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Elliptic Analogues of Multiple Zeta Values

Nils Matthes, Uni Hamburg 16th September 2014

Nils Matthes, Uni Hamburg Elliptic Analogues of Multiple Zeta Values 16th September 2014 1 / 16

Content

1 P1 \ {0, 1, ∞} and multiple zeta values 2 Elliptic parallel transport and elliptic multiple zeta values

Nils Matthes, Uni Hamburg Elliptic Analogues of Multiple Zeta Values 16th September 2014 2 / 16

Content

1 P1 \ {0, 1, ∞} and multiple zeta values 2 Elliptic parallel transport and elliptic multiple zeta values

Nils Matthes, Uni Hamburg Elliptic Analogues of Multiple Zeta Values 16th September 2014 3 / 16

Chen’s theorem for P1 \ {0, 1, ∞}

• M = P1 \ {0, 1, ∞}, a ∈ M
• Cπ1(M; a) group algebra, ε : Cπ1(M; a) → C augmentation map
• Cπ1(M; a)

completion w.r.t. to ker(ε) (augmentation ideal)

• ωKZ = dz

z X0 + dz z−1X1 ∈ Ω1(M) ⊗ CX0, X1

Parallel transport isomorphism (Chen)

Ta : Cπ1(M; a)

• ∼

=

− → CX0, X1 γ → 1 +

∞

• k=1
• γ

ωk

KZ

(well-defined because dωKZ + ωKZ ∧ ωKZ = 0).

• works also for Cπ1(M; b, a)

instead of Cπ1(M; a)

• :

Tb,a : Cπ1(M; b, a)

• ∼

=

− → CX0, X1

Nils Matthes, Uni Hamburg Elliptic Analogues of Multiple Zeta Values 16th September 2014 4 / 16

Chen’s theorem with tangential base points

• also works if a, b ∈ TzP1 \ {0}, z ∈ {0, 1, ∞}.
• specifically, let

− → 01 := ∂ ∂ξ ∈ T0P1, − → 10 := − ∂ ∂ξ ∈ T1P1.

Chen’s theorem with tangential base points

have an isomorphism T−

→ 10,− → 01 : Cπ1(M; −

→ 10, − → 01)

→ CX0, X1

γ → lim

t→0 t−X1Tγ(t),γ(1−t)(γ1−t t

)tX0.

Nils Matthes, Uni Hamburg Elliptic Analogues of Multiple Zeta Values 16th September 2014 5 / 16

Multiple zeta values

• [0, 1] ⊂ R canonical path from 0 to 1

Drinfel’d associator

dch := T−

→ 10,− → 01([0, 1]) =

• w∈X0,X1

ζ(w)w,

• We have

dch =

• w∈X0,X1

ζsh(w)w ∈ CX0, X1, where ζsh = (−1)rζ(k1, ..., kr) for w = Xk1−1 X1...Xkr−1 X1, k1 ≥ 2.

• In general, ζsh ∈ Q[MZV ].

Nils Matthes, Uni Hamburg Elliptic Analogues of Multiple Zeta Values 16th September 2014 6 / 16

Content

1 P1 \ {0, 1, ∞} and multiple zeta values 2 Elliptic parallel transport and elliptic multiple zeta values

Nils Matthes, Uni Hamburg Elliptic Analogues of Multiple Zeta Values 16th September 2014 7 / 16

Towards an elliptic analogue multiple zeta values

• Fix τ ∈ H = {ξ ∈ C | Im(τ) > 0}, E×

τ := C/(Z + Zτ) \ {0}.

• Want elliptic transport function.
• need an Eisenstein-Kronecker series Fτ(ξ, α) = θ′

τ(0)θτ(ξ+α)

θτ(ξ)θτ(α) ,

where θτ(ξ) standard odd elliptic theta function.

• Let ξ = s + rτ be the canonical coordinate on E×

τ , and consider

Ωτ(ξ, α) := e2πirαFτ(ξ, α) =

∞

• k=0

ω(k)αk−1

• Let ν = 2πidr and

J = νX0 − ad(X0)Ω(ξ, −ad(X0))(X1) ∈ Ω1(E×

τ ) ⊗Q QX0, X1.

(satisfies dJ + J ∧ J = 0)

Nils Matthes, Uni Hamburg Elliptic Analogues of Multiple Zeta Values 16th September 2014 8 / 16

Towards an elliptic analogue multiple zeta values

Elliptic parallel transport (Brown& Levin; 1110.6917)

For ρ, ξ ∈ E×

τ , we have an isomorphism

T ell

ρ,ξ : Cπ1(E× τ ; ρ, ξ) → CX0, X1

γ → 1 +

∞

• k=1
• γ

Jk

• Can also be defined for ρ, ξ ∈ T0Eτ \ {0}. Specifically, let

− → v = (−2πi)−1 ∂ ∂ξ ∈ (T0Eτ)× T ell

−− → v ,− → v :Cπ1(E× τ ; −−

→ v , − → v )

→ CX0, X1

γ → lim

t→0(−2πit)[X0,X1]T ell γ(1−t),γ(t)(γ1−t t

)(−2πit)−[X0,X1]

Nils Matthes, Uni Hamburg Elliptic Analogues of Multiple Zeta Values 16th September 2014 9 / 16

Relation with elliptic associators

• On E×

τ , have two canonical paths [0, 1], [0, τ].

Proposition (N.M.)

Let A(τ), B(τ) ∈ CX0, X1 denote the elliptic associators of Enriquez (Selecta 2014). We have T ell([0, 1]) = eπi[X0,X1]A(τ), T ell([0, τ]) = e−πi[X0,X1]B(τ).

• In this talk, we will consider only the elliptic associator A(τ).

Nils Matthes, Uni Hamburg Elliptic Analogues of Multiple Zeta Values 16th September 2014 10 / 16

An elliptic analogue of multiple zeta values

Elliptic analogue of multiple zeta values

Let w be a word in the letters X0, X1. Define Iw(τ) = T ell([0, 1])w ∈ C.

• The elliptic associator satisfies A(τ + 1) = A(τ)

the Iw(τ) admit expansions Iw(τ) =

• n∈N

anqn where q = e2πiτ.

• one can show: an ∈ Q[MZV, (2πi)−1].

Nils Matthes, Uni Hamburg Elliptic Analogues of Multiple Zeta Values 16th September 2014 11 / 16

Examples of elliptic analogues of multiple zeta values

Length 1

Id(τ) = (2πi)dBd

d!

Length 2

Let αd = 2(2πi)d

d!

, βd = (2πi)dBd

d!

Id1,d2(τ) =                        βd1βd2 2 if d1 + d2 ∈ 2Z αd2βd1 2πi

∞

• k=1
• k=ab

bd2 a qk if d1 odd and +αd1αd2 2(2πi)

∞

• k=1
• k=a(b1+b2)

b2=0

bd1

1 bd2 2

a qk d2 = 0 even

Nils Matthes, Uni Hamburg Elliptic Analogues of Multiple Zeta Values 16th September 2014 12 / 16

An elliptic analogue of multiple zeta values

• Q[eMZVτ] the Q-vector space spanned by the Iw(τ), for fixed τ

(algebra with the shuffle product).

Proposition (N.M.)

We have a surjection of Q-algebras Q[eMZVτ, (2πi)−1] → Q[MZV, (2πi)−1]

• n∈N

anqn → a0.

• Problem: how to describe explicitly a section of this surjection?
• Related problem: find a good "’numerology"’ for elliptic multiple

zeta values.

Nils Matthes, Uni Hamburg Elliptic Analogues of Multiple Zeta Values 16th September 2014 13 / 16

Elliptic multiple zeta values

• Now consider Iw(τ) as a complex function on the upper-half plane
• H. It is holomorphic for every w (Enriquez, Selecta 2014).
• Let Q[eMZV ] ⊂ O(H) be the Q-algebra spanned by the Iw.
• For a word w ∈ X0, X1 define its complexity c(w) as the number
• f X1’s appearing, and denote its length by l(w).
• Let

Q[eMZV ]c,l = SpanQ{Iw | c(w) = c l(w) = l}. Note that dc,l := dimQ Q[eMZV ]c,l < ∞.

Goal

understand Q[eMZV ]c,l; in particular compute dc,l.

Nils Matthes, Uni Hamburg Elliptic Analogues of Multiple Zeta Values 16th September 2014 14 / 16

First steps in computing dc,l

• c = 0
• Have

IXn

0 =

1 if n = 0 else hence d0,l = δ0,l for all l ∈ N.

• Q[eMZV ]0 = Q.
• c = 1
• Have

IXn

0 X1 =

• −2ζ(n)

if n is even else

• In general, IXm

0 X1Xn 0 ∈ Qπ2(m+n), i.e. in particular, they are all

constant.

• Q[eMZV ]1 = Q[π2].

Nils Matthes, Uni Hamburg Elliptic Analogues of Multiple Zeta Values 16th September 2014 15 / 16

First steps in computing dc,l

• c = 2
• Have

Q[eMZV ]2 = Q[eMZV ]even

2

• =Q[π2]
• Q[eMZV ]odd

2

.

• Q[eMZV ]odd

2

contains non-constant Iw’s.

• I can prove

l 4

• ≤ dl

2 − 1 ≤

l 2

• for l odd.

Conjecture

We have dl

2 =

l 3

• + 1

for all odd l

• verified with a computer up to length 200.

Nils Matthes, Uni Hamburg Elliptic Analogues of Multiple Zeta Values 16th September 2014 16 / 16