On the generators of a certain algebra of q-multiple zeta values Ulf Kühn - Universität of Hamburg Zeta Values, Modular Forms and Elliptic Motives II ICMAT Madrid, December 5, 2014 joint work with Henrik Bachmann 1 / 34
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Multiple zeta values Definition For natural numbers s 1 ≥ 2 , s 2 , ..., s l ≥ 1 the multiple zeta value (MZV) of weight s 1 + ... + s l and length l is defined by � 1 ζ ( s 1 , ..., s l ) = . n s 1 1 . . . n s l l n 1 >...>n l > 0 The rules for the product of infinite sums imply that the product of MZV can be expressed as a linear combination of MZV with the same weight (stuffle product). MZV can be expressed as iterated integrals. This gives another way (shuffle product) to express the product of two MZV as a linear combination of MZV. These two products give a large number of Q -linear relations (extended double shuffle relations) between MZV. Conjecturally these are all relations between MZV, e.g. shuffle stuffle ζ (2 , 3) + 3 ζ (3 , 2) + 6 ζ (4 , 1) = ζ (2) · ζ (3) = ζ (2 , 3) + ζ (3 , 2) + ζ (5) . 3 / 34
Dimension conjectures for MZ Broadhurst-Kreimer Conjecture The Q -algebra MZ of multiple zeta values is a free polynomial algebra, which is graded for the weight and filtered for the depth ("depth drop for even zetas"). The numbers g k,l of generators in weight k ≥ 3 and depth l are determined by � � � 1 + E ( x ) y x k y l = gr W,D BK ( x, y ) = dim Q MZ 1 − O ( x ) y + S ( x ) y 2 − S ( x ) y 4 k,l k,l � � � 1 = 1 + E ( x ) y � 1 − x k y l � g k,l k ≥ 3 ,l ≥ 1 with x 2 1 − x 2 = x 2 + x 4 + x 6 + ... "even zetas" , E ( x ) = x 3 1 − x 2 = x 3 + x 5 + x 7 + ... "odd zetas" , O ( x ) = x 12 (1 − x 4 )(1 − x 6 ) = x 12 + x 16 + x 18 + ... "cusp forms" . S ( x ) = 4 / 34
Dimension conjectures for MZ Zagier’s Conjecture The following identities hold: � � � 1 x k = gr W Zag ( x ) = dim Q k MZ 1 − x 2 − x 3 . k Zagier’s conjecture is implied by Broadhurst-Kreimer’s conjecture. In order to neglect the depth we just have to set y = 1 and get x 2 1 + Zag ( x ) = BK ( x, 1) = 1 + E ( x ) 1 1 − x 2 1 − O ( x ) = = 1 − x 2 − x 3 . x 3 1 − 1 − x 2 5 / 34
Dimension conjectures for MZ Theorem (Euler, Ihara-Kaneko-Zagier, Goncharov, Ihara-Ochiai, Brown, ...) Let l ≤ 3 , then the numbers g k,l of generators for MZ of weight k and length l are not bigger than implied by the Broadhurst-Kreimer conjecture, i.e. we have � x 3 g k, 1 x k ≤ x 2 + 1 − x 2 , k � k − 2 � � � x 8 g k, 2 x k ≤ x k = (1 − x 2 )(1 − x 6 ) , 6 k k> 0 k even � ( k − 3) 2 − 1 � x 11 (1 + x 2 − x 4 ) � � g k, 3 x k ≤ x k = (1 − x 2 )(1 − x 4 )(1 − x 6 ) . 48 k k> 0 k odd Idea of proof: For l = 1 this is a trivial consequence of Euler’s formula for even zetas. For l = 2 , 3 we bound the number of generators by the dimension of the double shuffle spaces DS ( k − l, l ) . 6 / 34
double shuffle space for MZ ( s 1 , ..., s l ) for all ( s 1 , ..., s l ) ∈ N l ("shuffle regularised There is a unique way to define ζ ∃ MZV") such that the generating series � ∃ ( s 1 , ..., s l ) x s 1 − 1 · ... · x s l − 1 F l ( x 1 , ..., x l ) = ζ , ∃ 1 l ( s 1 ,...,s l ) ∈ N l with notation f ♯ ( x 1 , ..., x j ) = f ( x 1 + ... + x j , ..., x j − 1 + x j , x j ) satisfies � ♯ � � � ♯ ( x 1 , ..., x j ) � � ♯ ( x j +1 , ..., x l ) = � � ∃ ∃ ∃ sh j ( x 1 , ..., x l ) , F F F � (1) j l − j n where sh j denotes the set of all shuffles of type ( j, l − j ) . If we consider (1) modulo products, then (assuming MZV are graded for the weight) we obtain for each generator of weight k and depth l a homogenous polynomial in the set � � deg f = k − l and f ♯ | sh j ( x 1 , ..., x l ) = 0 ∀ 1 ≤ j ≤ l − 1 } { f ∈ Q [ x 1 , ..., x l ] 7 / 34
double shuffle space for MZ Analogously there are stuffle regularised MZV and their generating series � F ∗ ζ ∗ ( s 1 , ..., s l ) x s 1 − 1 · ... · x s l − 1 l ( x 1 , ..., x l ) = 1 l ( s 1 ,...,s l ) ∈ N l satisfies (ignoring the terms with MZV of depth less than l ) � F ∗ j ( x 1 , ..., x j ) F ∗ l − j ( x j +1 , ..., x l − j ) = F ∗ � sh j ( x 1 , ..., x l ) + ... (2) l If we consider (2) modulo products and terms of lower depth, we obtain for each generator a homogenous polynomial in the set � � deg f = k − l and f | sh j ( x 1 , ..., x l ) = 0 ∀ 1 ≤ j ≤ l − 1 } { f ∈ Q [ x 1 , ..., x l ] Because of ζ ( s 1 , ..., s l ) ≡ ζ ∗ ( s 1 , .., s l ) modulo lower depth, we get ∃ Proposition Let the double shuffle space be defined by � � deg f = k − l, f ♯ | sh j = f | sh j = 0 ∀ j } , DS ( k − l, l ) = { f ∈ Q [ x 1 , ..., x l ] then we get for the number g k,l of algebra generators weight k and depth l for MZ the bound g k,l ≤ dim Q DS ( k − l, l ) . 8 / 34
generators for MZ in l=2: double shuffle space The spaces DS ( k − 2 , 2) are spanned by polynomials such that f ( x 1 + x 2 , x 2 ) + f ( x 1 + x 2 , x 1 ) = 0 and f ( x 1 , x 2 ) + f ( x 2 , x 1 ) = 0 ⇐ ⇒ f ( x 1 , x 2 ) = − f ( x 1 , x 1 − x 2 ) and f ( x 1 , x 2 ) = − f ( x 2 , x 1 ) . � 1 0 � Let ptp − 1 := and t = ( 0 1 1 0 ) and G = � ptp − 1 , t � . Then 1 − 1 DS ( k − 2 , 2) = Q [ x 1 , x 2 ] G k − 2 , where of the action of G on Q [ x 1 , x 2 ] is determined above. Now by Molien’s theorem the numbers of invariant polynomials of given degree is determined by � x 6 H Q [ x 1 ,x 2 ] G ( x ) = 1 det( g ) det(1 − g · x ) = (1 − x 2 )(1 − x 6 ) 12 g ∈ G = x 6 + x 8 + x 10 + 2 x 12 + 2 x 14 + ... 9 / 34
generators for MZ in l=3: double shuffle space � 0 0 1 � � 1 1 1 � and set H = � t, ptp − 1 , − 1 � , then Let t = , p = 0 1 0 1 1 0 1 0 0 1 0 0 DS ( k − 3 , 3) ⊂ Q [ x 1 , x 2 , x 3 ] H � 0 1 0 � � 0 1 0 � and G = � t, ptp − 1 , c 3 � . The main idea is to interprete the Let c 2 = , c 3 = 1 0 0 0 0 1 0 0 1 1 0 0 shuffle sh 1 = 1 + c 2 + c 3 , as the relative Reynolds operator from H to G , this is due to the key identity tc 2 = c − 1 . Then we get an exact sequence 3 | sh1 ⊕| p sh1 p − 1 � Q [ x 1 , x 2 , x 3 ] G ⊕ Q [ x 1 , x 2 , x 3 ] G p → 0 0 → DS ( · , 3) → Q [ x 1 , x 2 , x 3 ] H where G p = � t, ptp − 1 , pc 3 p − 1 � . Again by Molien’s theorem we get the result. Remark: The cases l ≥ 4 are only partially understood so far. 10 / 34
bi-brackets Definition For n ∈ N and natural numbers r 1 , . . . , r l ≥ 0 , s 1 , . . . , s l > 0 we call � u r 1 1 v s 1 1 . . . u r l l v s l σ s 1 ,...,s l r 1 ,...,r l ( n ) := l u 1 v 1 + ··· + u l v l = n u 1 > ··· >u l > 0 v 1 ,...,v l > 0 the bi-multiple divisor sum of n with bi index s 1 ,...,s l r 1 ,...,r l and with κ := r 1 !( s 1 − 1)! . . . r l !( s l − 1)! we define their generating series as � s 1 , . . . , s l � � := 1 ( n ) q n . κ · σ s 1 − 1 ,...,s l − 1 r 1 , . . . , r l r 1 ,..., r l n ∈ N For short hand we refer to this q -series as bi-brackets of length l and of weight w = s 1 + · · · + s l + r 1 + · · · + r l . Its upper weight is given by s 1 + · · · + s l and lower weight equals r 1 + · · · + r l . 11 / 34
bi-brackets The bi-brackets can also be written as � s 1 , . . . , s l � � n r 1 1 P s 1 − 1 ( q n 1 ) . . . n r l l P s l − 1 ( q n l ) = c · , (1 − q n 1 ) s 1 . . . (1 − q n l ) s l r 1 , . . . , r l n 1 > ··· >n l > 0 where the P k − 1 ( t ) are the Eulerian polynomials defined by � P k − 1 ( t ) d k − 1 t d . (1 − t ) k = Li 1 − k ( t ) = d> 0 Examples: P 2 ( t ) = t 2 + t , P 3 ( t ) = t 3 + 4 t 2 + t , P 0 ( t ) = P 1 ( t ) = t , � 1 , 1 � � q n 1 n 2 q n 2 = (1 − q n 1 )(1 − q n 2 ) , 0 , 1 n 1 >n 2 > 0 � 4 , 2 , 1 � � 1 ( q 3 n 1 + 4 q 2 n 1 + q n 1 ) · q n 2 · n 5 n 2 3 q n 3 1 = (1 − q n 1 ) 4 · (1 − q n 1 ) 2 · (1 − q n 1 ) 1 . 2 , 0 , 5 3! · 2! · 5! n 1 >n 2 >n 3 > 0 12 / 34
multiple divisor sums and modular forms For r 1 = · · · = r l = 0 we also write � s 1 , . . . , s l � � 1 σ s 1 − 1 ,...,s l − 1 ( n ) q n . = [ s 1 , . . . , s l ] =: 0 , . . . , 0 ( s 1 − 1)! . . . ( s l − 1)! n> 0 We call the coefficients σ s 1 − 1 ,...,s l − 1 ( n ) multiple divisor sums and their generating series [ s 1 , . . . , s l ] will be called brackets . In the case l = 1 we get the classical divisor sums σ k − 1 ( n ) = � d | n d k − 1 and � 1 σ k − 1 ( n ) q n . [ k ] = ( k − 1)! n> 0 These function appear in the Fourier expansion of classical Eisenstein series which are (quasi)-modular forms for SL 2 ( Z ) , for example G 2 = − 1 1 1 24 + [2] , G 4 = 1440 + [4] , G 6 = − 60480 + [6] . 13 / 34
BD and MD Definition By BD we denote the Q -vector space spanned by all bi-brackets and 1 and by MD ⊆ BD we denote the Q -vector space spanned by all brackets and 1 . The (bi)-brackets have a direct connection to multiple zeta values, since they are q -multiple zeta values: Theorem (B.-K. , Zudilin) Assume that s 1 > r 1 + 1 and s j ≥ r j + 1 for j = 2 , .., l . Then � s 1 , ..., s l � � � s 1 + ... + s l 1 lim 1 − q = r 1 ! · ... · r l ! ζ ( s 1 − r 1 , ..., s l − r l ) . r 1 , ..., r j q → 1 Remark: Another very interesting connection to MZV is given by the Fourier expansion of multiple Eisenstein series. 14 / 34
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