Ramanujan J (2016) 41:543–561 DOI 10.1007/s11139-016-9844-7

Parabolic cohomology and multiple Hecke L-values YoungJu Choie1 Dedicated to the memory of my teacher Marvin Knopp

Received: 27 January 2016 / Accepted: 19 August 2016 / Published online: 2 September 2016 © Springer Science+Business Media New York 2016

Abstract We derive various identities among the special values of multiple Hecke Lseries. We show that linear combinations of multiple Hecke L-values can be expressed as linear combinations of products of the usual Hecke L-series evaluated at the critical points. The period polynomials introduced here are values of 2-cocycles, whereas the classical period polynomials of elliptic modular forms come from the 1-cocycles. We derive the 2-cycle and the 3-cycle relations among them. Keywords Iterated Integral · Eichler Integral · Period polynomial · Multiple Hecke L-value · Parabolic cohomology · Critical values Mathematics Subject Classification 11F67 · 11F11

1 Introduction There has been intensive studies on multiple zeta values, originally defined by Euler [13]. It turns out that there are many interesting and surprising connections with various subjects including mathematical physics, ranging from periods of mixed Tate motives to values of Feynman integrals in perturbative quantum field theory [1–3,23]. The multiple zeta function is defined by, for kn ≥ 2, k2 , . . . kn−1 ≥ 1,

This work was partially supported by NRF-2016R1A2B1012330.

B 1

YoungJu Choie [email protected] Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, Korea

123

544

Y. Choie

ζ (k1 , k2 , . . . , kn ) =



1

k1 k2 0
· · · m knn

.

There are many known identities; in particular there are expressions of products of two simple zetas in terms of double zetas [8] including the following shuffle formulae: ζ (m)ζ (n) =

n  

m+n− j−1 m−1

j=1 m  

+



m+n− j−1 n−1

ζ ( j, m + n − j) 

ζ ( j, m + n − j), ∀m, n ∈ Z

(1.1)

j=1

ζ (m, n) =

   1 1 (1 + (−1)n )ζ (m)ζ (n) + (−1)n m+n − 1 ζ (m + n) m 2 2 m+n−3 2

−(−1)

n

   m+n−2i−1 m−1

+

 m+n−2i−1  n−1

ζ (2i)ζ (m + n − 2i)

i=1

(1.2) for m + n odd (see [8]). On the other hand, Manin generalized the classical theory by introducing “iterated non-commutative modular symbols” which also extends the definition of multiple zeta values via m-multiple iterated integrals [18–20]. Furthermore, multiple Hecke Lseries associated with higher weight cusp forms have been studied by further extending Manin’s iterated integral [9]. Its analytic properties and the linear relations among special values using shuffle relations have been discussed [5,9,10,14]. The period theory of modular forms concerns the special values of Hecke Lfunctions at the critical strip. This is the Eichler cohomology theory of Hecke cusp forms and extensively studied in various aspects (see, for instance, [6,7,11,15–17]). In this paper, we study the special values of multiple Hecke L-functions using the parabolic cohomology. These are values in the second parabolic cohomology group. One of the main applications is that we are able to show some linear combinations of the th order multiple Hecke L-values in terms of the lower order ones. Using inductive process, some linear combinations of the th order multiple Hecke L-values can be expressed in term of the critical values of the first-order Hecke L-series. Let H be the complex upper half-plane = S L 2 (Z) be the usual modular    and let (1) group, which is generated by T = 01 11 and S = 01 −1 0 acting on H. For any function f : H → C define

a b   a b  aτ + b −k , f |k c d (τ ) := (cτ + d) f c d ∈ S L 2 (R). cτ + d  The period polynomial of a cusp form f (τ ) = n≥1 a(n)q n ∈ Sk ((1)), q = e2πiτ , of even weight k on (1) is the polynomial of degree k − 2 defined by

123

Parabolic cohomology and multiple Hecke L-values



545

k−2  (k − 2)!L( f, n + 1) k−2−n z , (k − 2 − n)!(2πi)n+1 S −1 (i∞) n=0 (1.3) where L( f, s) denotes the L-series of f , which is an analytic continuation of  a(n) . Note that R(γ ; z), n≥1 n s

R(S; z) :=

i∞

f (τ )(τ − z)k−2 dτ = −

R(γ ; z) :=

i∞ γ −1 (i∞)

f (τ )(τ − z)k−2 dτ, ∀γ ∈ (1),

(1.4)

satisfies the first-order cocycle condition, that is, R(γ1 ; z)|2−k γ2 + R(γ2 ; z) − R(γ1 γ2 ; z) = 0, ∀γ j ∈ (1),

j = 1, 2, 3.

(1.5)

This gives the following “period relation” of R(S, z): R(S; z)|2−k S + R(S; z) = 0 R(S; z)|2−k U 2 + R1 (S; z)|2−k U + R(S; z) = 0, U = ST =

 0 −1  1 1

.(1.6)

R(T ; z) = 0 and this is called “parabolic” condition (see Knopp [15]). In fact, it is well known (see [16] or [15]) that the subspace

 Wk−2 := R(z) ∈ Vk−2 : R|2−k (I + S) = R|2−k (I + U + U 2 ) = 0

(1.7)

of Vk−2 , a space of polynomials of degree at most k − 2, is isomorphic to the first 1 ((1), V parabolic cohomology group Hpar k−2 ). In this paper, we study various identities among the special values of multiple Hecke L-series using the second parabolic cohomology group. To do that, it is also convenient to introduce period polynomials whose coefficient involves the special values of multiple Hecke L-values. Using relations among those polynomials, we derive identities among the special values of multiple Hecke L-series. The classical period relation in (1.6) can be regarded as the simplest case of our main result. The period polynomials introduced here are values of 2-cocycles, while the classical period polynomial in (1.3) is a 1-cocycle. We plan to investigate further properties of the period polynomials in several variables. Also a very recent work by Brown [2] shows a possible connection between the special values of multiple Hecke L-values and mixed motive. It will be interesting to explore this connection further. This paper is organized as follows: In Sect. 2, we introduce the period polynomial whose coefficients involve a special values of multiple Hecke L-series and study relations among them using the second cocycle condition. Section 3 treats double Hecke L-series and explain the analytic continuation and functional equation for a special type of Hecke L-series using the incomplete Gamma function. Explicit examples are given. Section 4 treats the triple and the higher cocycle relation among special values of multiple Hecke L-series.

123

546

Y. Choie

2 The period polynomials 2.1 Group cohomology Let us review group cohomology [4], which can be found in standard textbooks such as [4]. Let G be a group and let M be a right G-module. Set C 0 (G, M) = M and let m C (G, M) := {g : G m → M} be an abelian group. The coboundary operator defined by (dm f )(γ1 , γ2 , . . . , γm+1 ) = f (γ1 , . . . , γm )γm+1 + (−1)m+1 f (γ2 , . . . , γm+1 ) m  + (−1) j f (γ1 , . . . , γm+1− j γm+2− j , . . . , γm+1 ). j=1

The element of C m (G, M) is called an m-cochain. Let Z m (G, M) := ker(dm ) and B m (G, M) := Im(dm−1 ) for 0 < m ∈ Z which are called the space of the m-cocycles and the m-coboundaries, respectively. Then H m (G, M) = Z m (G, M)/B m (G, M) is a cohomology group.   Now take G = (1) and let ∞ := { 01 1 |  ∈ Z} be a parabolic sub∗ ((1), M) = group of (1). One can define a parabolic cohomology group H par ∗ ∗ Z par ((1), M)/B par ((1), M) as a subspace from the cohomology functor induced by the kernel of the restriction map C ∗ ((1), ·) → C ∗ (∞ , ·). Remark 2.1 (1) Note that, for the groups and for the kind of coefficient modules considered in this paper, those 2-cohomology groups and higher (and the 3-parabolic cohomology groups and higher) vanish. This is an application of transfer operator (see [4]) and Selberg’s Lemma. 1 (, cot) was studied in [15] when  is H -group. (2) H par 2.2 Period polynomial and parabolic cohomology Throughout this paper, we only consider integral weight modular form on the full group (1) = S L 2 (Z) for simplicity. Let Sk ((1)) be a space of cusp forms of evenweight k on (1). Take f j ∈ Sk j ((1)), j ∈ N with Fourier expansion f j (τ ) = n≥1 a j (n)e2πinτ and let w j := k j − 2. Consider a polynomial R2, (γ ; z) in z = (z 1 , . . . , z  ) of degree w j in each variable z j associated to the cusp forms f j ∈ Sk j ((1)) : for any γ , γ1 , γ2 ∈ (1), R2, (γ ; z) :=

i∞ γ −1 (i∞)



···

τ−1

i∞

123

f 1 (τ1 )(τ1 − z 1 )w1 dτ1



τ1

f 2 (τ2 )(τ2 − z 2 )w2 dτ2

i∞ w

f  (τ )(τ − z  ) dτ

(2.1)

Parabolic cohomology and multiple Hecke L-values

547

and P2, (γ1 , γ2 ; z) := R2, (γ1 ; z)|(−w1 ,−w2 ,...,−w ) γ2 + R2, (γ2 ; z)− R2, (γ1 γ2 ; z) (2.2) with the slash operator on the polynomial space defined by, for any

a b cd

∈ S L(2, R),

a b

R2, (γ ; z)|(−w1 ,−w2 ,...,−w ) c d

  az 1 + b az 2 + b az  + b wj := , ,..., . (cz j + d) R2, γ ; cz 1 + d cz 2 + d cz  + d

(2.3)

j=1

Remark 2.2 (1) The iterated integral of the type in (2.1) has been introduced by Manin when γ = S, z 1 = z 2 = · · · = z  = 0 (see [19, (2.12)]). (2) Two integrals in (2.1) and (2.2) converge since every cusp form f i (τ ) decays exponentially at cusp. It is obvious that R2, (γ ; z) and P2, (γ ; z) are in the space of polynomials V(w1 ,...,w ) with degree at most w j in each variable z j , j = 1, . . . . Using (2.1) we check the following:   Lemma 2.3 (1) For any n ∈ Z, R2, (T n ; z) = 0 with T = 01 11 (2) P2, (γ1 , γ2 ; z) γ −1 (i∞) 2 =

γ2−1 γ1−1 (i∞) τ−1

··· +

f 1 (τ1 )(τ1 − z 1 )w1 dτ1

···

i∞

τ1

γ2−1 (i∞)

f 2 (τ2 )(τ2 − z 2 )w2 dτ2

f  (τ2 )(τ − z  )w dτ

γ2−1 (i∞) −1 −1 γ2 γ1 (i∞)

γ2−1 (i∞) τ−1



w1

f 1 (τ1 )(τ1 − z 1 ) dτ1



τ1

f 2 (τ2 )(τ2 − z 2 )w2 dτ2

i∞

f  (τ2 )(τ − z  )w dτ . 

Proof This is immediate from Eq. (2.1).

Now we explain that the period polynomial R2, satisfies the second cocycle condition: let M be a set of polynomials in z = (z 1 , . . . , z  ) with degree w j , in each variable  z j , j = 1, . . . ,  and define a right action of (1) on M by, ∀φ ∈ M and ∀γ = ac db ∈ (1), φ · γ := (φ|(−w1 ,...,−w ) γ )(z 1 , , z  ) :=

wj 

(cz j + d)w j φ

j=1



az  + b az 1 + b ,..., . cz 1 + d cz  + d

The following relations can be obtained using the second coboundary condition and the integral representation in (2.2):

123

548

Y. Choie

Proposition 2.4 Let γ , γ j ∈ (1), j = 1, 2, 3 and n ∈ Z. (1) (cocycle condition) P2, (γ1 , γ2 ; z)|(−w1 ,...,−w ) γ3 + P2, (γ1 γ2 , γ3 ; z) = P2, (γ1 , γ2 γ3 ; z) + P2, (γ2 , γ3 ; z). (2) (parabolic   condition)   (i) P2, ( 01 n1 ,γ ; z) = P2, (γ , 01 n1 ; z) = 0.    (ii) P2, (γ1 , γ2 01 n1 ; z) = P2, (γ1 , γ2 ; z)|(−w1 ,−w2 ,...−w ) 01 n1 . Proof This follows because P2, (γ1 , γ2 ; z) = d1 (R2, (γ ; z)) and d2 ◦ d1 = 0, where di are the boundary maps defined in (2.1).     (i) P2, ( 01 n1 , γ ; z) = P2, (γ , 01 n1 ; z) = 0 is true using the integral representation in (2.2) and the fact M(i∞)  = i∞.   (ii) The relation P2, (γ1 , γ2 01 n1 ; z) = P2, (γ1 , γ2 ; z)|(−w1 ,...,−w) 01n1 follows from the condition (1) together with the fact P2, ( 01 n1 , γ ; z) =   cocycle P2, (γ , 01 n1 ; z) = 0.   0 −1 

 0 −1 

Since (1) is spanned by two elements S = 1 0 , T = 1 0 with two relations S 2 = U 3 = I, the polynomials R2, (γ ; z) and P2, (γ1 , γ2 ; z) in (2.1) and (2.2) satisfy the following relations: Corollary 2.5 (1) (L-value relation) P2, (S, S; z) = R2, (S; z)|(−w1 ,−w2 ,...−w ) S + R2, (S; z), S =

 0 −1  1 0

.

(2) (period relations) (i) P2, (S, S; z)|(−w1 ,...,−w ) S = P2, (S, S; z) (ii) (R2, (S; z) + R2, (S; z)|(−w1 ,...,−w ) T )|(−w1 ,...,w ) (I + U + U 2 ) = P2, (S, S; z)|(−w1 ,...,−w ) T · (I + U + U 2 ) Proof (1) Taking γ1 = γ2 = S in (2.2) with the relation S 2 = −I gives a desired identity. (2) These are the relations using parabolic conditions with the relations S 2 = −I and U 3 = −I.  Remark 2.6 (1) When  = 1, R2,1 (S; z) is the usual period polynomial R(S; z) of the cusp form f 1 , as defined in (1.3). (2) The period relations in Corollary 2.5—(1) and (2) are the well-known 2-cycle and 3-cycle classical period relations given in (1.6). (3) Note P2,1 (S, S; z) = 0 (the case when  = 1), but P2, (S, S; z) is not trivial if  ≥ 2.

123

Parabolic cohomology and multiple Hecke L-values

549

2.3 The th order Hecke L-series and critical values In this section we, show that the coefficients of R2, (S; z) and P2, (S, S; z) are the special values of multiple Hecke L-values. 2.3.1 Convergence Let us recall the convergence result of more general multiple L-series studied in [21]: let s be a complex variable and φk (s) :=

 ak (n) n≥1

ns

be a Dirichlet series with complex coefficients ak (n) (0 ≤ k ≤ r ). Consider r ((s1 , . . . , sr ); (φ1 , . . . , φr ))  a1 (m 1 )a2 (m 2 ) · · · ar (m r ) = s1 m (m + m 2 )s2 · · · (m 1 + m 2 + · · · + m r )sr 1 m ,...m ≥1 1 1

(2.4)

r

associated with φ1 , . . . , φr , where sk (1 ≤ k ≤ r ) are complex variables. Theorem 2.7 [21] If φk (s) is entire for 1 ≤ k ≤ r, then r ((s1 , . . . , sr ); (φ1 , . . . , φr )) is also entire. The following th order Hecke L-series has been studied in [9,18]: Definition 2.8 The th order Hecke L-series associated with cusp forms f j (τ ) =  2πinτ ∈ S ((1)), 1 ≤ j ≤ , is defined as, for (s , . . . , s ) ∈ C , kj 1  n≥1 a j (n)e L

 s1 , s2 , ..., s  f 1 , f 2 ..., f 

:=



n 1 ,n 2 ,...,n 

a1 (n 1 )a2 (n 2 ) · · · a (n  ) . (n 1 + n 2 + · · · + n  )s1 (n 2 + · · · + n  )s2 · · · (n −1 + n  )s−1 n s ≥1

When  = 1, we will use the usual notation L

 s1  f1

= L( f 1 , s1 ) =



a1 (n) n≥1 n s1 .

The series in (2.5) converges, has an analytic continuation, and is entire by replacing φ j in (2.4) by Hecke L-series associated to cusp forms f j .

123

550

Y. Choie

An integral representation of multiple Hecke L-series L as follows: fix a natural number  ≥ 1. L∗

 s1 , s2 , ..., s  f 1 , f 2 ..., f 

can be given

 s1 , s2 , ..., s  f 1 , f 2 ..., f 

 

 s , s , ..., s  (s j ) · L f11 , f22 ..., f s j (−2πi) j=1 i∞ i∞ dτ2 s1 dτ1 = f 1 (τ1 )τ1 f 2 (τ1 + τ2 )τ2s2 τ τ2 1 0 0 i∞ dτ ··· f  (τ1 + τ2 + · · · + τ )τs . τ 0 =

(2.5)

s  Again when  = 1, we use the usual conventional notation L ∗ f11 = L ∗ ( f 1 , s1 ) =  i∞ 1 f 1 (τ1 )τ1s1 dτ 0 τ1 .  s , s , ..., s  Note that the analytic properties of L ∗ f11 , f22 ..., f with respect to the variable  s , s , ..., s  (s1 , . . . , s ) ∈ C follow from that of L f11 , f22 ..., f . The above integral in (2.5) converges when Re(s j ) >> 0, ∀1 ≤ j ≤ , and has an analytic continuation for all s j since each cusp form f j decays exponentially at cusp. Next we define the critical values of multiple Hecke L-series. Definition 2.9 The critical values of L ∗ ( f ; s), in the sense of Deligne [12], associated with a cusp form f ∈ Sk ((1)) are the values of L ∗ ( f ; s) at the integral points s = ν in the range of 1 ≤ ν ≤ k −1. The critical values of the th order multiple Hecke L-series associated with cusp forms f j ∈ Sk j ((1)), 1 ≤ j ≤ , k j = w j + 2 ≥ 2, are the  s , s , ..., s  values of L ∗ f11 , f22 ..., f at the integral points (s1 , . . . , s ) = (m 1 , m 2 , . . . , m  ) ∈ Z in the range of 1 ≤ m  ≤ w + 1, 1 ≤ m −1 ≤ w + w−1 + 1, ..., 1 ≤ m 1 ≤ w + w−1 + · · · + w1 + 1. The integral points (m 1 , m 2 , . . . , m  ) in this range will be called the critical points. Remark 2.10 There is a “gamma factor” γ (s) (equal to an exponential function As ∗ times a finite product of terms ( s+m 2 ) with m ∈ Z.) such that the product L (s) = γ (s)L(s) has a meromorphic continuation with only finitely many poles in C and satisfies L ∗ (s) = wL ∗ (h − s) for some integer h > 0 and sign w = ±1. An integer m is called “critical” if neither m nor h − m is a pole of γ (s) according to Deligne and Zagier [12,22]. Theorem 3.2 shows that the critical values of multiple L-series as defined here are compatible with the critical values of the usual L-functions. Next two sections we prove that the coefficients of R2, (S; z)|(−w1 ,−w2 ,...−w ) S + R2, (S; z), as a polynomial in z = (z 1 , z 2 , . . . , z  ), are linear combinations with integral coefficients of the critical values of the -th order Hecke L-series(analytic continuation) using the usual Mellin transform from the integral representation

123

Parabolic cohomology and multiple Hecke L-values



i∞

R2, (S; z) = 0

f 1 (τ1 )(τ1 − z 1 )wi dτ1

···

551



0

f 2 (τ2 + τ1 )(τ2 + τ1 − z 2 )w2 dτ2 ..



i∞ 0

0

i∞

f  (τ + τ−1··· + τ1 )(τ + τ−1··· + τ1 − z  )w dτ

i∞

while those of P2, (S, S; z) are linear combinations of the product of lower order, say, the nth order, 1 ≤ n ≤  − 1, Hecke L-series evaluated at the critical points.

3 Double Hecke L-series In this section, we focus on the case of the double( = 2) Hecke L-series. Various identities by comparing the coefficients of two polynomials P2,2 (S, S; z 1 , z 2 ) and R2,2 (S, S; z 1 , z 2 ) coming from the second cocycle relation P2,2 (S, S; z 1 , z 2 ) = R2,2 (S, S; z 1 , z 2 )|(−w1 ,−w2 ) S + R2,2 (S, S; z 1 , z 2 ) (3.1) are derived. We also study its analytic properties such as the functional equation.

3.1 Identities  2πin j τ ∈ S ((1)), j = 1, 2. For Take two cusp forms, f j (τ ) = kj n j ≥1 a j (n j )e Re(s1 ), Re(s2 ) >> 0, consider the double Hecke L-series defined by

L

s2  f1 , f2

 s1 ,

:=

 n 1 ,n 2

a1 (n 1 )a2 (n 2 ) . s s2 (n 1 + n2) 1 n2 >0

(3.2)

The analytic continuation of this type multiple L-series is proved in [21] in a more general context(see Theorem 2.7). The integral representation of the above series is given by L∗

 s1 ,

s2  f1 , f2

= 0

=

i∞

dτ1 i∞ dτ2 f 2 (τ1 + τ2 )τ2s2 τ1 0 τ2   s , s ∗ 1 2

f 1 (τ1 )τ1s1

(s1 )(s2 ) L (−2πi)s1 +s2

(3.3)

f1 , f2

s , s  and L ∗ f11 , f22 is also entire in s1 and s2 ∈ C. Using the relation in (3.1), we prove the following identities:

123

552

Y. Choie

Theorem 3.1 For each 0 ≤ j1 ≤ w1 , and 0 ≤ j2 ≤ w2 , L ∗ ( f 1 ; w1 − j1 + 1)L ∗ ( f 2 ; w2 − j2 + 1) j2      j1 + j2 ∗ j1 + j2 −+1, +1 j2 L = (−1) f2 f1 ,  =0

+

w 2 − j2  =0

w2 − j2 



L∗



w1 +w2 − j1 − j2 −+1, +1 f1 , f2



.

Proof First note that

i∞

−R2,2 (S; z 1 , z 2 ) = −

f 2 (τ2 )(τ2 − z 2 )w2 dτ2  w1   w2   w − j  ı∞



(−1) j1 + j2

j1



2

j2

0 ≤ j1 ≤ w1 0 ≤ j2 ≤ w2 0 ≤  ≤ w2 − j2

×L ∗

τ1

f 1 (τ1 )(τ1 − z 1 ) dτ1

0

=



w1

w1 +w2 − j1 − j2 −+1, +1 f2 f1 ,



j



2

j

z 11 z 22 .

By following the notation of integral representation in (2.5), we get −R2,2 (S; z 1 , z 2 )|(−w1 ,−w2 ) S − R2,2 (S; z 1 , z 2 ) ⎧ w 2 − j2      w w ⎨ 1 2 j1 + j2 w2 − j2 = L ∗ w1 +w2 − jf11−, j2 −+1, +1 (−1) j1 j2 f  2 ⎩ 0≤ j ≤w =0

1 1 0 ≤ j2 ≤ w2

+

j2   =0

j2 



L∗



j1 + j2 −+1, +1 f2 f1 ,

⎫ ⎬ ⎭

j

j

z 11 z 22 .

On the other hand,

i∞

w1



i∞

P2,2 (S, S; z 1 , z 2 ) = − f 1 (τ1 )(τ1 − z 1 ) dτ1 f 2 (τ2 )(τ2 − z 2 )w2 dτ2 0 0   w   w2  ∗ j1 j2 ∗ =− (−1) j1 + j2 j11 j2 L ( f 1 ; w1 − j1 +1)L ( f 2 ; w2 − j2 +1)z 1 z 2 . 0 ≤ j1 ≤ w1 0 ≤ j2 ≤ w2

Therefore, from the relation in (3.1), we derive the result.



The identity in Theorem 3.1 can be regarded as a shuffle relation whose formula is analogous to the identity (1.1) of multiple zeta values. A variation of the integral representation given in (2.5) gives further identities among special values of Hecke L-series:

123

Parabolic cohomology and multiple Hecke L-values

553

Theorem 3.2 Take k1 = 2s1 + s2 + s4 + 2 and k2 = 2s3 + s4 + 2, for nonnegative integers sn , 1 ≤ n ≤ 4 and let f i ∈ Ski ((1)). Then we have the following identities: for each j, 0 ≤ j ≤ s2 , s3       s3   j+s1 +s2 L ∗ s1 +s2 − j+1, s3 +s4 −+1 + L ∗ s1 + j+1, s3 +s4 −+1 (−1)  f , f , f f 1

=0

⎡

= (−1) j+s1 +s2 ⎣

s4   s4 

=0



2

1

2

⎤ (−1) L ∗ ( f 1 ; s1 + s2 − j +  + 1)L ∗ ( f 2 , s3 + s4 −  + 1)⎦ .

Proof Let si , i = 1, 2, 3, 4 be nonnegative integers. Consider

∞

R2 (z) := 0

f 1 (τ1 )τ1s1 (τ1 − z)s2



τ1 ∞

f 2 (τ2 )τ2s3 (τ2 − τ1 )s4 dτ1 dτ2 .

Since k1 = s2 + s4 + 2s1 + 2 and k2 = s4 + 2s3 + 2, we have

1 (−1)s1 +s2 R2 (z) + z s2 R2 − z ∞ = (−1)s1 +s2 f 1 (τ1 )τ1k1 −s1 −s2 −2−s4 (τ1 − z)s2

0 0

f 2 (τ2 )τ2k2 −s3 −s4 −2 (τ2 − τ1 )s4 dτ1 dτ2 ∞ s ∞ s2 4    s2   s4  s +s − j+ j+s1 +s2  (−1) = f 1 (τ1 )τ1 1 2 dτ1 j (−1)  ×

j=0

=0

j=0

=0 ∗

0

 ∞ × f 2 (τ2 )τ2s3 +s4 − dτ2 z j s 0 s 2 4 s    s4  2 j+s1 +s2  (−1) = j  (−1)

×L ( f 1 ; s1 + s2 − j +  + 1)  ×L ∗ ( f 2 , s3 + s4 −  + 1) z j . On the other hand, s  s2 3      s3  ∗  s1 +s2 − j+1, s3 +s4 −+1  j j s2 R2 (z) = (−1) j z  L f2 f1 , =0

j=0

and s 

 s2 3  s2    s3  ∗  s1 + j+1; s3 +s4 −+1  j 1 = z R2 − z . j  L f2 f1 , z s2

j=0

=0

123

554

Y. Choie

So, (−1)s1 +s2 R2 (z) + z s2 R2 =



−1 z



s2 s3     s2    s3   j+s1 +s2 ∗ s1 +s2 − j+1, s3 +s4 −+1 (−1) L j  f2 f1 , =0

j=0

+L ∗



s1 + j+1; s3 +s4 −+1 f1 , f2



z j. 

This implies Theorem 3.2.

Remark 3.3 Note that the weights k1 = 2s1 + s2 + s4 + 2 and k2 = 2s3 + s4 + 2 in Theorem 3.2 satisfy 2 ≤ s2 + s4 + 2 ≤ k2 , 2 ≤ s1 + s2 + s4 + 2 ≤ k1 . This implies that all L ∗ -values in the theorem are critical values. By choosing a various si in Theorem 3.2, we get the following identities which are analogous to those for multiple zeta values (1.2) . Corollary 3.4 (1) Let k1 = s2 + s4 + 2 and k2 = s4 + 2 in Theorem 3.2. For each u ∈ Z≥0 , 0 ≤ u ≤ s1 , L∗



s2 −u+1, s2 +1 f1 , f2

=−

s4 



+ (−1)u L ∗



u+1, s4 +1 f1 , f2



( sv4 ) (−1)v L ∗ ( f 1 , s2 − u + v + 1)L ∗ ( f 2 , s4 − v + 1).

v=0

(2) If f 1 = f 2 ∈ Ss4 +2 ((1)), L

∗



1, s4 +1 f1 , f1



s4 1  =− · ( sv4 ) (−1)v L ∗ ( f 1 , v + 1)L ∗ ( f 1 , s4 − v + 1). 2 v=0

(3) If s2 = s4 = 0 and s1 = s3 and f 1 = f 2 ∈ S2s1 ((1)), we get s1   s1  =0



((−1)s1 + 1)L ∗



s1 +1, s1 −+1 f1 , f1



= (−1)s1 L ∗ ( f 1 ; s1 )L ∗ ( f 1 ; s1 ).

3.2 Functional equation Recently in [10], the functional equation and analytic continuation of the following special type of Double Hecke L-series  m,n≥1

123

a(n) + n)s2

m s1 (m

Parabolic cohomology and multiple Hecke L-values

555

have been studied where {a(n)}n≥1 is a sequence of complex numbers. Here we give a simpler functional equation by adding incomplete Gamma function in the case when a multiple Hecke L-series is coming from two cusp forms, in which we intend to study further. Theorem 3.5 Take f j ∈ Sk j ((1)), j = 1, 2 and  be any positive even integer. For s1 , s2 ∈ C, let ⎛ D⎝ ⎛ H⎝

2 

⎞ f j ; s1 , s2 ⎠ :=

j=1 2 



∞ 0

⎞

⎛

f j ; s1 , s2 ⎠ := D ⎝

j=1

f 1 (i y)y s1 2 

y ∞

f 2 (iv)v s2 (v − y) dvdy,

⎞

f j ; s1 , s2 ⎠ +



∞

×

f 1 (i y)y s1

1

j=1



∞

f 2 (iv)v s2 (v − y) dvdy.

0

Then H satisfies the following functional equation: ⎛ H⎝

2 

⎛

⎞ f j ; s1 , s2 ⎠ =

(k1 +k2 −2) 2 (−1)−

H⎝

j=1

⎞

2 

f j ; k1 − s1 −  − 2, k2 − s2 −  − 2⎠ .

j=1

Proof ⎛ D⎝

2 

⎞

1

=

f 1 (i y)y s1

×

y

y ∞

0



∞

f 1 (i y)y

0

j=1





f j ; s1 , s2 ⎠ :=

s1

y

∞

f 2 (iv)v s2 (v − y) dvdy

f 2 (iv)v s2 (v − y) dvdy +



∞

f 1 (i y)y s1

1 

f 2 (iv)v (v − y) dvdy ∞ y f 1 (i y)y k1 −s1 −−2 f 2 (iv)v k2 −s2 −−2 (y − v) dvdy = i −k1 −k2 +2 1 0 y ∞ s1 f 1 (i y)y f 2 (iv)v s2 (v − y) dvdy + 1 ∞ ∞ y −k1 −k2 +2 k1 −s1 −−2 f 1 (i y)y f 2 (iv)v k2 −s2 −−2 (y − v) dvdy =i 1 0 ∞ y ∞ f 1 (i y)y s1 f 2 (iv)v s2 (v − y) dvdy + f 1 (i y)y s1 + s2

∞

1

×

0 0

∞

1 

f 2 (iv)v (v − y) dvdy. s2

123

556

Y. Choie

So, ⎛ H⎝

j=1

⎛

⎞

2 

f j ; s1 , s2 ⎠ := D ⎝

2 

⎞ f j ; s1 , s2 ⎠ +

∞

f 1 (i y)y s1

1

j=1





∞

f 2 (iv)v s2 (v − y) dvdy ∞ y −k1 −k2 +2 k1 −s1 −−2 f 1 (i y)y f 2 (iv)v k2 −s2 −−2 (v − y) dvdy =i 1 0 y ∞ f 1 (i y)y s1 f 2 (iv)v s2 (v − y) dvdy + ×

0

1

0

satisfies the desired functional equation: ⎛ H⎝

2 

⎛

⎞

f j ; s1 , s2 ⎠ = i −k1 −k2 +2 H ⎝

j=1

2 

⎞ f j ; k1 − s1 −  − 2, k2 − s2 −  − 2⎠ .

j=1

 Remark 3.6 (1) Note that

∞



1

=

∞

f 1 (i y)y s1

f 2 (iv)v s2 (v − y) dvdy

0      j

j=0

×



(−1) j (−i)s2 +− j+1 L ∗ ( f 2 , s2 +  − j + 1)

a1 (m) (s1 + j + 1; 2π m) (2π m)s1 + j+1

m≥1

∞ using the incomplete Gamma function (s; x) = x t s−1 e−t dt. (2) The above functional equation can be generalized to higher order multiple Hecke L-series. For simplicity, we treat only double Hecke L-series case.

3.3 Examples Using the relations given in Theorem 3.2, the following relations can be derived. Example 3.7 (Linear relations) Take f 1 = f 2 = (τ ), where (τ ) = q



(1 − q n )24 , q = e2πiτ , Im(τ ) > 0.

n≥1

123

Parabolic cohomology and multiple Hecke L-values

5  5 =0

=



−L ∗

4  4 =0





L∗

8, 6− , 





6, 7− , 

+ L∗





+ L∗

557

7, 6− , 





4, 7− , 

+ L∗





5, 6− , 



+ L∗



4, 6− , 



.

Proof From the condition in the Remark 3.3 0 ≤ s2 + s4 ≤ 10, 0 ≤ s1 + s2 + s4 ≤ 10, 2s3 = 2s1 + s2 , 2s1 + s2 + s4 = 10, 2s3 + s4 = 10 : (1) we take s1 = 4, s2 = 2, s3 = 5, s4 = 0 and j = 0 to get

L ∗ (; 7)L ∗ (; 6) =

5  5 =0



L∗



7, 6− , 



+ L∗



5, 6− , 



.

(2) we take s1 = 5, s2 = 0, s3 = 5, s4 = 0 and j = 0 to get

L ∗ (; 6)L ∗ (; 6) = 2

5  5 =0



L∗



6, 6− , 



.

(3) we take s1 = 3, s2 = 4, s3 = 5, s4 = 0 and j = 0 to get

L ∗ (; 8)L ∗ (; 6) =

5  5 =0



L∗



8, 6− , 



+ L∗



4, 6− , 



.

(4) we take s1 = 2, s2 = 4, s3 = 4, s4 = 2 and j = 1 L ∗ (; 6) · L ∗ (, 7) − 2L ∗ (; 7)L ∗ (; 6) + L ∗ (; 8)L ∗ (; 5) 4     4   ∗  6, 7−  4, 7− = . −L ,  + L ∗ ,   =0

So, using the relations (1) to (4), we derive the relation.



123

558

Y. Choie

4 Triple and higher Hecke L-series and identities 4.1 Triple We consider the 3rd order Hecke L-series. Let f j ∈ Sk j ((1)), j ∈ N with Fourier  expansion f j (τ ) = n≥1 a j (n)e2πinτ . Define, for Re(s j ) >> 0, 1 ≤ j ≤ 3,

L

 s1 ,

s2 , s3  f1 , f2 , f3



=

n 1 ,n 2 ,n 3

a1 (n 1 )a2 (n 2 )a3 (n 3 ) . (n + n 2 + n 3 )s1 (n 2 + n 3 )s2 n s33 1 ≥1

(4.1)

It has an integral representation

L∗

 s1 ,

s2 , s3  f1 , f2 , f3

=

i∞

0

3 

s , s , s  (s j ) · L f11 , f22 , f33 s j (−2πi) j=1 i∞ i∞ dτ3 s1 dτ1 s2 dτ2 f 1 (τ1 )τ1 f 2 (τ1 + τ2 )τ2 f 3 (τ1 + τ2 + τ3 )τ3s3 τ1 0 τ2 0 τ3 :=

with an analytic continuation for every (s1 , s2 , s3 ) ∈ C3 . The following theorem states that a linear combination of the special values of the 3rd order multiple Hecke Lfunction can be written as a linear combination of product of the lower order multiple Hecke L-series evaluated at critical points: Theorem 4.1 For each 0 ≤ j ≤ w , 1 ≤  = 3, 

 0≤≤w2 − j2

w2 − j2 



+



L∗

w



3 − j3

w1 − j1 +w2 − j2 −+1, +1 f1 , f2



p



L ∗ ( f 3 ; w3 − j3 + 1)

· L ∗ ( f 1 ; w1 − j1 + 1)L ∗



0≤ p≤w3 − j3

w2 − j2 +w3 − j3 − p+1, p+1 f3 f2 ,



+L ∗ ( f 1 ; w1 − j1 + 1)L ∗ ( f 2 ; w2 − j2 + 1)L ∗ ( f 3 ; w3 − j3 + 1)   w − j   w − j +w − j − p  2 2 3 3 3 3 = p  0 ≤ p ≤ w3 − j3 0 ≤  ≤ w2 − j2 + w3 − j3 − p

L∗



w1 − j1 +w2 − j2 +w3 − j3 − p−+1, +1, p+1 f2 , f3 f1 ,

+(−1) j1 + j2 + j3



3

p

0 ≤ p ≤ j3 0 ≤  ≤ j2 + j3 − p

123

 j  j



2 + j3 − p





L∗



j1 + j2 + j3 − p−+1, +1, p+1 f2 , f3 f1 ,



.

Parabolic cohomology and multiple Hecke L-values

559

Proof First consider R2,3 (S; z 1 , z 2 , z 3 ) i∞ τ1 τ2 f 1 (τ1 )(τ1 − z 1 )w1 dτ1 f 2 (τ2 )(τ2 − z 2 )w2 dτ2 f 3 (τ3 )(τ3 − z 3 )w3 dτ3 = i∞ i∞ 0        w w2 w3 w2 − j2 +w3 − j3 − p w3 − j3 = (−1) j1 + j2 + j3 j11 j2 j3 p  0 ≤ j1 ≤ w1 , 0 ≤ j2 ≤ w2 0 ≤ j3 ≤ w3 , 0 ≤ p ≤ w3 − j3 0 ≤  ≤ w2 − j2 + w3 − j3 − p

  j j j 3 − j3 − p−+1, +1, p+1 z 11 z 22 z 33 . L ∗ w1 − j1 +w2 − j2 +w f1 , f2 , f3

Note that R2,3 (S; z 1 , z 2 , z 3 )|−w1 ,−w2 ,−w3 S + R2,3 (S; z 1 , z 2 ) ⎧ ⎨   w w w ⎪ 1 2 3 (−1) j1 + j2 + j3 = j1 j2 j3 ⎪ ⎩ 0 ≤ j1 ≤ w1 0 ≤ p ≤ w3 − j3 0 ≤ j2 ≤ w2

w

3 − j3 p

w

2 − j2 +w3 − j3 − p 



+

j

3

p

0 ≤ p ≤ j3 0 ≤  ≤ j2 + j3 − p



 L

∗



j2 + j3 − p 

0 ≤  ≤ w2 − j2 + w3 − j3 − p

w1 − j1 +w2 − j2 +w3 − j3 − p−+1, +1, p+1 f2 , f3 f1 ,



L∗



j1 + j2 + j3 − p−+1, +1, p+1 f2 , f3 f1 ,

⎫ ⎬ ⎪ ⎪ ⎭



j

j

j

z 11 z 22 z 33 .

On the other hand, Lemma 2.3 implies that −P2,3 (S, S; z 1 , z 2 , z 3 ) i∞ τ1 i∞ f 1 (τ1 )(τ1 − z 1 )w1 dτ1 f 2 (τ2 )(τ2 − z 2 )w2 dτ2 f 3 (τ3 )(τ3 − z 2 )w2 dτ3 = i∞ 0 0 i∞ i∞ τ2 + f 1 (τ1 )(τ1 − z 1 )w1 dτ1 f 2 (τ2 )(τ2 − z 2 )w2 dτ2 f 2 (τ2 )(τ2 − z 2 )w2 dτ2 0

+

i∞ 0

⎩

i∞

f 2 (τ2 )(τ2 − z 2 )w2 dτ2

0

i∞

   0 w1 w2 w3 j + j + j 1 2 3 (−1) j1 j2 j3



= ⎧ ⎨

i∞

0

f 1 (τ1 )(τ1 − z 1 )w1 dτ1

f 2 (τ3 )(τ3 − z 3 )w3 dτ3

0 ≤ j1 ≤ w1 , 0 ≤ j2 ≤ w2 , 0 ≤ j3 ≤ w3





0≤≤w2 − j2

+



0≤≤w2 − j2

w2 − j2 





w2 − j2 p

  j2 −+1,  L ∗ w1 − j1 +wf2 − L ∗ ( f 3 ; w3 − j3 + 1) , f 1



2

  j3 −+1,  L ∗ ( f 1 ; w1 − j1 + 1)L ∗ w2 − j2 +wf3 − , f 2

+L ∗ ( f 1 ; w1 − j1 + 1)L ∗ ( f 2 ; w2 − j2 + 1)L ∗ ( f 3 ; w3 − j3 + 1)

3

⎫ ⎬

j

j

j

z 1z 2z 3. ⎭ 1 2 3

123

560

Y. Choie

In conclusion, for each 0 ≤ j1 ≤ w1 , 0 ≤ j2 ≤ w2 and 0 ≤ j3 ≤ w3 ,      w2 − j2 − L ∗ w1 − j1 +wf21−, j2 −+1, f2  0≤≤w2 − j2



+

w

2 − j2



L ∗ ( f 1 ; w1 − j1 + 1)L ∗

p 0≤≤w2 − j2 +L ∗ ( f 1 ; w1 − j1 + 1)L ∗ ( f 2 ; w2



=



w2 − j2 +w3 − j3 −+1,  f3 f2 ,



− j2 + 1)L ∗ ( f 3 ; w3 − j3 + 1)

0 ≤ p ≤ w3 − j3 , 0 ≤  ≤ w2 − j2 + w3 − j3 − p

w

3 − j3

p

w

2 − j2 +w3 − j3 − p

+ (−1) j1 + j2 + j3







  3 − j3 − p−+1, +1, p+1 L ∗ w1 − j1 +w2 − j2 +w f2 , f3 f1 ,  j   j + j − p  ∗  j1 + j2 + j3 − p−+1, +1, 2 3 3 L p f2 , f1 , 

0 ≤ p ≤ j3 0 ≤  ≤ j2 + j3 − p

p+1 f3



.

 Remark 4.2 Note that the pattern established in the case of double and triple L-series continues to higher order l-series: linear combinations of multiple L-values equal linear combinations of products lower order multiple L-values. Acknowledgments The author would like to thank to Prof. F. Brown, Prof. R. Bruggeman, Prof. K. Mastumoto, and Prof. Y. Manin for their comments and valuable discussions. The author also thanks to referee for the valuable comments, which made our exposition much clearer.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

11.

12. 13.

Brown, F.: Mixed Tate motives over Z. Ann. Math. (2) 175(2), 949976 (2012) Brown, F.: Multiple modular values for S L 2 (Z). arXiv:1304.5342 (2014) Brown, F.: Motivic periods and P 1 \{0, 1, ∞}. In: 2014 ICM Proceedings, pp. 1–25 (2014) (Slides) Brown, K.: Cohomology of Groups. Springer, Berlin (1982) Bruggeman, R., Choie, Y.: Multiple period integrals and cohomology. Algebra Number Theory 10(3), 645–664 (2016) Bruggeman, R., Choie, Y., Diamantis, N.: Holomorphic automorphic forms and cohomology. Mem. AMS (accepted and to appear) (2016). arXiv:1404.6718 Bruggeman, R. Lewis, J., Zagier, D.: Period functions for Maass wave forms and cohomology. Mem. AMS (2015). doi:10.1090/memo/1118 Cartier, P.: On the double zeta values. IHES/M/11/21, Juillet (2011) Choie, Y., Ihara, K.: Iterated period integrals and multiple Hecke L-functions. Manuscr. Math. 142(1– 2), 245255 (2013) Choie, Y., Matsumoto, K.: Functional equations for double series of Euler type with coefficients, Functional equations for double series of Euler type with coefficients. Adv. Math. 292, 529557 (2016). Arxiv:1306.0987v2 [math.NT] Choie, Y., Zagier, D.: Rational period functions for P S L(2, Z). In: A tribute to Emil Grosswald: Number Theory and Related Analysis. Contemporary Mathematics, vol. 143, pp. 89–108. AMS, Providence (1993) Deligne, P.: Valeurs de fonctions L et periodes d’integrales. In: Proceedings of Symposia in Pure Mathematics (part 2), vol. 33, pp. 313-346 (1979) Euler, L.: Meditationes circa singulare serierum genus. Novi Comm. Acad. Sci. Petropol 20, 140–186 (1775). Reprinted in Opera Omnia ser. I, vol. 15, pp. 217–267, B. G. Teubner, Berlin (1927). Double zeta values and zeta functions, 71106. World Scientific, Hackensack

123

Parabolic cohomology and multiple Hecke L-values

561

14. Ihara, K.: Special values of multiple Hecke L-functions. J. Appl. Math. Comput. 40(1–2), 649658 (2012) 15. Knopp, M.: Some new results on the Eichler cohomology of automorphic forms. Bull. AMS 80, 607–632 (1974) 16. Kohnen, W., Zagier, D., Modular forms with rational periods. In: Modular Forms (Durham, 1983). Ellis Horwood Series in Mathematics and Its Applications: Statistics, Operational Research, pp. 197–249. Horwood, Chichester (1984) 17. Manin, Y.I.: Periods of cusp forms, and p-adic Hecke series (in Russian). Mat. Sb. (NS) 92(134), 378–401, 503 (1973) 18. Manin, Y.I.: Iterated Shimura integrals. Mosc. Math. J. 5(4), 869881, 973 (2005) 19. Manin, Y.I.: Iterated integrals of modular forms and noncommutative modular symbols. In: Algebraic Geometry and Number Theory. Progress in Mathematics, vol. 253, p. 565597. Birkhuser, Boston (2006) 20. Manin, Y.I.: Remarks on modular symbols for Maass wave forms. Algebra Number Theory 4(8), 10911114 (2010) 21. Matsumoto, K., Tanigawa, Y.: The analytic continuation and the order extimate of multiple Dirichlet series. J. Théor. Nr. Bordeaus 15(1), 267–274 (2003) 22. Zagier, D.: Values of zeta functions and their applications. In: First European Congress of Mathematics. Volume II: Progress in Mathematics, vol. 120, pp. 497–512. Birkhuser, Basel (1994) 23. Zagier, D.: Evaluation of the multiple zeta values (2,2,3,2,2). Ann. Math. 175, 9771000 (2012). doi:10. 4007/annals.2012.175.2.11

123