4 LECTURES ON JACOBI FORMS. 1. Plan

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1 4 LECTURES ON JACOBI FORMS YOUNGJU CHOIE Abstract. 1. Plan This lecture series is intended for graduate students or motivated undergraduate students. We introduce a concept of Jacobi forms and try to explain a various connection with other fields, such as quasi-modular forms and Mock modular forms if time allows. More specifically, I am planning to cover: 1 (Lecture 1 ) After introducing a concept of Jacobi forms we look at two expansions of Jacobi forms ; its Taylor series expansions and theta series expansions. The first lecture will focus on theta-expansion and a map between the space of Jacobi forms and the vector valued modular form of half integral weight. (Lecture ) We discuss the correspondence between the space of Jacobi form of integral weight and index 1 and that of modular form of half integral weight. We discuss the correspondence between Jacobi Eisenstein series and Cohen Eisenstein series. 3 (Lecture 3) The Taylor series expansion gives a map between the space Jacobi forms of integral weight and the space of modular forms of integral weight while the theta series expansion gives a map between the space Keynote: Jacobi forms, Theta expansions, Taylor expansions, Quasimodular forms, Mock modular forms, Mock Jacobi forms. This is a lecture note given in the meeting of A Workshop on Modular forms and Related Topics at Center for Advanced Mathematical Sciences and the Department of Mathematics, American University of Beirut from Feb 06-Feb 10, 01. Author thanks to Professor Raji Wissam for his invitation and kind hospitality. 1

2 YOUNGJU CHOIE of Jacobi forms and that of modular forms of half integral weight. In particular we will derive the explicit isomorphism in the case of the space of Jacobi forms of weight k, k integral, J k,1 (Γ(1)). Furthermore, as an application, we discuss Cohen-Rankin operator of ellipic modular form via Jacob form. We give examples. 4 (Lecture 4) Quasi-modular forms generalize classical modular forms and were introduced by Kaneko and Zagier in [?], where they identified quasimodular forms with generating functions counting maps of curves of genus g > 1 to curves of genus 1. More generally hey often appear as generating functions of counting functions in various problems such as Hurwitz enumeration problems, which include not only number theoretic problems but also those in certain areas of applied mathematics Unlike modular forms, derivatives of quasimodular forms are also quasimodular forms. The goal of this talk is to study various properties of quasimodular forms by using their connections with Jacobi-like forms and modular forms. We will briefly mention about the connection with pseudodifferential operators. For more details see [7]. I will briefully introduce a mock modular form and mock Jacobi forms as a holomorphic part of harmonic weak Maass forms. There are vast of recent research results, due to K.Brigmann, K. Ono, J Bruinier, S. Zwegers, etc.. Lecture 1.1. Definitions and Notations. Let us set up the following notations. Let H be the usual complex upper half plane and τ H, z C. Let Γ be any discrete subgroup of SL(, R). In this lecture we take Γ = Γ(1) = SL(, Z) =< ( ) ( ) > First Γ acts on H as a linear fractional transformation: Γ H H

3 4 LECTURES ON JACOBI FORMS 3 aτ + b (τ, γ = ( a c d b )) γτ = cτ + d Fix k, m Z. Consider a holomorphic function φ : H C C such that (1) (modular) For all γ = ( a b c d ) Γ, (cτ + d) k cz πim z e cτ+d φ(γτ, ) = φ(τ, z) cτ + d () (elliptic) For all (λ, µ) Z, e πim(λ τ+λz) φ(τ, z + λτ + µ) = φ(τ, z) (3) Since φ(τ, z +1) = φ(τ, z) and φ(τ, z +1) = φ(τ, z) implies that phi has a Fourier expansion φ(τ, z) = n,r Z c(n, r)qn ξ r, q = e πiτ, ψ = e πiz. The function φ(τ, z) is called a holomorphic Jacobi form (or simply a Jacobi form) of weight k and index m if the coefficients c(n, r) = 0 unless 4mn r 0. It is called a Jacobi cusp form if it satisfies the stronger condition that c(n, r) = 0 unless 4mn r > 0. Moreover, it is called a weak Jacobi form if it satisfies the weaker condition c(n, r) = 0 unless n 0. They form vector spaces and denote those by J cusp k,m (Γ) J k,m(γ) Jk,m weak (Γ) We can summarize the above discussion as the following way: The Jacobi group Γ J is defined as follows: Definition.1. Let Γ J := Γ Z = {[M, (λ, µ)] M Γ, λ, µ Z}. This set Γ J forms a group under a group law [M 1, (λ 1, µ 1 )][M, (λ, µ )] = [M 1 M, (λ, µ)) + (λ, µ )],

4 4 YOUNGJU CHOIE where ( λ µ ) = M t ( λ1 µ 1 ) and is called the Jacobi group. Note that the Jacobi group Γ J acts on H C j as, for each γ = [( a b c d ), (λ, µ)] ΓJ, λ, µ Z j, γ(τ, z) = ( aτ + b cτ + d, z + λτ + µ ). cτ + d Furthermore, for γ = [( a c d b ), (λ, µ)] ΓJ, k 1Z and m 1 Z, let j k,m (γ, (τ, z)) := (cτ + d) k cz πi( e cτ+d +λτλ+λz+µ). Let us define the usual slash operator on a function f : H C C : (f k,m γ)(τ, z) := j k,m (γ, (τ, z))f(γ(τ, z)), γ Γ J, Then one checks the following consistency condition(see also [?], Section I.1 ): (f k,m γ k,m γ )(τ, z) = (f k,m γγ )(τ, z), γ, γ Γ J. A Jacobi form of weight k and index m is a holomorphic function φ : H C C such that (φ k,m γ)(τ, z) = φ(τ, z), γ Γ J satisfying the proper condition of the Fourier expansion... Examples. Example.. For τ H, The Weierstrass -function (τ, z) := 1 z + 1 (z w) 1 w Then = 1 z + m,n Z,(m,n) (0,0) (τ + 1, z) = (τ, z) w Z+τZ,w 0 1 (z (m + nτ)) 1 (m + nτ) ( 1 τ, z τ ) = τ ( 1 z + 1 (z (m+nτ)) 1 (m+nτ) = τ (τ, z) (τ, z + λτ + µ) = (τ, z), (λ, µ) Z So (τ, z) is a (meromorphic) Jacobi form of weight k = and index m = 0.

5 4 LECTURES ON JACOBI FORMS 5 Example.3. Let Q : Z N Z be a positive definite integer valued quadratic form and B be the associated bilinear form. y 0 Z N, the theta series θ y0 (τ, z) := x Z N e πi(q(x)τ+b(x,y0)z) Then for any vector is a Jacobi form (on a congruence subgroup of SL(, Z) of weight k = N index Q(y 0 ) (Hint: to check modular one uses Possion summation formula. To check elliptic it is immediate using relation B(x, y) = Q(x + y) Q(x) Q(y).).3. Theta expansion. This can be derived from elliptic property: take φ(τ, z) = n,r c(n, r)qn ξ r Jk,m weak(γ) Then, for any (λ, µ) Z, and φ(τ, z) = n,r c(n, r)q n ξ r = e πim(λ +λ) φ(τ, z + λτ + µ) = q λm ξ λm n,r c(n, r)q n ξ r = n,r c(n, r)q n+λ m+λr ξ r+λm So, c(n, r) = c(n + λ m + λr, r + λm), i.e. c(n, r) = c(n, r ) whenever r r (mod m) and 4n m r = 4mn r That is to say that c(n, r) depends on the discriminant 4mn r and on the value of r (mod m). So we let Then c(n, r) = c r (4mn r ), c r (N) = c r (N)for r r (mod m) This gives us coefficients c µ (N) for all µ Z/mZ and all integers N satisfying N µ (mod 4m), namely, c µ (N) := c( N + r, r) for r Z, r µ (mod m). 4m

6 6 YOUNGJU CHOIE We further extend definition of c µ (N) by { } c( N+r 4m c µ (N) =, r), for N ν (mod 4m) 0, otherwise So φ(τ, z) = n,r c(n, r)q n ξ r = = = µ (mod m) r Z,r µ (mod m) n>0 µ (mod m) r Z,r µ (mod m) µ (mod m) Defining, for each µ (mod m), N c µ (N)q N 4m N c µ (4mn r )q n ξ r c µ (N)q N+r 4m ξ r µ (mod m) q r 4m ξ r we can write h µ (τ) := N θ m,µ (τ, z) := r µ c µ (N)q N 4m (mod m),r Z q r 4m ξ r Proposition.4. (theta expansion) φ(τ, z) = h µ (τ)θ m,µ (τ, z) µ (mod m) Remark.5. Recall h µ (τ) := N c µ (N)q N 4m where c µ (N) = { c( N+r 4m, r), for N µ (mod 4m) 0, otherwise So N can be N = 4mn r 0 or N = 4mn r > 0 or N + r > 0 according to where φ(τ, z) belongs to J cusp k,m (Γ) or J k,m(γ) or Jk,m weak(γ). }

7 4 LECTURES ON JACOBI FORMS 7 On the other hand, µ πi θ m,µ (τ + 1, z) = e 4m θm,µ (τ, z) and, using Poisson summation formula, θ m,µ ( 1 τ, z τ ) = ( τ mi ) 1 z e πim τ ν (mod m) θ m,ν (τ, z)e πi( νµ) m So the transformation law of h µ (τ) with those of φ(τ, z) and θ m,µ (τ, z) µ πi (.1) h µ (τ + 1) = e 4m hµ (τ) (.) h µ ( 1 τ ) = τ k mτ i Theorem.6. ν (mod m) e πi(νµ) m hν (τ) J k,m (Γ) {(h 0 (τ), h 1 (τ),.., h m 1 (τ)) h µ : H C satisfying (.1) and (.) and proper growth} Next we wish to identify the space of Jacobi forms with that of scalar valued modular forms, that is, usual elliptic modular forms. In the case when m = 1 this can be done easily and for general m it is not easy, however when m is prime this was done in Eichler-Zagier[8]. Note that the similar result exists in the case of harmonic weak Maass forms[6], but proofs are totally different since the space of Harmonic weak Maass forms is not finite dimensional. Hope we can discuss this in the last lecture. 3. Lecture 3.1. In the case of J k,1 (Γ). Let φ(τ, z) J k,1 (Γ), with Γ = Γ(1). Then φ(τ, z) = h µ (τ)θ 1,µ (τ, z) µ (mod ) = h 0 (τ)θ 1,0 (τ, z) + h 1 (τ)θ 1,1 (τ, z)

8 8 YOUNGJU CHOIE Since define h(τ) = c µ (N) = { N 0 (mod 4) So h(τ) = h 0 (4τ) + h 1 (4τ) h µ (τ) = µ c µ (N)q N 4 c( N+r 4, r) if r µ (mod ) 0, otherwise c 0 (N)q N + N 1 (mod 4) } c 1 (N)q N Using the transformation formula given in (.1) and (.) one checks (1) h(τ + 1) = h(τ) () h( τ ) = (4τ + 4τ+1 1)k 1 h(τ) Since Γ 0 (4) = {( a c d b ) Γ c 0 (mod 4)} =< ( ), ( ) > h(τ) M k 1 (Γ 0 (4)) Conversely, if h M + (Γ k 1 0 (4)) (here M + (Γ k 1 0 (4)) is Kohnen plus space) then the reversing the same calculation shows that h 0 an h 1, where h(τ) = h 0 (4τ) + h 1 (4τ), satisfies the proper transformation formula given in (.1) and (.) to see φ(τ, z) = h 0 (τ)θ 1,0 (τ, z) + h 1 (τ)θ 1,1 (τ, z) J k,1 (Γ). This gives an idea of the following result: Theorem 3.1. Let k be an even integer. Then M + k 1 (Γ 0 (4)) J k,1 (Γ(1)) with isomorphism given by a(n)q N N 0,3 (mod 4),N 0 n,r Z,4n r 0 a(4n r )q n ξ r This isomorphism is compatible with the Petersson schlar products, with the action of Hecke operators, and with the structures of M 1 and J,1 an modules over M Here, M = k Z M k and J,1 = k Z J k,1 denote the graded ring of modular forms and the graded ring of Jacobi forms of index 1, respectively.

9 4 LECTURES ON JACOBI FORMS 9 Proof See page 64 in [8] Example 3.. (see [8] Fix an integer k > 3. Consider Jacobi Eisenstein series E k,1 (τ, z) J k,1 (Γ(1)); E k,1 (τ, z) = 1 c,d Z,gcd (c,d)=1 λ Z (cτ + d) k e aτ+b πim(λ cτ+d +λ z cτ+d cz cτ+d ) It is known that this series converges absolutely and uniformly on every compact subset if k > 3 and zero if k odd. Consider its Fourier expansion, for k >, E k,1 (τ, z) = e k,1 (n, r)q n, ξ r Then it turns out that n,r Z,4n r 0 e k,1 (n, r) = H k 1(4n r ) ζ(3 4k) with H k 1 (N) = L N ( k) (called Cohen s function). Here, L D (s) := 0 if D 0, 1 (mod 4) ζ(s 1) if D = 0 L D0 (s) d f µ(d)( D 0 d )d s σ 1 s ( f ) if D 0, 1 (mod 4), D 0 d Where D = D 0 f, f N, D 0 is the discriminant of Q( D), that is, D 0 is D 1 or 0 (mod 4) otherwise 4D 0, and µ : N {0, 1, 1} is the Möbius function defined by 0 if n is not square-free µ(n) = 1 if n is square free and has a odd number of distinct prime factors 1 if n is square free and has an even number of distinct prime factors Further ( ) is the Kronecker symbol defined by, with unique prime factorization of n = e 0 l n i pe i i, distinct p i odd prime, e i Z 0, ( a n ) = (a )e 0 l ( a ) e i p 1 i=1

10 10 YOUNGJU CHOIE where ( a ) is the Legendre symbol. p Theorem says that M + k 1 (Γ 0 (4)) J k (Γ(1)) So the correspondence for E k,1 (τ, z) will be N 0,N 0,3 (mod 4) So we conclude that N 0,N 0,3 (mod 4) 1 ζ(3 4k) H k 1(N)q N 4n r 0 H k 1 (4n r )q n ξ r 1 ζ(3 4k) H k 1(N)q n M + (Γ k 1 0 (4)) 4. Lecture Taylor expansion in z. For simplicity let k 0 (mod ) and m be any positive integer, Take φ(τ, z) = n,r Z,4nm r 0 c(n, r)qn ξ r J k,m (Γ). Since it is holomorphic on H C we have φ(τ, z) = ν 0 χ ν (τ)z ν Since φ(τ, z) = ( 1) k φ(τ, z) and k even we may assume that φ(τ, z) = ν 0 χ ν (τ)z ν From the modular transformation law of φ(τ, z), for anyγ = ( a b c d ) Γ, we get φ( aτ + b cτ + d, z cτ + d ) = (cτ + d)k e πi cz cτ+d φ(τ, z) χ 0 (γτ) = (cτ + d) k χ 0 (τ) χ (γτ) = (cτ + d) k+ (χ (τ) + πim c cτ + d χ 0(τ))

11 4 LECTURES ON JACOBI FORMS 11 (4.1) χ ν (γτ) = (cτ + d) k+ν [χ ν (τ) + (πm c cτ + d )χ ν + 1! (πm c cτ + d ) χ ν ] Note that g 0 (τ) := χ 0 (τ) M k (Γ) (why? g (τ) := χ (τ) πim k χ 0(τ) M k+ (Γ) First note that χ 0(γτ) = (cτ + d) k = (cτ + d) k+d (χ 0(τ) + kc cτ + d χ 0(τ) g (γτ) = χ (γτ) πim k χ 0(γτ) = (cτ + d) k+ (χ (τ) + πim c cτ + d χ 0(τ) πim kc k cτ + d χ 0(τ) πim k χ 0(τ)) = (cτ + d) k+ (χ (τ) πim k 0(τ)) = g χ (τ)) In general, (4.) g ν (τ) := (πi) ν ( n 0 r Z ( 0 µ ν In terms of Fourier coefficients of φ (k + ν µ )! (k + ν )! ( mn) µ r ν µ )c(n, r))q n µ!(ν µ)! dirnu (k + ν )! (πi) (k + ν )! g ν(τ) = ( P ν,k (mn, r )c(n, r))q n n 0 r Z where P ν,k is a homogeneous polynomial of degree ν in r an n with coefficients depending on k and m, the first few being P 0,k = 1 P,k = kr mn P 4,k = (k + 1)(K + )r 4 1(k + 1)r mn + 1m n By inverting the formula in (4.), we have

12 1 YOUNGJU CHOIE, that is χ ν (τ) = 0 µ ν (πim) ν (k + ν µ 1)! g (µ) (ν µ) (k + ν µ 1)!µ! (τ) φ(τ, z) = ν 0 χ ν (τ)z ν = g 0 (τ)+( g (τ) + mg 0(τ) )(πiz) +( g 4(τ) k 4 + mg (τ) (k + ) + m g 0 (τ) k(k + 1) )(πiz) In fact, in the case of the weak Jacobi forms, first m+1 coefficients g 0 (τ),.g ν (τ),.., g m (τ) in z, where g ν (τ) M k+ν (Γ), ν = 0, 1,.., m: So Theorem 4.1. [8] For all k 0 (mod ) and m, Jk,m weak (Γ) M k (Γ) M k+ (Γ) M k+m (Γ) The isomorphism is given by φ(τ, z) (g 0, g,.., g m ) Remark 4.. The similar result holds for odd k One can do it better when the index m = 1 using the explicit generators E 4,1 (τ, z) and E 6,1 (τ, z); Theorem 4.3. Let k even. Then J k,1 (Γ) M k (Γ) S k+ (Γ) The isomorphism is given by φ(τ, z) (g 0 (τ), g (τ)) Here, S k+ (Γ) is the space of cusp forms of weight k + on Γ.

13 4 LECTURES ON JACOBI FORMS Application. Now, we take m = 1, a nontrivial modular form g 0 = g M k (Γ) and g = 0 Then (g, 0, 0, 0, 0) gives so that χ ν (τ) = (πi)ν (k 1)! (k + ν 1)!ν! g(ν) (τ) G(τ, z) = ν 0 and this satisfies, for any γ = ( a b c d ) Γ, ( (πi)ν (k 1)! (k + ν 1)!ν! g(ν) (τ))z ν z (4.3) G(γτ, cτ + d ) = (cτ + d)k e πi cz cτ+d G(τ, z) Similarly, take f M l (Γ) and consider F (τ, z) = ν 0 so that, for any γ = ( a b c d ) Γ, ( (πi)ν (l 1)! (l + ν 1)!ν! f (ν) (τ))z ν z (4.4) F (γτ, cτ + d ) = (cτ + d)l e πi cz cτ+d F (τ, z) Define H(τ, z) := G(τ, z) F (τ, iz) Then from (4.3) and (4.4) we check that, for any γ = ( a b c d ) Γ, (4.5) H(γτ, z cτ + d ) = (cτ + d)k+l H(τ, z) Again by comparing the coefficient of z in H(τ, z) = ν 0 h ν(τ)z ν using the transformation formula in (4.5) h ν (γτ) = (cτ + d) k+l+ν = h ν (τ), γ = ( a b c d ) Γ.

14 14 YOUNGJU CHOIE So, h ν (τ) = (πi) ( ν ( 1) r r+s=ν k + ν 1 r ) ( l + ν 1 s ) f (r) (τ)g (s) (τ) is in M k+l+ν (Γ) Usually, [g, f] (k,l) ν (τ) := ( ( 1) r r+s=ν k + ν 1 r ) ( l + ν 1 s ) f (r) (τ)g (s) (τ) is called the νth Rankin-Cohen s bracket. Example 4.4. Let E k (τ) M k (Γ) be an Eisenstein series. E 4 (τ) = n 1 σ 3 (n)q n = q +... E 6 (τ)1 504 n 1 σ 5 (n)q n = 1 504q +... and Then (τ) = q (1 q n ) 4 = q 4q + 5q S 1 (Γ) n=1 [E 4, E 6 ] 1 = 3456 [E 4, E 4 ] = 4800 See the next separated slide 5. Lecture 4

15 4 LECTURES ON JACOBI FORMS 15 References [1] K.Bringmann and K. Ono, The f(q) mock theta function conjecture and partition ranks. Invent. Math. 165 (006), no., [] K. Bringmann, K. Ono and R. Rhoades, Eulerian series as modular forms. J. Amer. Math. Soc. 1 (008), no. 4, [3] K.Brigmann and K.Ono, Dyson s ranks and Maass forms, ANN. Math. to appear (009) [4] J.Bruinier and K. Ono, Heegner divisors, L-functions, and Maass forms, to appear in Annals of Mathematics(009) [5] Y.Choie, A short note on the full Jacobi group, Proceedings of the AMS, Vol 13, No 9, Spe 1995, [6] B.Cho and Y. Choie, On harmonic Maass forms of half integral weight, to appear in Proc. of AMS, 01 [7] Y. Choie and M. H. Lee, Quasimodular forms, Jacobi-like forms, and Pseudodifferentail operators, Preprint [8] M. Eichler and D. Zagier, The theory of Jacobi forms, Progress in Math., vol. 55, Birkhäuser, Boston, [9] M. Kaneko and Zagier, A generalized Jacobi theta function and quasimodular forms, Progress in Math., vol. 19, Birkhäuser, Boston, 1995, pp [10] D. Zagier, Ramanujan s Mock Theta functions and their applications, Séminaire Bourbaki, 60éme année, , N o 986 [11] S. Zwegers, Mock Theta Functions, PH.D Thesis, Universiteit Utrecht, 00 Department of Mathematics and PMI, Pohang University of Science and Technology, Pohang, , Korea address: yjc@postech.ac.kr