Multiple Eisenstein series - PDF Free Download

Workshop on Periods and Motives - YRS Madrid 4th June 2012

Motivation

A particular 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

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

Definition For τ H and natural numbers s 1 3, s 2,..., s l 2 we define the multiple Eisenstein series of weight s 1 +... + s l and length l by G s1,...,s l (τ) := a 1... a l 0 a i Zτ+Z 1 a s 1 1.... 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))

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?

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 1 +...+a l =s 1 +...+s l with generalised divisor functions (πi) a 1+...+aj ζ(a j+1,..., a l ) σ a1 1,...,a j 1(n) σ s1,...,s l (n) := u 1 v 1 +...+u l v l =n u 1 >...>u l >0 v s 1 1... vs l l and the numbers α a 1,...,a l j Q can be computed algorithmically.

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 1+...+s l ζ(s 1,..., s l ) G s1,..,s l (τ) := ( 2πi) s 1+...+s l G s1,..,s l (τ) e.g. with further simplifications by explicit known MZV s G 4,4 (τ) = 1 29030400 + 1 36 n>0 ( σ 3,3 (n) 1 84 σ 1(n) + 1 ) 80 σ 3(n) q n

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) 281 2882880 σ 0(n) + 130399 ) 605404800 σ 37 2(n) 1330560 σ 4(n) q n n>0 1 ( ) 3600σ 4 (n) ζ(6, 4) + 293760σ 2 (n) ζ(7, 5) + 302400σ 2 (n) ζ(8, 4) q n c n>0 1 ( ) 1814400σ 0 (n) ζ(8, 6) + 2903040σ 0 (n) ζ(9, 5) + 27280σ 0 (n) ζ(10, 4) q n c n>0 1 ( 1 c n>0 168 σ 2,5(n) 1 120 σ 3,2(n) + 1 168 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) + 42525ζ(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) + 1125ζ(9) 64π 9 σ 5 (n) + 2835ζ(9) ) 4π 9 σ 3,1 (n) q n i ( 8505ζ(5)ζ(9) c n>0 16π 14 σ 0 (n) + 28755ζ(11) 64π 11 σ 3 (n) 135135ζ(13) ) 128π 13 σ 1 (n) q n, (c = 3! 4! 5!)

Fourier expansion - Sketch of the proof In the classical case (l = 1) one splits the sum of G k into two parts: G k (τ) = a 0 a Zτ+Z = ζ(k) + m>0 1 a k = a R 1 a k }{{} G R k := 1 a k }{{} G U k := + a U 1 (mτ + n) k n Z }{{} Ψ k (mτ):= and then uses the Lipschitz summation formula for the second part: to get Ψ k (τ) = n Z 1 (τ + n) k = ( 2πi)k (k 1)! G k (τ) = ζ(k) + ( 2πi)k (k 1)! d k 1 q d. d=1 σ k 1 (n)q n. n=1

Fourier expansion - Sketch of the proof For the case l = 2 one has to consider 4 = 2 2 sums: where G s1,s 2 (τ) = G R2 s 1,s 2 (τ) + G UR s 1,s 2 (τ) + G U 2 s 1,s 2 (τ) + G RU s 1,s 2 (τ) = ζ(s 1, s 2 ) + m>0 Ψ s1 (mτ)ζ(s 2 ) + m 1 >m 2 >0 Ψ s1 (m 1 τ)ψ s2 (m 2 τ) + m>0 Ψ s1,...,s l (τ) := n 1 >...>n l n i Z l i=1 1 (τ + n i ) s i. Ψ s1,s 2 (mτ),

Fourier expansion - Sketch of the proof For the Fourier expansion of G U 2 s 1,s 2 (τ) = m 1 >m 2 >0 Ψ s 1 (m 1 τ)ψ s2 (m 2 τ) we use the lemma: Lemma For s 1,..., s l > 1 we have Ψ s1 (m 1 τ)... Ψ sl (m l τ) = m 1 >...>m l >0 ( 2πi) s 1+...+s l (s 1 1)!... (s l 1)! σ s1 1,...,s l 1(n)q n. n>0 Proof: Use the Lipschitz summation formula l-times.

Fourier expansion - Sketch of the proof For the term G RU s 1,s 2 (τ) = m>0 Ψ s 1,s 2 (mτ) we use the following theorem: Theorem We have Ψ s1,...,s l (x) = k h=2 λ s 1,...,s l h Ψ h (x), where k = s 1 +... + s l and the coefficients λ s 1,...,s l of MZV s of weight k h. h are linear combinations Proof: Use the partial fraction decomposition. (Compare to the Result in the talk of O. Bouillot) Combining the Lemma and Theorem also works for the general case l 2.

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 1 +... + s l. Proof: Easy induction using stuffle relation.

Modularity, cusp forms By the double shuffle relations and Eulers formula ζ(2k) = λ π 2k for λ Q one can show: 25 3 5 757 ζ(12) 29 3 5 2 7 691 ζ(6, 3, 3) 29 3 2 5 7 691 ζ(4, 5, 3) + 28 3 2 5 2 691 ζ(7, 5) = some sum of products of even Zeta values by double shuffle relations, sorry the coeficients are at home... = 0 because of Euler s formula. But in the context of multiple Eisenstein series we get: Theorem 25 3 5 757 29 3 2 5 7 691 G 12 29 3 5 2 7 691 G 4,5,3 + 28 3 2 5 2 691 G 6,3,3 G 7,5 = S 12 (SL 2 (Z)).

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 we have 0 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 The Algebra spanned by the multiple Eisenstein series will be studied in another PhD-project in Hamburg.

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) = 2 7 691 3 2 11 σ 1(n) 43 691 2 3 3 2 5 σ 3(n) + 757 2 3 3 3 5 7 11 σ 11(n) 2 22 7 691 3 + 2 5 691 3 σ 3,3 (n) + 2 7 691 σ 6,4 (n) + 24 5 7 691 3 5 691 3 691 2 3 3 7 σ 5(n) σ 2,2 (n) + 22 5 691 σ 3,1 (n) 3 σ 4,2 (n) + 22 7 691 σ 5,1 (n) 3 σ 3,4,2 (n) 24 5 7 691 σ 5,2,2 (n). (τ) = q q i=1(1 n ) 24 = τ(n)q n = q 24q 2 + 252q 3 1472q 4 +... n>0

"Non convergent" multiple Eisenstein series For l = 2 Kaneko, Gangl & Zagier give an extended definition for the Double Eisenstein series G 3,1, G 2,2,.... 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

"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 σ 1 (n)q n n>0 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:

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). Proof: Following a suggestion by Zagier we use the generating function of G 2,...,2 and the theory of Jacobi-like forms.

The multiple Eisenstein series G 2,...,2 Examples: G 2,2 = G 2,2 + 1 4 D1 G2, G 2,2,2 = G 2,2,2 + 1 8 D1 G2,2 + 1 96 D2 G2, G 2,2,2,2 = G 2,2,2,2 + 1 12 D1 G2,2,2 + 1 240 D2 G2,2 + 1 5760 D3 G2. More details on this talk are in my master thesis. It is available on my webpage http://www.math.uni-hamburg.de/home/bachmann/