Multiple Eisenstein series

Transcription

1 Heilbronn Workshop on String Theory and Arithmetic Geometry University of Bristol - 5th September 2012

2 Multiple zeta-values Definition For natural numbers s 1 2, s 2,..., s l 1 the multiple zeta-value (MZV) of weight s s l and length l ist defined by ζ(s 1,..., s l ) = n 1 >...>n l >0 1 n s ns l l 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.

3 Multiple zeta-values Example: ζ(2, 3)+3ζ(3, 2)+6ζ(4, 1) shuffle = ζ(2) ζ(3) stuffle = ζ(2, 3)+ζ(3, 2)+ζ(5). = 2ζ(3, 2) + 6ζ(4, 1) But there are more relations between MZV. e.g.: ζ(2, 1) = ζ(3). double shuffle = ζ(5). 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.

4 Generalized divisor sums Definition For integers s 1,..., s l 0 we define the generalized divisor sum of length l by σ s1,...,s l (n) := v s vs l l. u 1 v u l v l =n u 1 >...>u l >0 v 1,...,v l >0 In the case l = 1 this is just the classical divisor sum σ s1 (n) = d n ds1.

5 Generalized divisor sums Definition For integers s 1,..., s l 0 we define the generalized divisor sum of length l by σ s1,...,s l (n) := v s vs l l. u 1 v u l v l =n u 1 >...>u l >0 v 1,...,v l >0 In the case l = 1 this is just the classical divisor sum σ s1 (n) = d n ds1. These functions also fulfill a lot of relations. For example for all n N we have σ 1,0 (n) 1 2 σ 1(n) + n σ 0 (n) = 1 2 σ 2(n) or σ 7 = 1680σ 3,3 (n) + 21σ 3 (n) σ 1 (n).

6 Modular forms Definition A holomorphic function f : H C is a modular form of weight k if it satisfies ( ) ( ) aτ + b f = (cτ + d) k a b f(τ), SL cτ + d c d 2 (Z) and if it has a Fourier expansion of the form f(τ) = n=0 a nq n with a n C and q = e 2πiτ. If a 0 = 0 then f is called cusp form.

7 Modular forms Definition A holomorphic function f : H C is a modular form of weight k if it satisfies ( ) ( ) aτ + b f = (cτ + d) k a b f(τ), SL cτ + d c d 2 (Z) and if it has a Fourier expansion of the form f(τ) = n=0 a nq n with a n C and q = e 2πiτ. If a 0 = 0 then f is called cusp form. For even k > 2 the Eisenstein series of weight k defined by G k (τ) := 1 2 (m,n) Z 2 (m,n) (0,0) 1 ( 2πi)k = ζ(k)+ (mτ + n) k (k 1)! σ k 1 (n)q n n=1 are modular forms of weight k. These functions vanish for odd k (and there are no non trivial modular forms of odd weight).

8 A particular order on lattices Before we can define the multiple Eisenstein series we need an order on lattices: Given τ H we consider the lattice Zτ + Z, then for lattice points a 1 = m 1 τ + n 1 and a 2 = m 2 τ + n 2 we write a 1 a 2 if (m 1 m 2, n 2 n 1 ) P := R U with R = { (0, n) Z 2 n > 0 } and U = { (m, n) Z 2 m > 0 }. m U nr

9 Classical Eisenstein series are ordered sums With this order on Zτ + Z one gets for even k > 2: G k (τ) := a 0 a Zτ+Z 1 a k = 1 2 (m,n) Z 2 (m,n) (0,0) 1 (mτ + n) k. Using this modified definition for G k we get in fact for all k > 2: G k (τ) = ζ(k) + ( 2πi)k (k 1)! σ k 1 (n)q n. n=1

10 Definition For τ H and natural numbers s 1 3, s 2,..., s l 2 we define the multiple Eisenstein series of weight s s l and length l by G s1,...,s l (τ) := a 1... a l 0 a i Zτ+Z 1 a s as l l Remark The Stuffle relations are fulfilled because the formal sum manipulations are the same as for MZV. Shuffle relations hold as long as they are consequences of the formal partial fraction decomposition. (For l = 2 these functions were studied by Gangl, Kaneko & Zagier (2006))

11 Questions We have G s1,...,s l (τ) = G s1,...,s l (τ + 1). How does the Fourier expansion look like? Modularity (for even k)? modified definitions of G s1,...,s l in the cases where MZV are defined, e.g. for all s 1 2, s 2,..., s l 1? Stuffle? Shuffle?

12 Fourier expansion of multiple Eisenstein series Theorem For s 1 3 and s 2,..., s l 2 one has: G s1,...,s l = ζ(s 1,..., s l ) + n>0 a n q n, where a n = α a 1,...,a l j 1 j l a a l =s s l with generalized divisor sums (πi) a aj ζ(a j+1,..., a l ) σ a1 1,...,a j 1(n) σ s1,...,s l (n) := u 1 v u l v l =n u 1 >...>u l >0 v s vs l l and the numbers α a 1,...,a l j Q can be computed algorithmically.

13 Examples G 4,4 (τ) = ζ(4, 4) + ( ) ( 2πi) 4+4 σ 3,3 (n) + 20( 2πi) 2 ζ(6)σ 1 (n) + ( 2πi)4 ζ(4)σ 3 (n) q n. 3! 3! 2 n>0 Notation: ζ(s 1,..., s l ) := ( 2πi) s s l ζ(s 1,..., s l ) G s1,..,s l (τ) := ( 2πi) s s l G s1,..,s l (τ) e.g. with further simplifications by explicit known MZV s G 4,4 (τ) = n>0 ( σ 3,3 (n) 1 84 σ 1(n) + 1 ) 80 σ 3(n) q n

14 Examples G 4,5,6 (τ) = ( 2πi) 15 G 4,5,6 (τ) = ζ(4, 5, 6) + ( 1)4+5+6 c ( σ 3,4,5 (n) σ 0(n) ) σ 37 2(n) σ 4(n) q n n>0 1 ( ) 3600σ 4 (n) ζ(6, 4) σ 2 (n) ζ(7, 5) σ 2 (n) ζ(8, 4) q n c n>0 1 ( ) σ 0 (n) ζ(8, 6) σ 0 (n) ζ(9, 5) σ 0 (n) ζ(10, 4) q n c n>0 1 ( 1 c n>0 168 σ 2,5(n) σ 3,2(n) σ 3,4(n) + 1 ) 240 σ 4,5(n) q n i ( ζ(5) c n>0 20π 5 σ 1(n) + ζ(5) 14π 5 σ 3(n) ζ(5) 80π 5 σ 5(n) 3ζ(5) ) π 5 σ 3,5(n) q n i ( 45ζ(5) 2 c n>0 32π 10 σ 4(n) + 25ζ(7) 64π 7 σ 1(n) + 21ζ(7) 32π 7 σ 3(n) 105ζ(7) ) 64π 7 σ 5(n) q n i ( 315ζ(7) c n>0 8π 7 σ 1,5 (n) 315ζ(7) 4π 7 σ 3,3 (n) 2835ζ(5)ζ(7) 16π 12 σ 2 (n) ζ(7) ) 2 128π 14 σ 0 (n) q n i ( 189ζ(9) c n>0 16π 9 σ 1(n) 945ζ(9) 16π 9 σ 3(n) ζ(9) 64π 9 σ 5 (n) ζ(9) ) 4π 9 σ 3,1 (n) q n i ( 8505ζ(5)ζ(9) c n>0 16π 14 σ 0 (n) ζ(11) 64π 11 σ 3 (n) ζ(13) ) 128π 13 σ 1 (n) q n, (c = 3! 4! 5!)

15 Fourier expansion - Glimpse of the proof Nice combinatorics but lengthy. Unfortunately I have no time to explain it now. But you can ask me later.

16 Modularity, easy case Because of of the stuffle relation we have for example G 2 4 = 2G 4,4 + G 8 so G 4,4 is a modular form of weight 8. In general we have Theorem If all s 1,..., s l are even and all s j > 2, then we have G sσ(1),...,s σ(l) M k (SL 2 (Z)), σ Σ l where the weight k is given by k = s s l. Proof: Easy induction using stuffle relation.

17 Modularity, cusp forms By the double shuffle relations and Eulers formula ζ(2k) = λ π 2k for λ Q one can show: ζ(12) ζ(6, 3, 3) ζ(4, 5, 3) ζ(7, 5) = 40ζ(4) ζ(6) 2 = 0 because of Euler s formula.

18 Modularity, cusp forms By the double shuffle relations and Eulers formula ζ(2k) = λ π 2k for λ Q one can show: ζ(12) ζ(6, 3, 3) ζ(4, 5, 3) ζ(7, 5) = 40ζ(4) ζ(6) 2 = 0 because of Euler s formula. But in the context of multiple Eisenstein series we get: Theorem G G 4,5, G 6,3,3 G 7,5 = S 12 (SL 2 (Z)).

19 Modularity & cusp forms Proof: The first identity of the above MZV Relation hold also for multiple Eisensten series. It follows from the Stuffle relation and partial fraction decompositions which replaces Shuffle. But in the second identity, i.e., the place when Eulers formula is needed, one gets the "error term", because in general whenever s 1 + s 2 12 the following function doesn t vanish G s1 G s2 ζ(s 1)ζ(s 2 ) ζ(s 1 + s 2 ) G s 1 +s 2 S s1 +s 2 (SL 2 (Z)). So the failure of Euler s relation give us the cusp forms Remark There are many more such linear relations which give cusp forms

20 Modularity, cusp forms From such identities we get new relations between Fourier coefficients of modular forms and generalized divisor sums, e.g.: Corollary - Formula for the Ramanujan τ -function For all n N we have τ(n) = σ 1(n) σ 3(n) σ 11(n) σ 3,3 (n) σ 6,4 (n) σ 5(n) σ 2,2 (n) σ 3,1 (n) 3 σ 4,2 (n) σ 5,1 (n) 3 σ 3,4,2 (n) σ 5,2,2 (n). (τ) = q q i=1(1 n ) 24 = τ(n)q n = q 24q q q n>0

21 "Non convergent" multiple Eisenstein series For l = 2 Kaneko, Gangl & Zagier give an extended definition for the Double Eisenstein series G 3,1,G 2,1, G 2,2,.... Natural question: What should G 2,...,2 in general be? We want our multiple Eisenstein series to fulfill the same linear relations as the corresponding MZV (modulo cusp forms), therefore we have to imitate the following well known result in the context of multiple Eisenstein series: Theorem For λ n := ( 1) n 1 2 2n 1 (2n + 1) B 2n we have ζ(2n) λ n ζ(2,..., 2) = 0. }{{} n

22 "Non convergent" multiple Eisenstein series Ansatz: Define G 2,...,2 to be the function obtained by setting all s i to 2 in the formula of the Fourier Expansion. e.g. G 2 (τ) = ζ(2) + ( 2πi) 2 n>0 σ 1 (n)q n, G 2,2 (τ) = ζ(2, 2) + ( 2πi) 4 n>0 ( σ 1,1 (n) 1 ) 8 σ 1(n) q n. Will this give the "right" definition of G 2,...,2?

23 "Non convergent" multiple Eisenstein series Ansatz: Define G 2,...,2 to be the function obtained by setting all s i to 2 in the formula of the Fourier Expansion. e.g. G 2 (τ) = ζ(2) + ( 2πi) 2 n>0 σ 1 (n)q n, G 2,2 (τ) = ζ(2, 2) + ( 2πi) 4 n>0 ( σ 1,1 (n) 1 ) 8 σ 1(n) q n. Will this give the "right" definition of G 2,...,2? No, because with this definition the function G 2n λ n G 2..., 2 }{{} n / S 2n (SL 2 (Z)) is not a cusp form. It is not modular but quasi-modular, this property is used for the following modified definition:

24 The multiple Eisenstein series G 2,...,2 Theorem Let X n (τ) := (2πi) 2l G 2,..., 2 (τ), D := q d dq and G 2,..., 2 } {{ } n } {{ } n n 1 (τ) := X n (τ) + j=1 then we have with λ n Q as above (2n 2 j)! 2 j j! (2n 2)! Dj X n j (τ). G 2n λ n G 2,...,2 S 2n (SL 2 Z).

25 The multiple Eisenstein series G 2,...,2 Examples: G 2,2 = G 2, D1 G2, G 2,2,2 = G 2,2, D1 G2, D2 G2, G 2,2,2,2 = G 2,2,2, D1 G2,2, D2 G2, D3 G2.... In weight 12 we get G 12 λ 6 G 2,2,2,2,2,2 = which gives another expression for τ(n) in σ 11 (n) and σ 1,...,1 (n) (see last slide).

26 Sketch of the proof First show following identity for the generating function of X n Φ(τ, T ) := X n (τ)( 4πiT ) n = exp 2 ( 1) l (2l)! E 2l(τ)( 4πiT ) l n 0 l 1 where E k (τ) = B k 2k + n>0 σ k 1(n)q n. Using the modularity of the E k for k > 2 one sees easily that Φ is a Jacobi-like form of weight 0, i.e.: ( ) ( ) ( ) aτ + b Φ cτ + d, T ct a b (cτ + d) 2 = exp Φ(τ, T ), SL (cτ + d) c d 2 (Z). One can show that coefficients of such functions always give rise to modular forms as above.

27 Application New formular for fourier coefficients of cusp forms, e.g.: ( τ(n) = 7560 n n n n n 73 ) σ 1 (n) ( n n n n ) σ 1,1 (n) 864 ( n n n 479 ) σ 1,1,1 (n) 9 ( ) n2 288n σ 1,1,1,1 (n) + (768n 7040) σ 1,1,1,1,1 (n) σ 1,1,1,1,1,1 (n) σ 11(n) Reminder: σ 1,..., 1 (n) = }{{} l u 1v u l v l =n u 1>...>u l >0 v 1... v l.