Elliptic Analogues of Multiple Zeta Values Nils Matthes, Uni Hamburg - PowerPoint PPT Presentation

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 P 1 \ { 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 P 1 \ { 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 P 1 \ { 0 , 1 , ∞} • M = P 1 \ { 0 , 1 , ∞} , a ∈ M • C π 1 ( M ; a ) group algebra, ε : C π 1 ( M ; a ) → C augmentation map � completion w.r.t. to ker( ε ) (augmentation ideal) • C π 1 ( M ; a ) • ω KZ = dz z − 1 X 1 ∈ Ω 1 ( M ) ⊗ C �� X 0 , X 1 �� dz z X 0 + Parallel transport isomorphism (Chen) ∼ = � T a : C π 1 ( M ; a ) − → C �� X 0 , X 1 �� ∞ � � ω k γ �→ 1 + KZ γ k =1 (well-defined because dω KZ + ω KZ ∧ ω KZ = 0 ). � instead of C π 1 ( M ; a ) � • works also for C π 1 ( M ; b, a ) : ∼ = � � T b,a : C π 1 ( M ; b, a ) − → C �� X 0 , X 1 �� 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 ∈ T z P 1 \ { 0 } , z ∈ { 0 , 1 , ∞} . • specifically, let 01 := ∂ 10 := − ∂ − → ∂ξ ∈ T 0 P 1 , − → ∂ξ ∈ T 1 P 1 . Chen’s theorem with tangential base points have an isomorphism 01 : C π 1 ( M ; − → 10 , − → � → C �� X 0 , X 1 �� T − 01) → 10 , − → t → 0 t − X 1 T γ ( t ) ,γ (1 − t ) ( γ 1 − t ) t X 0 . γ �→ lim t 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 − 01 ([0 , 1]) = ζ ( w ) w, → 10 , − → w ∈� X 0 ,X 1 � • We have � ζ sh ( w ) w ∈ C �� X 0 , X 1 �� , dch = w ∈� X 0 ,X 1 � where ζ sh = ( − 1) r ζ ( k 1 , ..., k r ) for w = X k 1 − 1 X 1 ...X k r − 1 X 1 , k 1 ≥ 2 . 0 0 • In general, ζ sh ∈ Q [ MZV ] . Nils Matthes, Uni Hamburg Elliptic Analogues of Multiple Zeta Values 16th September 2014 6 / 16

Content 1 P 1 \ { 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 + Z τ ) \ { 0 } . • Want elliptic transport function. • need an Eisenstein-Kronecker series F τ ( ξ, α ) = θ ′ τ (0) θ τ ( ξ + α ) θ τ ( ξ ) θ τ ( α ) , where θ τ ( ξ ) standard odd elliptic theta function. • Let ξ = s + rτ be the canonical coordinate on E × τ , and consider ∞ � Ω τ ( ξ, α ) := e 2 πirα F τ ( ξ, α ) = ω ( k ) α k − 1 k =0 • Let ν = 2 πidr and J = νX 0 − ad( X 0 )Ω( ξ, − ad ( X 0 ))( X 1 ) ∈ Ω 1 ( E × τ ) ⊗ Q Q �� X 0 , X 1 �� . (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 : C π 1 ( E × � → C �� X 0 , X 1 �� τ ; ρ, ξ ) ∞ � � J k γ �→ 1 + γ k =1 • Can also be defined for ρ, ξ ∈ T 0 E τ \ { 0 } . Specifically, let v = ( − 2 πi ) − 1 ∂ − → ∂ξ ∈ ( T 0 E τ ) × τ ; −− → v , − → v : C π 1 ( E × T ell � → C �� X 0 , X 1 �� v ) −− → v , − → t → 0 ( − 2 πit ) [ X 0 ,X 1 ] T ell γ (1 − t ) ,γ ( t ) ( γ 1 − t )( − 2 πit ) − [ X 0 ,X 1 ] γ �→ lim t 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 ) ∈ C �� X 0 , X 1 �� denote the elliptic associators of Enriquez (Selecta 2014). We have T ell ([0 , 1]) = e πi [ X 0 ,X 1 ] A ( τ ) , T ell ([0 , τ ]) = e − πi [ X 0 ,X 1 ] 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 X 0 , X 1 . Define I w ( τ ) = T ell ([0 , 1]) w ∈ C . • The elliptic associator satisfies A ( τ + 1) = A ( τ ) � the I w ( τ ) admit expansions � a n q n I w ( τ ) = n ∈ N where q = e 2 πiτ . • one can show: a n ∈ Q [ MZV, (2 πi ) − 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 I d ( τ ) = (2 πi ) d B d d ! Length 2 Let α d = 2 (2 πi ) d β d = (2 πi ) d B d , d ! d !  β d 1 β d 2 if d 1 + d 2 ∈ 2 Z    2       ∞ b d 2  α d 2 β d 1 � �  a q k if d 1 odd and I d 1 ,d 2 ( τ ) = 2 πi  k =1 k = ab   ∞ b d 1 1 b d 2  + α d 1 α d 2 � �   2 q k d 2 � = 0 even   2(2 πi ) a    k =1 k = a ( b 1+ b 2) b 2 � =0 Nils Matthes, Uni Hamburg Elliptic Analogues of Multiple Zeta Values 16th September 2014 12 / 16

An elliptic analogue of multiple zeta values • Q [ eMZV τ ] the Q -vector space spanned by the I w ( τ ) , for fixed τ (algebra with the shuffle product). Proposition (N.M.) We have a surjection of Q -algebras Q [ eMZV τ , (2 πi ) − 1 ] → Q [ MZV, (2 πi ) − 1 ] � a n q n �→ a 0 . n ∈ N • 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 I w ( 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 I w . • For a word w ∈ � X 0 , X 1 � define its complexity c ( w ) as the number of X 1 ’s appearing, and denote its length by l ( w ) . • Let Q [ eMZV ] c,l = Span Q { I w | c ( w ) = c l ( w ) = l } . Note that d c,l := dim Q Q [ eMZV ] c,l < ∞ . Goal understand Q [ eMZV ] c,l ; in particular compute d c,l . Nils Matthes, Uni Hamburg Elliptic Analogues of Multiple Zeta Values 16th September 2014 14 / 16

First steps in computing d c,l • c = 0 • Have � 1 if n = 0 I X n 0 = 0 else hence d 0 ,l = δ 0 ,l for all l ∈ N . • � Q [ eMZV ] 0 = Q . • c = 1 • Have � − 2 ζ ( n ) if n is even I X n 0 X 1 = 0 else 0 ∈ Q π 2( m + n ) , i.e. in particular, they are all • In general, I X m 0 X 1 X n constant. • � Q [ eMZV ] 1 = Q [ π 2 ] . Nils Matthes, Uni Hamburg Elliptic Analogues of Multiple Zeta Values 16th September 2014 15 / 16

First steps in computing d c,l • c = 2 • Have � Q [ eMZV ] 2 = Q [ eMZV ] even Q [ eMZV ] odd . 2 2 � �� � = Q [ π 2 ] • Q [ eMZV ] odd contains non-constant I w ’s. 2 • I can prove � l � l � � ≤ d l 2 − 1 ≤ 4 2 for l odd. Conjecture We have � l � d l 2 = + 1 3 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