The elliptic KZB-associator - Elliptic multiple zeta values

Recall that L

denotes the free Lie algebra on the set { x

, x

}, completed for the degree (where x

₀

and x

₁

both have degree one).

Proposition 2.3.1. There exists a unique solution G : (0, 1) → exp( L

) to the differential equation

dG(s) =





− ad(x

)F

(s, ad(x

))(x

)





G(s)ds, (2.39)

such that G(s) ∼ ( − 2πis)

^−[x⁰^,x¹^]

as s → 0, where the branch of the logarithm is chosen such that log( ± πi/2) = ± i. Likewise, there exists a unique solution H : (0, 1) → exp( L

) to the differential equation

dH(s) =





2πix

ds − τ ad(x

)e

^2πis·ad(x⁰⁾

F

(s · τ, ad(x

))(x

)





H(s)ds, (2.40) such that H(s) ∼ ( − 2πis)

^−[x⁰^,x¹^]

as s → 0.

Proof: It follows from Proposition A.2.2.(vi) that for all 0 < ε << 1, the function G

(s) = exp

^{" Z} ^ε

ad(x

)F

(s, ad(x

))(x

)

( − 2πiε)

⁻^[x⁰^,x¹^]

(2.41) solves (2.39). Moreover, by Proposition A.2.6, the limit

G(s) := lim

_ε→0

G

(s) (2.42)

exists, and as a limit of solutions to (2.39), is also a solution to (2.39) on all of (0, 1).

To see that it has the correct asymptotic behavior, note that G(s) ∼ lim

ε→0

e

⁻^log(s)[x⁰^,x¹^]+log(ε)[x⁰^,x¹^]

e

⁻log(−2πiε)[x0,x1]

= e

⁻log(−2πis)[x0,x1]

(2.43) as s → 0. A similar argument shows that the function

H(s) := lim

ε→0

exp

^{" Z} ^ε

− 2πix

+ τ ad(x

)e

^2πis^·ad(x⁰⁾

F

(s · τ, ad(x

))(x

)

ds

× ( − 2πiε)

^−[x⁰^,x¹^]

(2.44)

solves (2.40), and satisfies H(s) ∼ ( − 2πis)

^−[x⁰^,x¹^]

as s → 0.

Remark 2.3.2. The asymptotic condition that G(s) ∼ ( − 2πis)

^−[x⁰^,x¹^]

, H(s) ∼

( − 2πis)

^−[x⁰^,x¹^]

as s → 0 can be seen as an analogue of an initial condition for an

ordinary differential equation in the presence of regular singularities. Expanding

2.3. The elliptic KZB-associator

(−2πis)

⁻^[x⁰^,x¹^]

as a formal exponential introduces log(−2πi)-terms in the formulas above. If one passes from E

= C /( Z + Z τ ) to the Tate curve C

^×

/q

via the exponential map ξ 7→ e

^2πiξ

and transports the differential equation in Proposition 2.3.1 to the Tate curve, then one would get rid of the log( − 2πi)-terms.

We extend G(s) to (0, 2) \ { 1 } by analytic continuation around the point 1 ∈ C in the negative direction, i.e. along a path whose image is contained in { a +bτ | b ≥ 0}.

Likewise, we extend H(s) to (0, 2) \ { 1 } along a path whose image is contained in { a +bτ | a ≥ 0 } . Since both { a +bτ | a ≥ 0 } and { a+bτ | b ≥ 0 } are simply connected the analytic continuation does not depend on the choice of path.

Definition 2.3.3. The elliptic KZB-associator is the triple (Φ

, A(τ ), B(τ)), where Φ

is the Drinfeld associator and A(τ ), B(τ) ∈ Chh x

, x

ii are formal series defined by

A(τ) = G(s)

⁻¹

G(1 + s), B(τ ) = H(s)

⁻¹

H(1 + s). (2.45) The above definition is the original definition of the elliptic KZB associator, as given in [31], Section 6. For our purposes, however, it will be useful to modify the definition of A(τ) and B(τ ) slightly by setting

A(τ) := e

^−πi^ad(x⁰^)(x¹⁾

A(τ), B(τ) := e

^πi^ad(x⁰^)(x¹⁾

B(τ). (2.46) Remark 2.3.4. We have given the definition of the elliptic KZB associator using ex-plicit iterated integrals, essentially following [31]. A more conceptual way of defining it, which also clarifies the difference between A(τ ), B(τ ) and its underscored variants A(τ), B(τ) can be given as follows. Consider the tangent vector − → v

₀

:= ( − 2πi)

⁻¹_∂ξ^∂

at 0 ∈ E

_τ

(which equals the tangent vector −

_∂z^∂

at 1 on the Tate curve C

^×

/q

, where q = e

^2πiτ

, z = e

^2πiξ

). As in (2.33), we have a morphism

T

−→^ellv

: π

₁

(E

_τ^×

; − → v ) → Chh x

₀

, x

₁

ii , (2.47) obtained by integrating the elliptic KZB form ω

KZB

, where π

(E

_τ^×

; − → v ) is the fun-damental group of E

_τ^×

with respect to the “tangential base point” − → v at zero ([26],

§15). This is a free group on two generators α, β, which correspond to the two natural homology cycles on an elliptic curve. We then have the identities

T

−→^ellv

(α) = A(τ ), T

−→^ellv

(β) = B(τ). (2.48) On the other hand, if instead of the fundamental group π

₁

(E

_τ^×

; − → v ) one considers the fundamental torsor of paths π

₁

(E

_τ^×

; − → v , −− → v ) from − → v to −− → v , then

T

−→^ellv ,−−→v

(α) = A(τ ), T

−→^ellv ,−−→v

(β) = B (τ ). (2.49)

This definition of the elliptic KZB associator is in some sense analogous to the definition of the Drinfeld (KZ) associator Φ

as the image of the natural straight line path [0, 1] under the map π

₁

( P

\ { 0, 1, ∞} ; − → 1

, − − → 1

) → Chh x

₀

, x

₁

ii obtained by integrating the KZ form ω

_KZ

(cf. Example A.2.7, or [29], §5.16).

Proposition 2.3.5. The series A(τ) and B(τ) satisfy the following properties.

(i) We have the explicit formulae A(τ ) = lim

ε→0

( − 2πiε)

^ad(x⁰^)(x¹⁾

× exp

^{" Z} ^1−ε

− ad(x

)F

(s, ad(x

))(x

)ds

( − 2πiε)

⁻^ad(x⁰^)(x¹⁾

, (2.50)

B(τ ) = lim

ε→0

( − 2πiε)

^ad(x⁰^)(x¹⁾

× exp

^{" Z} ^(1−ε)

2πix

− τ ad(x

)e

^2πis^ad(x⁰⁾

F

_τ

(sτ, ad(x

))(x

)

ds

× ( − 2πiε)

⁻^ad(x⁰^)(x¹⁾

. (2.51)

(ii) They are exponentials of Lie series A(τ), B(τ) : H → L

, i.e.

A(τ) = exp(A(τ )), B(τ ) = exp(B(τ)). (2.52) These Lie series satisfy

A(τ) ≡ − x

mod [ L

, L

] (2.53) B(τ) ≡ 2πix

− τ x

₁

mod [ L

, L

]. (2.54) Proof: (i) follows directly from the formulas for G(t) and H(t) given in Proposition 2.3.1, using the composition of paths formula for iterated integrals (Proposition A.2.2.(i)). For (ii), first note that since G(t) and H(t) (thus also G(1 + s) and H(1 + s)) are solutions to initial value problems, they are group-like by Proposition A.2.3, hence so are A(τ) and B(τ ) as products of group-like series (where we also note that e

^±πi^ad(x⁰^)(x¹⁾

is also group-like). But a group-like series is the exponential of a Lie series (cf. Proposition A.1.9).

Finally, by Proposition 2.2.3, the image of ω

KZB

in the abelianization L

/p, where p denotes the commutator of L

is given by

ω

_KZB^ab

= − 2πidr · x

+ dξ · x

. (2.55)

2.3. The elliptic KZB-associator

Hence

A(τ ) = log(A(τ)) = log(e

⁻^πi^ad(x⁰^)(x¹⁾

· G

⁻¹

(t)G(1 + t)) ≡ −

^Z ⁰

x

₁

ds +

^Z ⁰

1+t

x

₁

ds

= − x

₁

mod p, (2.56) and likewise

B(τ ) = log(B(τ )) = log(e

^πi^ad(x⁰^)(x¹⁾

· H

⁻¹

(t)H(1 + t))

≡ −

Z ₀

− 2πix

+ τ x

₁

ds +

^Z ⁰

1+t

− 2πix

+ τ x

₁

ds (2.57)

= 2πix

− τ x

₁

mod p.

We end this section by mentioning that A(τ ) and B(τ ) also satisfy a certain

dif-ferential equation on the upper half-plane, which relates them to iterated integrals

of Eisenstein series [20, 57] and also to a Lie algebra of special derivations on the

fundamental Lie algebra of a once-punctured elliptic curve [60, 61, 75]. We postpone

its discussion to Chapter 5, where it is put into its natural context.

Chapter 3 Elliptic multiple zeta values

In this chapter, we begin our study of the coefficients of Enriquez’ elliptic KZB associator, which are called elliptic multiple zeta values. This name is justified by the fact that the elliptic KZB associator is an elliptic analogue of the Drinfeld associator (cf. Section 1.3), whose coefficients are precisely the multiple zeta values.

Elliptic multiple zeta values come in two types, namely A-elliptic and B-elliptic multiple zeta values, corresponding to the two “elliptic” parts A(τ ) and B(τ) of the elliptic KZB associator (cf. Definition 2.3.3). In this chapter, we first define A-elliptic multiple zeta values, study their properties, and give many examples. Then, we introduce our version of B-elliptic multiple zeta values, and study its relation to the variant of B-elliptic multiple zeta values defined by Enriquez. The difference between the two versions is mainly that our version of the B-elliptic multiple zeta values is given by homotopy invariant iterated integrals, while Enriquez’s version is not.

3.1 Definition and first properties of A-elliptic multiple zeta values

The A-elliptic multiple zeta values to be introduced in this section have first been defined by Enriquez [32].

Recall the definition of the formal differential one-form

Ω

(ξ, α) = e

^2πirα

F

(ξ, α), ξ = s + rτ, r, s ∈ R , (3.1) where F

denotes the Kronecker series (cf. (2.20)).

Proposition 3.1.1. For all λ, µ ∈ C

^×

, the limit lim

t→0

(λt)

^ad(x⁰^)(x¹⁾

exp

^{" Z} ^1−t

ad(x

)Ω

(ξ, ad(x

))(x

)

(µt)

⁻^ad(x⁰^)(x¹⁾

(3.2)

exists.

Proof: If ξ is real, then r(ξ) = 0. Therefore

ad(x

)Ω

(ξ, ad(x

))(x

) = F

(ξ, ad(x

))(x

)dξ. (3.3) From Proposition 2.1.2, F

_τ

(ξ, α) has a simple pole at ξ = 0 with residue 1. Thus, the existence of the limit follows from Proposition A.2.6

Definition 3.1.2. For integers k

, . . . , k

≥ 0, define the A-elliptic multiple zeta value I

(k

, . . . , k

; τ ) to be the coefficient of ad

^k¹

(x

)(x

) . . . ad

^kⁿ

(x

)(x

) in

lim

t→0

( − 2πit)

^ad(x⁰^)(x¹⁾

exp

^{" Z} ¹⁻^t

ad(x

)Ω

(ξ, ad(x

))(x

)

( − 2πit)

⁻^ad(x⁰^)(x¹⁾

, (3.4) which is contained in Chh x

₀

, x

₁

ii . The weight of I

(k

, . . . , k

) is the sum k

₁

+. . .+k

, and its length is n.

Remark 3.1.3. If k

₁

, k

6 = 1, then ω

^(k¹⁾

and ω

^(kⁿ⁾

have no poles at 0 and at 1, and I

(k

, . . . , k

) is equal to the bona fide convergent iterated integral

I

(k

, . . . , k

; τ) =

^Z ¹

ω

^(k¹⁾

. . . ω

^(kⁿ⁾

=

^Z ¹

f

^(k¹⁾

(ξ

)dξ

. . . f

^(kⁿ⁾

(ξ

)dξ

, (3.5) where the functions f

^(k)

have been introduced in Section 2.1.

An important property of A-elliptic multiple zeta values is their Fourier expansion.

Proposition 3.1.4. Every A-elliptic multiple zeta value has a Fourier expansion

X∞ n=0

a

q

ⁿ

, q = e

^2πiτ

. (3.6)

Proof: By Proposition 2.1.3, the functions f

^(k)

(ξ) (which implicitly depend on τ ) have Fourier expansions in q = e

^2πiτ

. Using the equality (3.5), this Fourier expansion passes to the A-elliptic multiple zeta values by integration.

The Fourier coefficients of A-elliptic multiple zeta values turn out to be Q [(2πi)

⁻¹

]-linear combinations of multiple zeta values, which can be computed explicitly, see Section 3.3.

Definition 3.1.5. Define the Q -vector space of A-elliptic multiple zeta values to be

EZ

= h I

(k

, . . . , k

; τ ) | k

, . . . , k

≥ 0 i

. (3.7)

3.1. Definition and first properties of A-elliptic multiple zeta values

In analogy to the case of multiple zeta values, we define for k, n ≥ 0 the Q -vector subspace

L

ⁿ

( EZ

^Ak

) = h I

(k

, . . . , k

; τ) | k

₁

+ . . . + k

= k, r ≤ n i

⊂ EZ

. (3.8) We will sometimes also use the notation L

( EZ

) :=

^P_k≥0

L

( EZ

^Ak

) for the space of all A-elliptic multiple zeta values of length at most n.

Proposition 3.1.6. For all k, k

⁰

, n, n

⁰

≥ 0 , we have

L

( EZ

^Ak

) L

n⁰

( EZ

^Ak⁰

) ⊂ L

n+n⁰

( EZ

^Ak+k⁰

), (3.9) i.e. EZ

is a bi-filtered Q -subalgebra of O ( H ), the C -algebra of holomorphic func-tions on H . More precisely, we have

I

(k

, . . . , k

; τ )I

(k

r+1

, . . . , k

_r+s

; τ) =

σ∈Σ(r,s)

I

(k

σ(1)

, . . . , k

_σ(r+s)

; τ ) (3.10) where Σ(r, s) denotes the set of (r, s)-shuffles, i.e. the set of permutations σ of { 1, . . . , r +s } , such that σ

⁻¹

is strictly increasing on { 1, . . . , r } and on { r +1, . . . , r + s } .

Proof: As a quotient of Jacobi theta functions, which are holomorphic functions of τ, the Kronecker series F

_τ

(ξ, α) is holomorphic in the variable τ as well. This implies that A-elliptic multiple zeta values are also holomorphic in τ, being integrals of holo-morphic functions. Equation (3.10) follows from the definition of I

(k

, . . . , k

; τ) and the fact that (3.4) is group-like, and hence its coefficients satisfy the shuffle product formula (by Proposition A.1.7).

Proposition 3.1.7. For all k

₁

, . . . , k

≥ 0 , we have the reflection relation

I

(k

, . . . , k

; τ) = ( − 1)

^k¹^+...+kⁿ

I

(k

, . . . , k

; τ) (3.11) Proof: By the inversion of paths formula for iterated integrals (cf. Proposition A.2.2.(ii)), we have for every 0 < t << 1

Z ₁₋t

ω

^(kⁿ⁾

. . . ω

^(k¹⁾

= ( − 1)

ⁿ^Z ^t

1−t

ω

^(k¹⁾

. . . ω

^(kⁿ⁾

. (3.12) It follows from the symmetry properties of the Kronecker series (Proposition 2.1.2) that under the substitution ξ 7→ 1 − ξ, we have

ω

^(k_ξ7→1−ξⁱ⁾

= ( − 1)

^kⁱ⁺¹

ω

^kⁱ

, (3.13)

hence

( − 1)

ⁿ^Z ^t

1−t

ω

^(k¹⁾

. . . ω

^(kⁿ⁾

= ( − 1)

^k¹^+...+kⁿ^Z ¹⁻^t

ω

^(k¹⁾

. . . ω

^(kⁿ⁾

. (3.14) Now I

(k

, . . . , k

; τ ) is the coefficient of ad

^kⁿ

(x

)(x

) . . . ad

^k¹

(x

)(x

) in

lim

t→0

( − 2πit)

^ad(x⁰^)(x¹⁾

exp

^{" Z} ¹⁻^t

ad(x

)Ω

(ξ, ad(x

))(x

)

( − 2πit)

⁻^ad(x⁰^)(x¹⁾

. (3.15) and by (3.4), ( − 1)

^k¹^+...+kⁿ

I

(k

, . . . , k

; τ ) is the coefficient of

lim

t→0

( − 2πit)

^ad(x⁰^)(x¹⁾

exp

^{" Z} ^1−t

ad(x

)Ω

(ξ, ad(x

))(x

)

( − 2πit)

⁻^ad(x⁰^)(x¹⁾

#op

, (3.16) where a supscript op denotes the opposite multiplication on Chh x

, x

ii , defined by (F · G)

^op

:= G · F . Comparing coefficients yields the result.

3.1.1 Explicit examples in lengths one and two

Using the Fourier expansion of the Kronecker series, one can give explicit formulas of elliptic multiple zeta values of length one, and relate them to even single zeta values.

Proposition 3.1.8. We have

I

(k; τ) =







− 2ζ(k) if k is even, 0 if k is odd.

(3.17)

In particular, I

(0; τ ) = − 2ζ(0) = − 2

−

¹₂

= 1.

Proof: For even k, we have from Proposition 2.1.3.(v) I

(k; τ ) =

^Z ¹

ω

^(k)

=

^Z ¹

f

^(k)

(ξ)dξ

=

^Z ¹

− 2ζ(k) − 2 (2πi)

(k − 1)!

X∞

m=1

cosh(2πimξ)

^X^∞

n=1

n

^k⁻¹

q

^mn

dξ

= − 2ζ(k), (3.18)

while for odd k with k 6 = 1 I

(k; τ ) =

^Z ¹

ω

^(k)

=

^Z ¹

f

^(k)

(ξ)dξ

=

^Z ¹

− 2 (2πi)

(k − 1)!

X∞

m=1

sinh(2πimξ)

^X^∞

n=1

n

^k−1

q

^mn

dξ (3.19)

3.1. Definition and first properties of A-elliptic multiple zeta values

= 0.

For k = 1, use the Fourier expansion of f

⁽¹⁾

together with (3.4) to obtain I

(1; τ ) = lim

_t→0

log( − 2πit) − log( − 2πit) +

^Z ¹⁻^t

f

⁽¹⁾

(ξ)dξ

= lim

_t→0^Z ¹⁻^t

πi coth(πiξ) − 4πi

^X^∞

m=1

sinh(2πimξ)

^X^∞

n=1

q

^mn

dξ (3.20)

= 0.

The preceding proposition already implies our first theorem concerning the algebraic structure of elliptic multiple zeta values.

Theorem 3.1.9. We have

L

( EZ

^Ak

) =







Q · π

if k is even

{ 0 } else. (3.21)

In particular,

D

^ell_k,1

=







1 if k ≥ 2 is even

0 else. (3.22)

In principle, the Fourier expansion method can be used in higher lengths as well, however, the resulting formulas become quite long and cumbersome. Later on, a different representation of A-elliptic multiple zeta values in terms of iterated integrals of Eisenstein series will be introduced.

For the case of length two elliptic multiple zeta values, we have the following partial result. See Propositions 4.1.2 and 4.1.1 for more precise results.