τ) = G 2 (τ) (τ), where (τ) = q n>0 ( q n ) 24 = n>0 τ(n)qn is the famous cusp form of weight 2. Comparing the coefficients of both sides, we obtain 23 τ(n) = σ (n) σ 5(n) σ 3(n) σ (n) 2 69 ρ 5,5 (n), 3

27 Double zeta values, double Eisenstein series, and modular forms of level 2 7 where ρ,l (n) := a+b=n a,b>0 u a,v b a u > b v u v l. Incidentally, the Ramanujan congruence τ(n) σ (n) (mod 69) is clearly seen from this. Secondly tae (0, 0, 7, 28, 0, 20, 0, 0, 0), which gives the relation G 4,8 (τ) G 5,7 (τ) and the formula 69G 7,5 (τ) = G 2 (τ) (τ) τ(n) = σ (n) σ 7(n) σ 5(n) σ 3(n) σ (n) ρ 3,7(n) ρ 4,6 (n) 3820 ρ 6,4 (n). 3 We may tae yet other vectors in the left ernel of Q () 2 (the dimension is 6) and may deduce similar ind of formulas for τ(n). Remar 3 Interestingly enough, the matrix Q () appears when we write motivic double zeta values in terms of certain basis elements f 3, f 5,... using coproduct structure described in F. Brown s recent important papers [4,5]. One of the present authors has found the same relation between triple Eisenstein series and motivic triple zeta values. Or a variant (minor matrix) of Q () appears in the wor of Baumard and Schneps [3] on a relation of double zeta values and period polynomials. Each of these should be related with each other, but we have not figured out the exact relations. References. Apostol, T.M.: Modular functions and Dirichlet series in number theory. Springer, Berlin (976) 2. Araawa, T., Kaneo, M.: Multiple L-values. J. Math. Soc. Jpn. 56(4), (2004) 3. Baumard, S., Schneps, L.: Period polynomial relations between double zeta values. preprint (20) 4. Brown, F.: Mixed Tate motives over Z. Ann. Math. 75 2, (202) 5. Brown, F.: On the decomposition of motivic multiple zeta values. Galois-Teichmüller theory and arithmetic geometry. Advanced Studies in Pure Mathematics, pp (202) 6. Fuuhara, S., Yang, Y.: Period polynomials and explicit formulas for Hece operators on Ɣ 0 (2). Math. Proc. Camb. Philos. Soc. 46(2), (2009) 7. Gangl, H., Kaneo, M., Zagier, D.: Double zeta values and modular forms. Automorphic forms and Zeta functions. World Scientific Publishing, New Yor, pp (2006) 8. Imamoḡlu, Ö., Kohnen, W.: Representations of integers as sums of an even number of squares. Math. Ann. 333(4), (2005) 23

28 8 M. Kaneo, K. Tasaa 9. Kaneo, M.: Double zeta values and modular forms. In: Kim, H.K., Taguchi, Y. (eds.) Proceedings of the Japan Korea joint seminar on Number Theory, Kuju, Japan (2004) 0. Kohnen, K., Zagier, D.: Modular forms with rational periods. In Modular forms (Durham, 983), Series in mathematics and its applications: statistics, operational research and computational mathematics, pp Horwood, Chichester (984). Komori, Y., Matsumoto, K., Tsumura, H.: Multiple zeta values and zeta-functions of root systems. In: Proceedings of the Japan Academy, Series A, vol. 87, pp (20) 2. Tasaa, K.: On a conjecture for representations of integers as sums of squares and double shuffle relations. preprint (20) 23