Shuffle regularized multiple Eisenstein series and the Goncharov - PowerPoint PPT Presentation

Shuffle regularized multiple Eisenstein series and the Goncharov coproduct Henrik Bachmann - University of Hamburg joint work with Koji Tasaka (PMI, POSTECH) Numbers and Physics (NAP2014) ICMAT Madrid, 17 September 2014 Henrik Bachmann - University of Hamburg Shuffle regularized multiple Eisenstein series and the Goncharov coproduct

Content of this talk In this talk we want to explain a connection between the Fourier expansion of multiple Eisenstein series and the Goncharov coproduct given on a space of formal iterated integrals. This enables us to define shuffle regularized multiple Eisenstein series G ✁ . Multiple zeta values Multiple Eisenstein series and their Fourier expansion Formal iterated integrals and the Goncharov coproduct Shuffle regularized multiple zeta values and certain q -series Shuffle regularized multiple Eisenstein series Open questions Henrik Bachmann - University of Hamburg Shuffle regularized multiple Eisenstein series and the Goncharov coproduct

Multiple zeta values Definition For natural numbers n 1 , ..., n r − 1 ≥ 1 , n r ≥ 2 , the multiple zeta value (MZV) of weight N = n 1 + ... + n r and length r is defined by 1 � ζ ( n 1 , ..., n r ) = . m n 1 1 . . . m n r r 0 <m 1 <...<m r By MZ N we denote the space spanned by all MZV of weight N and by MZ the space spanned by all MZV. The product of two MZV can be expressed as a linear combination of MZV with the same weight (stuffle relation). e.g: ζ ( r ) · ζ ( s ) = ζ ( r, s ) + ζ ( s, r ) + ζ ( r + s ) . MZV can be expressed as iterated integrals. This gives another way (shuffle relation) to express the product of two MZV as a linear combination of MZV. These two products give a number of Q -relations (double shuffle relations) between MZV. Henrik Bachmann - University of Hamburg Shuffle regularized multiple Eisenstein series and the Goncharov coproduct

Multiple zeta-values Example: shuffle stuffle ζ (3 , 2) + 3 ζ (2 , 3) + 6 ζ (1 , 4) = ζ (2) · ζ (3) = ζ (2 , 3) + ζ (3 , 2) + ζ (5) . double shuffle = ⇒ 2 ζ (2 , 3) + 6 ζ (1 , 4) = ζ (5) . But there are more relations between MZV. e.g.: ζ (1 , 2) = ζ (3) . These follow from the "extended double shuffle relations" where one use the same combinatorics as above for " ζ (1) · ζ (2) " in a formal setting. The extended double shuffle relations are conjectured to give all relations between MZV. Henrik Bachmann - University of Hamburg Shuffle regularized multiple Eisenstein series and the Goncharov coproduct

Classical Eisenstein series For even k > 2 the Eisenstein series of weight k defined by ∞ ( lτ + m ) k = ζ ( k ) + ( − 2 πi ) k k ( τ ) := 1 1 � � G ♠ σ k − 1 ( n ) q n 2 ( k − 1)! ( l,m ) ∈ Z 2 n =1 ( l,m ) � =(0 , 0) are modular forms of weight k . These functions vanish for odd k (and there are no non trivial modular forms of odd weight) since one sums over all lattice points. Similary if one would define the riemann zeta value as a sum over all integer, i.e. ζ ♠ ( k ) := 1 1 � 2 n k n ∈ Z n � =0 then these series would vanish for odd k and ζ ♠ ( k ) = ζ ( k ) for even k . We now want to define a multiple version of these series where we also have odd Eisenstein series in length one. For this we define an order on the lattice Z τ + Z by defining what we mean by positive lattice points. Henrik Bachmann - University of Hamburg Shuffle regularized multiple Eisenstein series and the Goncharov coproduct

A particular order on lattices Let Λ τ = Z τ + Z be a lattice with τ ∈ H := { x + iy ∈ C | y > 0 } . We define an order ≺ on Λ τ by setting λ 1 ≺ λ 2 : ⇔ λ 2 − λ 1 ∈ P for λ 1 , λ 2 ∈ Λ τ and the following set which we call the set of positive lattice points P := { lτ + m ∈ Λ τ | l > 0 ∨ ( l = 0 ∧ m > 0) } = U ∪ R l U m R Henrik Bachmann - University of Hamburg Shuffle regularized multiple Eisenstein series and the Goncharov coproduct

Classical Eisenstein series are ordered sums With this order on Λ τ one gets for even k > 2 : λ k = 1 1 1 � � ( lτ + m ) k = G ♠ G k ( τ ) := k ( τ ) . 2 0 ≺ λ ( l,m ) ∈ Z 2 ( l,m ) � =(0 , 0) Since we are not summing over all lattice points the odd Eisenstein series don’t vanish anymore and we get for all k : ∞ G k ( τ ) = ζ ( k ) + ( − 2 πi ) k � σ k − 1 ( n ) q n . ( k − 1)! n =1 This order now allows us to define a multiple version of these series in the obvious way. Henrik Bachmann - University of Hamburg Shuffle regularized multiple Eisenstein series and the Goncharov coproduct

Multiple Eisenstein series Definition For integers n 1 , . . . , n r − 1 ≥ 2 and n r ≥ 3 , we define the multiple Eisenstein series G n 1 ,...,n r ( τ ) on H by 1 � G n 1 ,...,n r ( τ ) = . λ n 1 1 · · · λ n r r 0 ≺ λ 1 ≺···≺ λ r λ i ∈ Z τ + Z It is easy to see that these are holomorphic functions in the upper half plane and that they fulfill the stuffle product, i.e. it is for example G 3 ( τ ) · G 4 ( τ ) = G 4 , 3 ( τ ) + G 3 , 4 ( τ ) + G 7 ( τ ) . Remark The condition n r ≥ 3 is necessary for absolutely convergence of the sum. By choosing a specific way of summation we can also restrict this condition to get a definition of G n 1 ,...,n r ( τ ) with n r = 2 which also satisfies the stuffle product. Henrik Bachmann - University of Hamburg Shuffle regularized multiple Eisenstein series and the Goncharov coproduct

Multiple Eisenstein series - Fourier expansion Since G s 1 ,...,s r ( τ + 1) = G s 1 ,...,s r ( τ ) we have a Fourier expansion: Theorem (B. 2012) There exists a family of arithmetically defined q -series g n 1 ,...,n r ( q ) ∈ Q [2 πi ][[ q ]] such that for n 1 , . . . , n r ≥ 2 the Fourier expansion of G n 1 ,...,n r is given by G n 1 ,...,n r ( τ ) = ζ ( n 1 , . . . , n r ) � � ξ ( r − 1) ξ ( r − 2) + g k 2 ( q ) + g k 2 ,k 3 ( q ) k 1 k 1 k 1 + k 2 = N k 1 + k 2 + k 3 = N k 1 ,k 2 ≥ 2 k 1 ,k 2 ,k 3 ≥ 2 � ξ (1) + · · · + k 1 g k 2 ,...,k r ( q ) + g n 1 ,...,n r ( q ) , k 1 + ··· + k r = N k 1 ,...,k r ≥ 2 where the ξ ( d ) ∈ MZ k are Q -linear combinations of multiple zeta values of weight k and k length less than or equal to d . The length r = 2 case was originally considered by Gangl, Kaneko and Zagier. From now on we also write G n 1 ,...,n r ( q ) instead of G n 1 ,...,n r ( τ ) . Henrik Bachmann - University of Hamburg Shuffle regularized multiple Eisenstein series and the Goncharov coproduct

Multiple Eisenstein series - Fourier expansion - Multitangent functions Let 1 � Ψ n 1 ,...,n r ( x ) := ( x + m 1 ) n 1 · · · ( x + m r ) n r . m 1 < ··· <m r This series absolutely converges for n i ∈ Z > 0 with n 1 , n r ≥ 2 , and is called multitangent function . In the case r = 1 we also refer to these series as monotangent function . Theorem (Bouillot 2011, B. 2012) For n 1 , . . . , n r ≥ 2 and N = n 1 + · · · + n r the multitangent function can be written as N � Ψ n 1 ,...,n r ( x ) = α N − j Ψ j ( x ) j =2 with α N − j ∈ MZ N − j . Proof idea: Use partial fraction decomposition. Henrik Bachmann - University of Hamburg Shuffle regularized multiple Eisenstein series and the Goncharov coproduct

Multiple Eisenstein series - Fourier expansion - Multitangent functions Example: 1 � Ψ 2 , 3 ( x ) = ( x + m 1 ) 2 ( x + m 2 ) 3 m 1 <m 2 � 1 3 � � = + ( m 2 − m 1 ) 3 ( x + m 1 ) 2 − ( m 2 − m 1 ) 4 ( x + m 1 ) m 1 <m 2 � � 1 2 3 � ( m 2 − m 1 ) 2 ( x + m 2 ) 3 + ( m 2 − m 1 ) 3 ( x + m 2 ) 2 + ( m 2 − m 1 ) 4 ( x + m 2 ) m 1 <m 2 = 3 ζ (3)Ψ 2 ( x ) + ζ (2)Ψ 3 ( x ) . Since one can derive explicit formulas for the partial fraction expansions of 1 ( x + m 1 ) n 1 ... ( x + m r ) nr this reduction of multitangent into monotangent can be done explicitly. Henrik Bachmann - University of Hamburg Shuffle regularized multiple Eisenstein series and the Goncharov coproduct

Multiple Eisenstein series - Fourier expansion - The function g For integers n 1 , . . . , n r ≥ 1 , define � v n 1 − 1 · · · v n r − 1 q m 1 v 1 + ··· + m r v r , g n 1 ,...,n r ( q ) := c · 1 r 0 <m 1 <...<m r 0 <v 1 ,...,v r ( − 2 πi ) n 1+ ··· + nr where c = ( n 1 − 1)! ··· ( n r − 1)! . Proposition For n 1 , . . . , n r ≥ 2 these series can be written as an ordered sum of monotangent functions: � g n 1 ,...,n r ( q ) = Ψ n 1 ( m 1 τ ) . . . Ψ n r ( m r τ ) . 0 <m 1 < ··· <m r The q -series g n 1 ,...,n r ( q ) divided by ( − 2 πi ) n 1 + ··· + n r are the generating series of multiple divisor sums [ n r , . . . , n 1 ] ∈ Q [[ q ]] which were studied in a joint paper with U. Kühn. Henrik Bachmann - University of Hamburg Shuffle regularized multiple Eisenstein series and the Goncharov coproduct

Multiple Eisenstein series - Fourier expansion Summing over 0 ≺ λ 1 ≺ · · · ≺ λ r is by definition equivalent to summing over all λ 1 , . . . , λ r with λ i − λ i − 1 ∈ P = U ∪ R ( λ 0 := 0) . Since λ i − λ i − 1 can be either in U or in R we can split up the sum in the definition of the MES into 2 r terms. For w 1 , . . . , w r ∈ { U, R } we define 1 � G w 1 ...w r n 1 ,...,n r ( τ ) = . λ n 1 1 · · · λ n r r λ 1 ,...,λ r ∈ Λ τ λ i − λ i − 1 ∈ w i With this we get � G w 1 ...w r G n 1 ,...,n r ( τ ) = n 1 ,...,n r ( τ ) . w 1 ,...,w r ∈{ U,R } Henrik Bachmann - University of Hamburg Shuffle regularized multiple Eisenstein series and the Goncharov coproduct