Exact Formula for Jacobi-Eisenstein Series of Square Free Discriminant Lattice Index

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Bull. Korean Math. Soc. 57 (2020), No. 2, pp. 481–488 https://doi.org/10.4134/BKMS.b190342 pISSN: 1015-8634 / eISSN: 2234-3016

EXACT FORMULA FOR JACOBI-EISENSTEIN SERIES OF SQUARE FREE DISCRIMINANT LATTICE INDEX

Ran Xiong

Abstract. In this paper we give an exact formula for the Fourier coef- ﬁcients of the Jacobi-Eisenstein series of square free discriminant lattice index. For a special case the discriminant of lattice is prime we show that the Jacobi-Eisenstein series corresponds to a well known Eisenstein series of modular forms.

1. Introduction Jacobi forms of lattice index, whose theory can be viewed as an extension of the theory of classical Jacobi forms, play important roles in many mathematical fields, such as the theory of orthogonal modular forms, the theory of vertex op- erator algebras. A classical example of Jacobi form is Jacobi-Eisenstein series. In [1], Ajouz defined Jacobi-Eisenstein series of lattice index and studied its basic properties. In [6], Mocanu gave the first formula of Fourier expansion for Jacobi-Eisenstein series of lattice index and, for the trivial case, she showed that the Fourier coefficients in fact are special values of Dirichlet L-functions up to L finite Euler factors. In [7], Woitalla considered this for the lattice 1≤j≤N A1 where A1 is the scalar lattice (Z, (x, y) → 2xy). In this short paper we give an exact formula for Fourier coefficients of Jacobi-Eisenstein series of square free discriminant lattice index. This type lattice has occurred in several problems. This paper is organized as follows: In Section 2 we review some basic facts for Jacobi forms of lattice index briefly. An exact formula for Jacobi-Eisenstein series of square free discriminant index is given and proved in Section 3. Fi- nally, for the case the discriminant of lattice is prime we refer that the Jacobi- Eisenstein series corresponds to a well known Eisenstein series of elliptic modular forms.

Received April 1, 2019; Accepted August 23, 2019. 2010 Mathematics Subject Classiﬁcation. 11F50, 11M36. Key words and phrases. Jacobi forms of lattice index, Fourier coeﬃcients of Jacobi- Eisenstein series. This work is supported by the Fundamental Research Funds for the Central Universities of China (grant no. 22120180508).

c 2020 Korean Mathematical Society 481 482 R. XIONG

2. Jacobi forms of lattice index Throughout this paper, for a complex number t, write e(t) := e2πit, q = e(τ) for τ of the complex upper half plane H. For a prime p, a rational number m, denote νp(m) the p-adic valuation of m. We put Γ = SL2(Z) and, for a positive a b integer N > 1, set the congruence subgroup Γ0(N) := γ = c d ∈ Γ: N | c . We call a pair L = (L, β) a positive definite even lattice if L is a Z-module of finite rank rL, equipped with a positive definite integral quadratic form β. Denote by L] the dual lattice ] L := {y ∈ L ⊗Z Q : β(y, x) ∈ Z for all x ∈ L}. The discriminant form of L is defined as ] DL := L /L, x + L → β(x) + Z . Note that card L]/L = det(L) := det(F ) where F is the Gram matrix corre- ] ] sponding to a Z-basis of L. For r ∈ L , let ωr be its order in L /L. Define the discriminant of L as r ( b L c (−1) 2 det(L), rL even, ∆ := r L b L c (−1) 2 2 det(L), rL odd.

It is well known that ∆L ≡ 0, 1 mod 4. For a ∈ Z, D ∈ Q such that ∆LD ∈ Z, ∆LD denote χL(D, a) := a and in particular put χL(a) := χL(1, a). Let L = (L, β) be a positive deﬁnite even lattice and k an integer. The space Jk,L(Γ) of Jacobi forms of weight k and index L consists of all holomorphic functions φ(τ, z) on H × (L ⊗Z C) which satisfy a b (1) For any A = c d ∈ Γ and h = (x, y) ∈ L × L, the following identities hold: −cβ(z) aτ + b z φ(τ, z) = φ| A (τ, z) := (cτ + d)−ke φ , , k,L cτ + d cτ + d cτ + d φ(τ, z) = φ|k,Lh (τ, z) := e (τβ(x) + β(x, z)) φ (τ, z + xτ + y); (2) The Fourier expansion of φ is of the form

X n r φ(τ, z) = cφ(n, r) q ζβ. ] n∈Z,r∈L β(r)−n≤0 ] r Here and in the following, for z ∈ L ⊗Z C and r ∈ L we write ζβ := e(β(r, z)). By [1, Chapter 2, Proposition 2.4.3], the quality of cφ(n, r) depends on the value of β(r) − n and r mod L. In the following we write D := β(r) − n and Cφ(D, r) = cφ(n, r) as usual. Deﬁne the Jacobi theta series ϑL(τ, z) as

X β(x) x ϑL(τ, z) := q ζβ . x∈L JACOBI-EISENSTEIN SERIES 483

rL For an even integer k > 2 +2, the Jacobi-Eisenstein series Ek,L(τ, z) is deﬁned as X Ek,L(τ, z) := ϑL(τ, z) k,LA, A∈Γ∞\Γ 1 n where Γ∞ = {± [ 0 1 ]: n ∈ Z}. We have that Ek,L(τ, z) ∈ Jk,L(Γ). 3. Fourier expansion of Jacobi-Eisenstein series of square free discriminant index

From now on we assume that ∆L, the discriminant of L is square free. This 2 L apx implies that rL is even and DL ≈ Z/pZ, p . With notations as p|∆L p prime before the main result of this paper is stated as follows: Theorem 3.1. Let L = (L, β) be a positive deﬁnite even lattice of square free rL discriminant. Then for even integer k with k > 2 + 2, the Jacobi-Eisenstein series Ek,L(τ, z) of weight k and index L has the following Fourier expansion X n r Ek,L(τ, z) = ϑL(τ, z) + c(n, r)q ζβ, ] n∈Z,r∈L β(r)

p|∆L p-ωr

Proof. The application Γ∞A 7→ ±(0, 1)A gives a bijection 2 Γ∞\Γ → (c, d) ∈ Z : gcd(c, d) = 1 /{±1} therefore by the deﬁnition of Jacobi-Eisenstein series, e aτ+b β(x) + β(x,z) − cβ(z) X X X cτ+d cτ+d cτ+d E (τ, z) = ϑ (τ, z) + , k,L L (cτ + d)k c≥1 d∈Z x∈L gcd(c,d)=1 a b where a, b are chosen such that c d is in Γ. Using the identity aτ + b β(x, z) cβ(z) c x a β(x) + − = − β z − + β(x) cτ + d cτ + d cτ + d cτ + d c c we obtain e aτ+b β(x) + β(x,z) − cβ(z) X cτ+d cτ+d cτ+d (cτ + d)k x∈L 484 R. XIONG

X 1 X X d x = e(aβ(x))F τ + , z − , ck c c c≥1 d mod c x∈L/cL gcd(c,d)=1 where e − β(z+y) X X τ+h F (τ, z) := . (τ + h)k h∈Z y∈L Applying Possion summation we obtain

X n r F (τ, z) = ω(n, r) q ζβ ] n∈Z,r∈L with Z β(z) ω(n, r) = τ −ke − e (−(nτ + β(r, z)))dτdz ×(L⊗ ) τ R ZR Z Z β(z + rτ) = τ −ke(Dτ)dτ e − dz. =τ=v L⊗ τ ZR r /2 The inner integral equals to τ√L . By [5, Page 19], irL/2 det(L)

r r k− L k k− L −1 (2π) 2 i |D| 2 ω(n, r) = p rL det(L)Γ k − 2 if D < 0 and 0 otherwise. Summing up we ﬁnd

r r k− L k k− L −1 X (2π) 2 i |D| 2 n r Ek,L(τ, z) = ϑL(τ, z) + L(γn,r, k)q ζ , (1) p rL β ] det(L)Γ k − 2 n∈Z,r∈L β(r)

P γn,r (c) where L(γn,r, k) := c≥1 ck with X X γn,r(c) := e (d(β(x) + β(r, x) + n)) d∈Z x∈L/cL gcd(c,d)=1

(For the expression of γn,r(c) we replaced x by dx). Note that γn,r(c) is mul- tiplicative in c. Thus ν Y X γn,r(p ) L(γ , k) = L (γ , k) with L (γ , k) := . n,r p n,r p n,r pkν p ν≥0 By [3, Theorem 7], [3, Theorem 11],

Y 1 X rL 1−k+ 2 Lp(γn,r, k) = rL d χL(d). L(χL, k − ) p ∆L 2 - d |∆LD| JACOBI-EISENSTEIN SERIES 485

Applying the functional equation for L-function we have

r r k− L k k− L −1 (2π) 2 i |D| 2 Y Lp(γn,r, k) pdet(L)Γ k − rL 2 p-∆L (2) r d L e 2(−1) 4 X rL k− 2 −1 = rL d χL(|∆L|D/d). L(χL, 1 − k + 2 ) d |∆LD|

Now we calculate Lp(γn,r, k) for p | ∆L. If prime p does not divide ωr, then

ν X 2 X (3) γn,r(p ) = epν (−dωr D) epν (d(β(x))). d mod pν x∈L/pν L gcd(d,pν )=1

2 We have that the quadratic form β is Zp-isomorphic to the form x1 + ··· + a0 x2 +pa x2 where a0 ∈ such that a a0 = 2rL det(L)/p (see [4, Chapter p rL−1 p rL p Z p p 11, Theorem 2]). Thus we calculate the inner sum X epν (dβ(x)) x∈L/pν L

rL−2   Y X 2 X 0 2 X 2 e ν (dx ) e ν (da x ) e ν (da px ) =  p j  p p rL−1 p p rL j=0 ν ν ν xj mod p xrL−1 mod p xrL mod p s rL−1 s ν(r −1) rL−1 0 L d ap −1 ν+1 apd −1 = p 2 × p 2 pν pν pν pν−1 pν−1

s rL−1s νr +1 0 L apap apd −1 −1 = p 2 pν p pν pν−1

s rL−1s νr +1 L det(L)/p apd −1 −1 = p 2 . pν p pν pν−1

Moreover,

s rL−1s −1 −1 pν pν−1 ν(r −1) ν−1 s L b ν c(r −1) s b ν−1 c −1 −1 2 L −1 −1 2 = p p p p νr −1 s L s b ν c+b ν−1 c −1 −1 −1 2 2 = p p p 486 R. XIONG

rL ! s b ν c+b ν−1 c+1 (−1) 2 −1 −1 2 2 = pν p p

r s L +1 ! −1 (−1) 2 = . p pν

Therefore s νr +1 X L −1 −∆L/p apd e ν (dβ(x)) = p 2 . p p pν p x∈L/pν L Inserting this into (3) and applying [1, Lemma 2.1.14] we ﬁnd

 2 ! r 2 νp(ωr D) ν( L +1) −∆L/p apωr D/p  2 ν ν p ν , p kpD, (4) γn,r(p ) = p p  0, otherwise.

If p | ωr, then the quadratic polynomial β(x) + β(r, x) is Zp-isomorphic to the form x2 + ··· + a0 x2 + pa x2 + 2(r x + ··· + a0 r x + a r x ), 1 p rL−1 p rL 1 1 p rL−1 rL−1 p rL rL ν where r1, . . . , rrL ∈ Z with p - rrL . For ν ≥ 1 and each u mod p , the equation px2 + 2a r x ≡ u mod pν has exact one solution in /pν , which implies rL p rL rL Z Z that X 2 X e ν d(px + 2a r x ) = e ν (du) = 0. p rL p rL rL p ν ν xrL mod p u mod p ν Therefore γn,r(p ) = 0 for ν ≥ 1. By above discussions, for p | ∆L, if p | ωr, then Lp(γn,r, k) = 1 and if p - ωr, then rewrite the ﬁrst item of (4) in terms of the character associated with lattice we obtain

rL 2 νp(|∆LD|)(1−k+ ) νp(∆ D) νp(∆LD) Lp(γn,r, k) = 1 + p 2 χL(ap∆LD/p L , p)χL(−1/p, p ).

Inserting (2) and the formula for Lp(γn,r, k) with p | ∆L into (1) we complete the proof of Theorem 3.1.

4. Corresponding to Eisenstein series of modular forms

In this section we suppose that the discriminant of L is prime p thus DL ≈ 2 apx Z/pZ, x → p . In [1, Chapter 5] Ajouz constructed a surjective map from rL Jk,L(Γ) to M rL (p, χL), the space of modular forms of weight k − on Γ0(p) k− 2 2 with character χL. Explicitly the application

χL(ap) Ω: Jk,L(Γ) → M rL (p, χL) k− 2 JACOBI-EISENSTEIN SERIES 487 given by

X n r X X n cφ(n, r)q ζβ → Cφ(−n/p, x)q ] n≥0 ] n∈Z,r∈L x∈L /L β(r)≤n −n/p≡β(x) mod Z is an isomorphism, where  

χL(ap)  X  r n M rL (p, χL) = f(τ) ∈ Mk− L (p, χL): f(τ) = af (n)q . k− 2 2  χL(−n)6=−χL(ap) 

Let the two Eisenstein series of M rL (p, χL) be k− 2

rL 1,χL 2 X X k− 2 −1 n E rL (τ) = 1 + r d χL(d)q , k− L 2 L(1 − k + , χL) 2 n>0 d|n

rL χL,1 X X k− 2 −1 n E rL (τ) = d χL(n/d)q . k− 2 n>0 d|n

χL(ap) By [2, (14)], the unique Eisenstein series of M rL (p, χL) is k− 2

χL(ap) 1,χL 2χL(−ap) χL,1 E rL (τ) := E rL (τ) + r E rL (τ). k− k− L k− 2 2 L(1 − k + 2 , χL) 2

χL(ap) We now show that Ω(Ek,L(τ, z)) = E rL (τ). It is deduced by the table k− 2 r d L e in [1, Page 92] that (−1) 4 χL(−1) = χL(ap). By this we rewrite the Fourier expansion of Ek,L(τ, z) as

X β(x) x X n r Ek,L(τ, z) = q ζβ + c(n, r)q ζβ x∈L ] n∈Z,r∈L β(r)

The condition D ∈ Z is equivalent to r ∈ L, and for r∈ / L, the equation ] β(x) = −D has exact 1 + χL(−apD) solutions in L /L. Therefore Ω Ek,L(τ, z) 2 = 1 + rL L(1 − k + 2 , χL) 488 R. XIONG   r r X X k− L −1 X k− L −1 n ×  d 2 χL(d) + χL(−ap) d 2 χL(n/d) q n>0,p|n d|n d|n

2χL(−ap) X rL k− 2 −1 n + rL 1 + χL(−apn) d χL(n/d)q L(1 − k + 2 , χL) n>0,p-n

1,χL 2χL(−ap) χL,1 = E rL (τ) + r E rL (τ) k− L k− 2 L(1 − k + 2 , χL) 2 χL(ap) = E rL (τ). k− 2

Acknowledgement. The author would like to thank the referees for their valuable comments on this work.

References [1] A. Ajouz, Hecke operators on Jacobi forms of lattice index and the relation to elliptic modular forms, Ph.D. thesis, University of Siegen, 2015. [2] J. H. Bruinier and M. Bundschuh, On Borcherds products associated with lattices of prime discriminant, Ramanujan J. 7 (2003), no. 1-3, 49–61. https://doi.org/10.1023/ A:1026222507219 [3] J. H. Bruinier and M. Kuss, Eisenstein series attached to lattices and modular forms on orthogonal groups, Manuscripta Math. 106 (2001), no. 4, 443–459. https://doi.org/10. 1007/s229-001-8027-1 [4] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, third edition, Grundlehren der Mathematischen Wissenschaften, 290, Springer-Verlag, New York, 1999. https://doi.org/10.1007/978-1-4757-6568-7 [5] M. Eichler and D. Zagier, The theory of Jacobi forms, Progress in Mathematics, 55, BirkhäuserBoston, Inc., Boston, MA, 1985. https://doi.org/10.1007/978-1-4684- 9162-3 [6] A. Mocanu, Poincaréand Eisenstein series for Jacobi forms of lattice index, arXiv1712. 08174v2, 2018. [7] M. Woitalla, Calculating the Fourier coefficients of Jacobi-Eisenstein series, arXiv1705. 04595v2, 2017.

Ran Xiong School of Mathematical Sciences Tongji University Shanghai. 200092, P. R. China Email address: [email protected]