Iterated period integrals and multiple Hecke $ L $-functions

arXiv:1206.5068v1 [math.NT] 22 Jun 2012

Iterated period integrals and multiple Hecke L-functions YoungJu Choie∗ and Kentaro Ihara†

Abstract In this paper we express the multiple Hecke L-function in terms of a linear combination of iterated period integrals associated with elliptic cusp forms, which is introduced by Manin around 2004. This expression generalizes the classical formula of Hecke L-function obtained by the Mellin transformation of a cusp form. Also the expression gives a way of the analytic continuation of the multiple Hecke L-function.

1 Introduction Recently, in [9, 10], Manin introduces a generalization of the period integrals of elliptic cusp forms by means of the iterated path integrals on the complex upper-half plane and discusses several related topics. For example, he studies the iterated version of Mellin transformation and its functional equation, the properties of non-commutative generating series of the iterated integrals, an analogy of the classical period theory and its interpretation in terms of non-abelian group cohomology, and so on. The result which we would like to focus in his paper is an expression of the iterated period integral in terms of special values of a multiple Dirichlet series in their convergent region (see §3.2 in [9]). In this paper first we define the multiple Hecke L-function, which is essentially the same as the Dirichlet series which he introduced, and show the analytic continuation of the function. Next, as a main result, we give the expression of the iterated period integral in terms of a linear combination of L-functions, which holds in arbitrary region, and generalizes Manin’s expression. This expression also generalizes the classical formula (1) below given by Mellin transformation. As a consequence, one can write the iterated period integral as a Q-linear combination of special values of L-function. Let H = {z ∈ C | Im z > 0} be the complex upper-half plane. Assume that f (z) is a holomorphic function on H satisfying following two conditions: (i) (Fourier expansion) f (z) is periodic with period one and has a Fourier expansion of m 2πiz whose coefficients {c } have at most polythe form f (z) = ∑∞ m m =1 c m q , where q = e nomial growth in m when m → ∞: cm = O(m M ) for some M > 0. ∗ This

work was partially supported by Priority Research Centers Program NRF 2009-0094069. work was supported by Priority Research Centers Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant 2009-0094069). Mathematics Subject Classification; Primary 11E45, Secondary 11M32. † This

1

(ii) (Cusp conditions) For any γ ∈ SL2 (Z), there exists a constant C > 0 and an integer k such that ( f |k γ)(z) := j(γ, z)−k f (γz) = O(e2πiCz ), z → i∞,
where j(γ, z) = cz + d and γz = ( az + b)/(cz + d) for γ = ac db . For example, let Γ be a congruence subgroup of SL2 (Z) containing the translation
matrix 10 11 and Sk (Γ) be the space of holomorphic Γ-cusp forms of weight k. Then f (z) ∈ Sk (Γ) satisfies both (i) and (ii). The Mellin transformation of f (z) gives following expression: Z 0

i∞

f (z) zs−1 dz = −Γ(s) L( f , s)

(1)

R +∞ −s where Γ(s) = 0 e−t ts−1 dt is the Gamma function and L( f , s) := (−2πi )−s ∑∞ m =1 c m m for Re s ≫ 0 is the Hecke L-function attached to f which is normalized by (−2πi )−s for the sake of convenience. (We take the non-positive imaginary axis as a branch cut in the z-plane, so that arg(z) ∈ [−π/2, 3π/2) to define the complex power throughout this paper.) (Following [9, 10], we chose i∞ as the base point of paths. The minus sign in RHS of (1) happens for this choice.) The condition (i) guarantees the convergence of the L-function in Re s ≫ 0 and also (ii) implies the convergence of the integral in (1) in s ∈ C. For f ∈ Sk (Γ), the special values defined by the integral in (1) at critical points s = 1, . . . , k − 1 are called periods of f and play a fundamental role in the period theory. Shokurov [13] constructs some varieties, so called Kuga-Sato varieties, and interprets these numbers as periods of relative homology of the varieties. See also e.g. [7, 8] for the period theory. The equation (1) gives not only the interpretation of the periods as the special values of L-function but the way of the analytic continuation of L-functions. Manin generalizes the integral in (1) as follows. For r = 1, . . . , n, let f r be holomorphic functions on H satisfying the conditions (i) and (ii) above (for instance f r ∈ Skr (Γ) for integers kr ) and sr be complex variables. For points a, z ∈ H := H ∪ P1 (Q), we fix a path joining a to z on H and approaching a and z vertically if a or z is in P1 (Q). Then we consider the iterated integral along the path: Iaz



s1 , ..., s n f 1 , ..., f n



:=

Z z a

f 1 (z1 )z1s1 −1 dz1

Z z1 a

f 2 (z2 )z2s2 −1 dz2 · · ·

Z z n −1 a

f n (zn )zsnn −1 dzn .

(2)

The integral converges again because of cusp conditions of f r ’s and defines a holomorphic function in z ∈ H and in (s1 , . . . , sn ) ∈ Cn . For a connection between these integrals and the motif theory, see the recent paper [5] of Ichikawa. The purpose of this paper is to state a generalization of (1) in the case of the iterated integrals. In §2, we introduce the multiple Hecke L-functions (Definition 1) and generalize the identity (1) to an iterated case (Theorem 1). The LHS in (1) will be replaced by Manin’s iterated integral. A linear combination of the multiple Hecke L-functions will appear in RHS. As a consequence, one can write the iterated integral as a Q-linear combination of special values of L-function. Conversely the special values of L-function also can be expressed by the sum of iterated integrals (Corollary 1). In §3, we give proofs of these 2

theorems. In last section, we discuss the further properties of some functions which will be introduced in §2 for the proofs. In the last part of the Introduction, it is worth mentioning that the integral (2) can be seen as an analogy of the multiple polylogarithm (MPL): Lin1 ,...,nr (z) :=

zm1 ∑ n1 nr = m1 >···> m r >0 m1 · · · mr

Z z 0

η1 η2 · · · ηn ,

(3)

where |z| < 1, nl are positive integers with n = ∑ nl and ηl (z) are holomorphic 1-forms j on P1 (C) \ {0, 1, ∞} defined by ηl (z) = dz/(1 − z) or dz/z if l = ∑m=1 nm for some j = 1, . . . , r or otherwise, respectively. If n1 > 1, the limits of (3) as z → 1 exist and are called multiple zeta values (MZVs). It is known that MZVs have a rich theory and a geometric origin associated to P1 (C) \ {0, 1, ∞} or the mixed Tate motifs over Z. See [3, 4, 14]. As mentioned in [9, 10], the iterated integral (2) can be seen as an analogy of (3). The 1-forms on H replaced with those on P1 (C) \ {0, 1, ∞}. In this reason, we expect a rich theory behind the values defined by (2) such as MZVs. As a consequence of results in this paper, those values will be able to interpreted as the special values of multiple Hecke L-function. For a related topic, see [6].

2 Periods and multiple Hecke L In this section we state our main results. The proofs will be given in §3. Recall that f r (r = 1, . . . , n) are holomorphic functions on H satisfying the conditions (i) and (ii) in §1. Suppose that these functions have the Fourier expansions of the forms f r (z) = (r ) m ∑∞ m =1 c m q . Definition 1 (multiple Hecke L-function) For s ∈ C with Re s ≫ 0 and for any integers αr ≥ 1, we define L ( s) = L



s, α2 , ..., αn f 1 , f 2 , ..., f n



( 1)

:= (−2πi )

−( s + α2 +···+ αn )

∑ m1 >···> m n >0 ( m n +1 : = 0)

( 1)

= (−2πi)

−( s + α2 +···+ αn )

∑ l1 ,...,ln >0

(n)

c m 1 − m 2 · · · c m n − m n +1 m1s m2α2 · · · mαnn

(n)

c l1 · · · c l n

(l1 + · · · + ln )s (l2 + · · · + ln )α2 · · · lnαn

(r )

.

Since Fourier coefficients cm have polynomial order in m for any r, there exists a constant (n) ( 1) M > 0 such that |cm1 −m2 · · · cmn −mn+1 | = O(m1M ). This implies the absolute convergence of L(s) in Re(s) > M + 1. In the following theorem, we claim that L(s) can be extended to an entire function on C for every fixed positive integers α2 , . . . , αn . However the analytic continuation of this kind of Dirichlet series had been studied by Matsumoto-Tanigawa [11] in a general 3

context. They can regard other parameters α2 , . . . , αn as complex variables and show the analytic continuation to Cn by using the Mellin-Barnes formula. Our method in the next section is very simple and different from theirs, but may not treat other parameters as variables. Theorem 1 For any fixed positive integers αr for 2 ≤ r ≤ n, the function L(s) can be extended holomorphically to C. Further, the following equations hold for any s ∈ C:   s, α2 , ..., αn 0 (i) Ii∞ = Γ(s,α2,...,αn ) × f 1 , f 2 , ..., f n   n s + j2 , α2 − j2 + j3 , ..., αn − jn + jn +1 s + j2 −1 αl −1 + jl −1 , (4) ∑ ( j2 ) ∏ ( j ) L f1 , f2, ..., fn

(ii) L



0≤ jr < αr + jr+1 (2≤∀r ≤ n) jn +1 : =0



s, α2 , ..., αn f 1 , f 2 , ..., f n

=

l

l =3

1 Γ(s,α2,...,αn )

× n

∑ 0≤ jr < αr (2≤∀r ≤ n) jn +1 : =0

0 (s+ jj22 −1) ∏(−1) jl (αl j−l 1) Ii∞ l =2





s + j2 , α2 − j2 + j3 , ..., αn − jn + jn +1 f1 , f2, ..., fn

.

(5)

where Γ(s,α2 ,...,αn ) = (−1)n Γ(s)Γ(α2 ) · · · Γ(αn ). Corollary 1 For any integers αr ≥ 1, we have 0 (i) Ii∞

(ii) L





α1 , ..., αn f 1 , ..., f n

α1 , ..., αn f 1 , ..., f n





= Γ(α1 ,...,αn )

=

n

∏ (α − +j j −1) L

∑

l 1

l

0≤ jr < αr + jr+1 l =2 (2≤∀r ≤ n) j1 = jn +1 : =0

1 Γ(α1 ,...,αn )

∑

l

n



0 ∏(−1) jl (αl j−l 1) Ii∞

0≤ jr < αr l =2 (2≤∀r ≤ n) j1 = jn +1 : =0

α1 − j1 + j2 , ..., αn − jn + jn +1 f1 , ..., fn





α1 − j1 + j2 , ..., αn − jn + jn +1 f1 , ..., fn

,



.

3 Proof Corollary 1 follows directly from Theorem 1 by substituting s = α1 . Hence we will prove Theorem 1. The procedure of the proof of (i) is as follows: The LHS of (i) is the Mellin transformation of the function     α , ..., α −, α , ..., α Faz f1 , f22, ..., f nn := f1 (z) Iaz f22, ..., f nn −, α , ..., α for a = i∞. In the first step, we define an alternative family of functions Feaz ( f1 , f22, ..., f nn ) (Definition 2) then describe Faz by a linear combination of Feaz ’s (Proposition 1). In the second step, we compute the Fourier expansion and Mellin transformation of Feaz (Proposition 2, 3). In particular, the Mellin transformation of Feaz gives the analytic continuation

4

of L(s). Last, we will compute the Mellin transformation of Faz by combining the results in the first and second steps. For the proof of (ii), we use an inversion description of Feaz in terms of Faz ’s in Proposition 1. Definition 2 Let αr be positive integers for r = 1, . . . , n. For a ∈ H, we define holomorphic functions e Iaz and Feaz in z ∈ H by Z z1 Z z Z z n −1   z α1 , ..., αn α1 − 1 e Ia f1 , ..., f n := f1 (z1 )(z1 − z) dz1 ··· f n (zn )(zn − zn−1 )αn −1 dzn , a

Feaz

a



−, α2 , ..., αn f 1 , f 2 , ..., f n



: = f 1 ( z) e Iaz

a



α2 , ..., αn f 2 , ..., f n



,

a ∈ H.

The integral converges again by virtue of the cusp conditions of f r ’s. In this definition, the parameters αr should be positive integers, otherwise the integral depends on the choice of path because the factor (zr − zr −1 )αr −1 has a singularity on HR at zr = zr −1 . By definiz α−1 dz is ez α tion Feaz ( − 1 f 1 ) = f 1 ( z) is independent on a. The integral Ia ( f ) = a f ( z1 )( z1 − z) z e known as the Eichler integral of f when f ∈ Sk (Γ) and α = k − 1. In this sense, Ia is an iterated version of the Eichler integral. (A modular property of e Iaz will be stated in the next section.) Proposition 1 For z, a ∈ H, we have

(i) Faz



(ii) Feaz

−, α2 , ..., αn f 1 , f 2 ..., f n





−, α2 , ..., αn f 1 , f 2 ..., f n



−, α2 − j2 + j3 , ..., αn − jn + jn +1 f1, f2 ..., fn





−, α2 − j2 + j3 , ..., αn − jn + jn +1 f1 , f2 ..., fn



n



∏ (αl + jlj+l 1−1)zj2 Feaz

∑

=

0≤ jr < αr + jr+1 l =2 (2≤∀r ≤ n) jn +1 : =0

∑

=

n

∏(−1) jl (αl j−l 1)zj2 Faz

0≤ jr < αr l =2 (2≤∀r ≤ n) jn +1 : =0

,

(6)

.

(7)

Proof. Induction on n. For (i), when n = 2 it is easy. For n > 2, by using the inductive hypothesis, we have Z z     −, α , ..., α z −, α2 , ..., αn z2α2 −1 Faz2 f2 , f33 ..., f nn dz2 Fa f1 , f2 ..., f n = f1 (z)

= f 1 ( z)

Z z a

a

z2α2 −1

∑

n

j

∏ (αl + jlj+l 1−1)z23 Feaz2

0≤ jr < αr + jr+1 l =3 (3≤∀r ≤ n) jn +1 : =0



−, α3 − j3 + j4 , ..., αn − jn + jn +1 f2, f2 ..., fn



dz2 .

By the following binomial expansion α + j3 −1

z2 2

= (z2 − z + z)α2 + j3 −1 =

∑ 0≤ j2 < α2 + j3

(α2 +jj23 −1)(z2 − z)α2 − j2 + j3 −1 z j2

we get (i). (ii) follows directly from the binomial expansions. 5



z and F ez are given by Proposition 2 (Fourier expansion) The Fourier expansions of e Ii∞ i∞ z e Ii∞ z Fei∞





α1 , ..., αn f 1 , ..., f n



( 1)

(n)

( 1)

(n)

c m 1 − m 2 · · · c m n − m n +1 m Γ(α1 ,...,αn ) · = q 1, ∑ α1 αn α +···+ α n (−2πi) 1 · · · m m n m1 >···> m n >0 1 ( m n +1 : = 0)

−, α2 , ..., αn f 1 , f 2 , ..., f n



c m 1 − m 2 · · · c m n − m n +1 m Γ(α2 ,...,αn ) q 1. = · ∑ α2 αn α +···+ α n (−2πi) 2 · · · m m n 2 m1 >···> m n >0 ( m n +1 : = 0)

z and F ez are periodic functions in z of period one. In particular e Ii∞ i∞

Proof. One can prove these by comparing the derivative of both-hand sides in z, and by an inductive argument.  Proposition 3 (Mellin transform) Z 0

i∞ Z 0 i∞

z e Ii∞

z Fei∞



α1 , ..., αn f 1 , ..., f n





zs−1 dz = Γ(s,α1,...,αn ) L

−, α2 , ..., αn f 1 , f 2 , ..., f n





s + α1 , α2 , ..., αn f 1 , f 2 , ..., f n

zs−1 dz = Γ(s,α2,...,αn ) L





s, α2 , ..., αn f 1 , f 2 , ..., f n

,



.

(8)

In particular, by eq. (8) we can extend L(s) to an entire function. Proof. By applying the Mellin transformation to the equations in Proposition 2, we easily get both expressions. Since the LHS of eq. (8) and 1/Γ(s) are entire, L(s) can be extended entirely.  Proof of Theorem 1. In Proposition 3, we have already proved the analytic continuation of L(s). For (i), by using eq. (6) and (8), 0 Ii∞



s, α2 , ..., αn f 1 , f 2 , ..., f n

=

Z 0

i∞

z Fi∞

n

0≤ jr < αr + jr+1 l =2 (2≤∀r ≤ n) jn +1 : =0

−, α2 , ..., αn f 1 , f 2 , ..., f n

Z 0

i∞

Feaz





zs−1 dz

−, α2 − j2 + j3 , ..., αn − jn + jn +1 f1, f2 ..., fn

n

∏ (α + jj+ −1)Γ(s+ j ,α − j + j ,...,α − j + j + ) L

∑

=



∏ (αl + jlj+l 1−1)

∑

=



l

0≤ jr < αr + jr+1 l =2 (2≤∀r ≤ n) jn +1 : =0

l 1

2

2

2

n

3

n

n 1

l





zs+ j2 −1 dz

s + j2 , α2 − j2 + j3 , ..., αn − jn + jn +1 f1, f2 , ..., fn



.

The elementary equation below completes the proof of (i): n

n

∏ (α + jj+ −1)Γ(s+ j ,α − j + j ,...,α − j + j + ) = Γ(s,α ,...,α ) (s+ jj −1) ∏ (α − +j j −1). l

l =2

l 1

2

2

2

3

n

n

n 1

2

n

2

2

l

For (ii), apply the Mellin transformation to eq. (7), and use eq. (8). 6

l 1

l =3

l

l



4 Further properties of e Iaz and Feaz

In this section, we show several further properties of the functions e Iaz and Feaz defined in the previous section.

4.1 Alternative expression Rz Since the function a f (w)(w − z)α−1 dw in z gives the α-th anti-derivative of f (z) up to a constant multiple in general, e Iaz has an alternative simple expression in terms of an iterated integral: Proposition 4 For any positive integers αr we have   α , ..., α e Iaz f11, ..., f nn = (−1)α1 +···+αn Γ(α1 ,...,αn )

×

Z z

|a

dz1

Z z1 a

··· {z

Z z•

α1 − 1

a

dz• }

Z z• a

f 1 (z• )dz• · · ·

Z z•

|a

dz• · · · {z

Z z•

α n −1

a

dz• }

Z z• a

f n (z• )dz• .

4.2 Modular properties and its applications In this subsection we show modular properties of e Iaz and Feaz and discuss its applications. Let αr be positive integers for r = 1, . . . , n. We fix the positive integers kr by kr = αr + αr +1 for r = 1, . . . , n with αn+1 := 1 throughout this subsection. Conversely αr are expressed by kr ’s as αr = (−1)n−r +1 + ∑nj=r (−1) j−r k j . Proposition 5 For a ∈ H and γ ∈ GL2+ (Q) (+ means positive determinant), we have  α , ..., α    α , ..., α n 1 e Iγz −1 a f1 |k γ, ..., f n |k n γ ( In ) Iaz f11, ..., f nn |(−α1 +1) γ = e 1    −,  α2 , ..., αn −, α , ..., α ( Fn ) Feaz f1 , f22, ..., f nn |(α1 +1) γ = Feγz −1 a f1 |k γ, f2 |k γ, ..., f n |k n γ 1

2

where the slash action is defined in the usual manner: ( f |k γ)(z) := (det γ)k/2 j(γ, z)−k f (γz) for any function f on H. Proof. First we check ( F1 ) then show implications ( Fn ) ⇒ ( In ) and ( In−1 ) ⇒ ( Fn ) for all n. ( F1 ) is trivial: f 1 | ( α1 + 1) γ = f 1 | k 1 γ

because of the choice of k1 = α1 + 1. For ( Fn ) ⇒ ( In ), we have   Z z    α , ..., α  −, α , ..., α e Iaz f11, ..., f nn |(−α1 +1) γ = Feaz f1 , f22, ..., f nn (z1 − z)α1 −1 dz1  γ (− α1 +1) a Z γz   − α1 +1 −, α , ..., α Feaz1 f , f 2, ..., f n (z1 − γz)α1 −1 dz1 = (det γ) 2 j(γ, z)α1 −1 1

a

= (det γ)

− α1 +1 2

j(γ, z)α1 −1

Z z

γ −1 a

γz Fea 1



7

2

n

−, α2 , ..., αn f 1 , f 2 , ..., f n



(γz1 − γz)α1 −1 d(γz1 ).

(9)

Substituting the formulas γz1 − γz = (det γ) j(γ, z1 )−1 j(γ, z)−1 (z1 − z),

d(γz1 ) = (det γ) j(γ, z1 )−2 dz1

and the induction hypothesis ( Fn ):    −, α1 +1 −, α , ..., α γz Fea 1 f1 , f22, ..., f nn = (det γ)− 2 j(γ, z)α1 +1 Feγz1−1 a f1 |k γ, 1

α2 , ..., αn f 2 |k2 γ, ..., f n |k n γ

into (9), we obtain the first implication: Z z  α , ...,  −,  α2 , ..., αn 1 Iγz −1 a f1 |k γ, ..., (9) = Feaz1 f1 |k γ, f2 |k γ, ..., f n |k n γ (z1 − z)α1 −1 dz1 = e γ −1 a

1

1

2

For ( In−1 ) ⇒ ( Fn ),       α , ..., α  −, α , ..., α Iaz f22, ..., f nn Feaz f1 , f22, ..., f nn |(α1 +1) γ = f 1 (z) e 

= (det γ)

α1 +1 2

j(γ, z)

−( α1 +1)

( α1 + 1)

γz f1 (γz) e Ia



α2 , ..., αn f 2 , ..., f n

2

αn f n |k n γ

 α , ..., α  n 2 j(γ, z)−(α1 +α2 ) f 1 (γz) e Iγz −1 a f2 |k γ, ..., f n |k n γ 2  −,  α , ..., α  α , ..., n 2 = ( f1 |(α1 +α2 ) γ)(z) e Iγz −1 a f2 |k γ, ..., f n |k n γ = Feγz −1 a f1 |k γ, f2 |k2 γ, ...,

(10) = (det γ)

αn f n |k n γ



.

γ

By using the assumption ( In−1 ):    α , ..., − α2 +1 2 γz α2 , ..., αn e Ia = (det γ)− 2 j(γ, z)−α2 +1 e Iγz −1 a f2 |k γ, ..., f 2 , ..., f n

we have





.



,

(10)

α1 + α2 2

2

1

This completes the proof. We show two applications of Proposition 5.

2

αn f n |k n γ



.



4.2.1

Functional equations
Let w N = N0 −01 (Fricke involution). Suppose that f r are w N -eigenfunctions of weight   α , α , ..., α kr : fr |kr w N = ε r f r , ε r ∈ {±1} for r = 1, . . . , n. Put g(z) := e Icz f11, f22, ..., f nn for c = √iN R0 (w N -fixed point) and set Λ(s, g) := i∞ g(z)zs−1 dz.

Theorem 2 It holds that

Λ(s, g) = (−1)α1 −1+s N where ε =

− α1 +1 −s 2

∏rn=1 ε r .

εΛ(−α1 + 1 − s, g)

Proof. For any holomorphic function f (z) on H satisfying the condition (ii) in §1, one can R0 show by the standard method that the Mellin transformation Λ(s, f ) := i∞ f (z)zs−1 dz satisfies α Λ(s, fb) = (−1)s−α N 2 −s Λ(α − s, f )

where α is any integer and fb(z) = ( f |α w N )(z). If we substitute g to f and −α1 + 1 to α, then we obtain the theorem because gb = εg is true by virtue of Proposition 5.  8

4.2.2

Twisted iterated integral

M z+ m For any Dirichlet character χ mod M we set f χ (z) := ∑m =1 χ ( m ) f ( M ). It is well-known [12] that a holomorphic function f on H is Γ0 ( N )-invariant of weight k: f |k γ = f for ∀γ ∈ Γ0 ( N ) if and only if it is periodic with period one and satisfies
σ = 01 −01 f χ k σ = χ(−1) f χ ,

for all Dirichlet characters χ (mod Nc) for each c ∈ {1, . . . , N }, where χ is the complex conjugate of χ. Suppose fr ∈ Skr (Γ0 ( N )) and χr are Dirichlet characters of mod Nc with 1 ≤ c ≤ N for r = 1, . . . , n. Define two functions by  α , ..., α   α1 , ..., αn  n 1 Gχ ( z ) : = e Iiz f χ1 , ..., f nχn , Gχ ( z ) : = e Iiz f χ1 , ..., f nχn 1

1

and denote their Mellin transformations by Λ(s, Gχ ) and Λ(s, Gχ ) respectively. By Proposition 5 and above fact, one can show that Gχ (−α +1) σ = χ1 (−1) · · · χn (−1) Gχ . 1

Hence we have the following identity: Theorem 3

Λ(s, Gχ ) = (−1)s χ1 (−1) · · · χn (−1)Λ(1 − α1 − s, Gχ ).

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