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A CHARACTERIZATION OF THE MODULAR UNITS 7 0 AMANDAFOLSOM 0 2 Abstract. We provide an exact formula for the complex exponents n a in the modular product expansion of the modular units, and deduce a J characterization of the modular units in terms of the growth of these 6 exponents, answering a question of W.Kohnen. 2 ] T 1. Introduction N . h Let Φ(τ) be the modular product defined by t a ∞ m Φ(τ)= κqβ (1 qn)c(n) (1) − [ nY=1 1 where q = e2πiτ, τ , β Q, and c(n),κ C. In [1], Borcherds shows a ∈ H ∈ ∈ v duality between the exponents c(n) of the modular products Φ(τ) and the 6 coefficients a(j) in the Fourier expansions f(τ) = a(k)qm of weakly 7 k≥m 7 holomorphicmodularformsofprescribedweightandPlevel. In[2]theauthors 1 provide an exact formula for the exponents c(n) when Φ(τ) is a weight k 0 7 meromorphicmodularformonΓ(1) := SL2(Z)whosefirstFouriercoefficient 0 is 1, in terms of the unique modular functions j (τ), d N, holomorphic on d / ∈ h the upper half plane with Fourier expansion t H a ∞ m j (τ) = q−d+ a (m)qm d d . : mX=1 v i The exact expansion in [2] in this setting is given by X r c(n) = 2k+n−1 eτordτ(Φ) µ(n/d)jd(τ), (2) a τ∈ΓX(1)\H∗ Xd|n where the numbers e are defined by τ 1/2 τ = i e =  1/3 τ = (1+i√3)/2 τ  1 otherwise , ord (Φ) refers to the orderof Φ at τ, µ(n) is the usual Möbius function, τ 1 AmandaFolsom,Max-Planck-Institutfu¨rMathematik, Bonn,Germany [email protected] 1 2 AMANDAFOLSOM and ∗ denotes the compactification of the quotient space Γ(1) . The mod- \H ular units may be characterized as those meromorphic modular functions (meromorphic modular forms of weight 0) with divisors supported in the cusps. We consider the product expansions (1) of the modular units, and provide an exact formula for the exponents c(n), and from this deduce a characterization of the modular units in terms of the growth of the exponents, answering a question of W. Kohnen. In what follows, for a ring R, we let R∗ denote the multiplicative group of R, we let Z = Z/ℓZ, ZC = Z (Z∗ 0 ), and for an integer n let n ℓ ℓ ℓ \ ℓ ∪{ } denote the equivalence class of n modulo ℓ. We let q := q1/ℓ, ℓ 1, and ℓ ≥ consider modular units of level ℓ, that is, modular units with respect to the principal congruence subgroups Γ(ℓ) := γ Γ(1) γ 1 mod ℓ . { ∈ | ≡ } Theorem 1. Let u(τ) = Φ(τ/ℓ) be a modular unit of level ℓ = pf, p prime, p = 2,3, f N. Then 6 ∈ 1 n c(n) = µ(d) t (3) k n dk Xd|n Xk|n (cid:16) (cid:17) d where n m(n,s)e(ǫ(n)ms/ℓ) (n,p) = 1  sX∈Zℓ       n m(n,s)e(ǫ(n)ms/ℓ) p n,ℓ ∤ n tm(n) =  sX∈Z∗ℓ |   2n m(0,s)cos(ms/ℓ) ℓ n,  |  s∈ZX∗ℓ/{±1}    e(z) = e2πiz, 1 if n j mod ℓ for some j, 1 j (ℓ 1)/2 ǫ(n) = ≡ ≤ ≤ − (4) (cid:26) 1 if n j mod ℓ for some j, (ℓ 1)/2 < j ℓ 1, − ≡ − ≤ − and m = m(r,s) is a set of integers indexed by { a }a∈Tℓ∗ 1 T∗ = a= (a ,a ) = (r/ℓ,s/ℓ) Z2/Z2 ord(a) = ℓ (5) ℓ { 1 2 ∈ ℓ | } A CHARACTERIZATION OF THE MODULAR UNITS 3 satisfying m r2 m s2 m rs 0 mod ℓ (6) a a a ≡ ≡ ≡ X X X a∈T∗ a∈T∗ a∈T∗ ℓ ℓ ℓ and m 0 mod 12 (7) a ≡ X a∈T∗ ℓ where ord(a) = min n Z≥0 n a Z2 . { ∈ | · ∈ } Theorem 2. A meromorphic modular form u(τ) = Φ(τ/ℓ) of weight zero on Γ(ℓ) is a modular unit if and only if c(n) (loglogn)2 u ≪ for all n 1, where the implied constant depends only on u(τ). ≥ Theorem 2 is proved more generally by W. Kohnen in [3] for modular forms of weight k on congruence subgroups Γ Γ(1). Following the state- ⊆ ment of the theorem in [3] (Theorem 1 p. 66) the author remarks “It might be interesting to investigate if [Theorem 1 p. 66 [3]] could also be proved [in the case weight k = 0 using the theory of the modular units].” Indeed we respond to the above remark of Kohnen and apply the theory of the modular units to give an exact formula for the modular exponents c(n) in Theorem 1 (not given in [3]) which allows us to prove Theorem 2. 2. Modular units Much of the theory of the modular units has been developed by Kubert and Lang [4], who provide a description of the modular units in terms of Siegel functions. The Siegel functions are defined using Klein forms t (τ), a a R2, τ and are given by ∈ ∈ H t (τ) = e−ηa(τ)a·(τ,1)/2σ (τ) a a where σ and η are the usual Weierstrass functions. The Siegel functions g (τ) are defined by a g (τ) = t (τ)∆(τ)1/12 a a where ∆(τ) is the discriminant function. The modular units of a particular level ℓ form a group, and a major result of Kubert and Lang provides a description of the modular unit groups of level ℓ = pf, p prime, p = 2,3, 6 f N, in terms of the Siegel functions. ∈ 4 AMANDAFOLSOM Theorem 3. (Kubert, Lang [4].) For prime power ℓ = pf, p = 2,3 prime, 6 f N, the modular units of level ℓ consist of products ∈ gma (8) a Y a∈T∗ ℓ of Siegel functions g , where m is a set of integers satisfying the a { a}a∈Tℓ∗ quadratic relations (6) and (7). WeremarkthatchoosingdifferentrepresentativesinT∗ changestheSiegel ℓ function by a root of unity, so it is understood that the theorem of Kubert and Lang is stated modulo constants. From the q product expansion for − the function σ (τ), one may obtain the q product expansion for the Siegel a − functions ∞ ga(τ) = q21B2(a1)e(a2(a1 1)/2) (1 qn−1+a1e(a2))(1 qn−a1e( a2))(9) − − − − − nY=1 where B (x) = x2 x+1/6 is the second Bernoulli polynomial. We will use 2 − this theory to prove Theorems 1 and 2. 3. Proofs We let ZC be the set of equivalence classes [z] in ZC, defined by ℓ ℓ f ZC = [z] z ZC,[z] = [w] z+w 0mod ℓ ℓ { | ∈ ℓ ⇔ ≡ } f and let T∗ be represented by ℓ 1 1 1 T∗ = Z∗/ 1 Z ZC Z∗ ( 0 Z∗/ 1 ). ℓ ∼ ℓ ℓ {± }× ℓ ∪ ℓ ℓ × ℓ ∪ ℓ { }× ℓ {± } (cid:0) (cid:1) (cid:0) (cid:1) f Let u(τ) be a modular unit of level ℓ, ℓ = pf, p prime, p = 2,3, f N. 6 ∈ Then there exist m satisfying (6) and (7) such that u(τ) has an { a}a∈Tℓ∗ expression as given in (8). By applying the product expansion (9) for the Siegel functions, we see that u(τ) ∞ u(τ) = ξqα [(1 qn−1+a1e(a ))(1 qn−a1e( a ))]ma 2 2 − − − Ya nY=1 A CHARACTERIZATION OF THE MODULAR UNITS 5 where α Q, ξ C. We compute ∈ ∈ log(ξqαu(τ)−1) ℓ = m /m qℓm(n−1+a1)e(ma )+qℓm(n−a1)e( ma ) a ℓ 2 − 2 aX∈T∗nX≥1mX≥1 (cid:16) (cid:17) ℓ m(ℓ(n−1)+r) m(ℓn−r) = m(r,s)/m q e(sm/ℓ)+q e( sm/ℓ) ℓ ℓ − nX≥1 mX≥1 (r,sX)∈Z∗ℓ/{±1}×Zℓ (cid:16) (cid:17) m(ℓ(n−1)+r) m(ℓn−r) + m(r,s)/m q e(sm/ℓ)+q e( sm/ℓ) ℓ ℓ − nX≥1mX≥1(Xr,s)∈ZfC×Z∗ (cid:16) (cid:17) ℓ ℓ + m(0,s)/m qm(n−1)ℓe(ms/ℓ)+qmℓne( ms/ℓ) ℓ ℓ − nX≥1mX≥1 s∈ZX∗/{±1} (cid:16) (cid:17) ℓ = m(n,s)/m e(ǫ(n)ms/ℓ)qmn ℓ mX≥1 Xn≥1 sX∈Zℓ (n,p)=1 + m(n,s)/m e(ǫ(n)ms/ℓ)qmn ℓ mX≥1 Xn≥1 sX∈Z∗ℓ p|n, ℓ∤n + 2m(0,s)/mcos(ms/ℓ)qℓmn ℓ mX≥1nX≥1s∈ZX∗/{±1} ℓ + m(0,s)/m e(ms/ℓ) (10) mX≥1s∈ZX∗/{±1} ℓ where ǫ(n) defined as in (4). We will apply the theta operator Θ with ℓ respect to the parameter q , defined by ℓ df Θ (f)= q . ℓ ℓ dq ℓ Using (10) we find Θ (u) ℓ = αℓ n m(n,s)e(ǫ(n)ms/ℓ)qmn u − ℓ mX≥1 Xn≥1 sX∈Zℓ (n,p)=1 n m(n,s)e(ǫ(n)ms/ℓ)qmn − ℓ mX≥1 Xn≥1 sX∈Z∗ℓ p|n, ℓ∤n 2ℓn m(0,s)cos(ms/ℓ)qℓmn − ℓ mX≥1nX≥1 s∈ZX∗/{±1} ℓ = αℓ t (n)qmn m − mX≥1Xn≥1 = αℓ t (d)qn (11) n/d − nX≥1Xd|n 6 AMANDAFOLSOM where n m(n,s)e(ǫ(n)ms/ℓ) (n,p)= 1  sX∈Zℓ       n m(n,s)e(ǫ(n)ms/ℓ) p n,ℓ ∤n tm(n) =  sX∈Z∗ℓ |   2n m(0,s)cos(ms/ℓ) ℓ n.  |  s∈ZX∗ℓ/{±1}    On the other hand, u(τ) has a modular productexpansion of the form given in (1). We compute log (1 qn)−c(n) = c(n)/m qmn (cid:18) − ℓ (cid:19) ℓ nY≥1 nX≥1mX≥1 so that Θ (u) ℓ = β nc(n)qmn u − ℓ nX≥1mX≥1 = β dc(d)qn. (12) − ℓ nX≥1Xd|n Comparing (11) and (12), we find αℓ = β and t (n/d) = dc(d). d X X d|n d|n By Möbius inversion, we find 1 n c(n) = µ(d) t . k n dk Xd|n Xk|n (cid:16) (cid:17) d This proves Theorem 1. To prove Theorem 2, suppose first u(τ) = Φ(τ/ℓ) is a modular unit of level ℓ = pf, p prime, p = 2,3, f N. Then by Theorem 1, the modular 6 ∈ A CHARACTERIZATION OF THE MODULAR UNITS 7 exponents c(n) are of the form given in (3). Thus, 1 c(n) t (k) n/dk | | ≤ n Xd|n Xk|n (cid:12)(cid:12) (cid:12)(cid:12) d (cid:12) (cid:12) = t (k) n/dk Xd|n Xk|nd (cid:12)(cid:12) (cid:12)(cid:12) (k,p)=1(cid:12) (cid:12) + t (k) n/dk Xd|n Xk|nd (cid:12)(cid:12) (cid:12)(cid:12) p|k, ℓ∤k(cid:12) (cid:12) + t (k) n/dk Xd|n Xk|nd (cid:12)(cid:12) (cid:12)(cid:12) ℓ|k (cid:12) (cid:12) k m(k,s) ≤ | | Xd|n Xk|nd sX∈Zℓ (k,p)=1 + k m(k,s) | | Xd|n Xk|nd sX∈Z∗ℓ p|k, ℓ∤k + 2k m(0,s) | | Xd|n Xk|nd s∈ZX∗ℓ/{±1} ℓ|k ℓ k+φ(ℓ) k+φ(ℓ) /2 2k u u u ≤ M M M Xd|n Xk|nd Xd|n Xk|nd Xd|n Xk|nd (k,p)=1 p|k, ℓ∤k ℓ|k where = max m , so that u a M a∈T∗{| |} ℓ c(n) ℓ k u | | ≤ M Xd|n Xk|n d = ℓ σ (n/d) u 1 M X d|n ℓ 2dloglogd u ≤ M X d|n 2ℓ loglogn σ (n) u 1 ≤ M · 4ℓ (loglogn)2 u ≤ M hence c(n) (loglogn)2. Conversely, suppose Φ(τ/ℓ) is a weight 0 mod- u ≪ ular form of level ℓ = pf, ℓ prime, ℓ = 2,3, f N, with exponents c(n) 6 ∈ satisfying c(n) (loglogn)2. Then for all τ , by (12) we see that u ≪ ∈ H 8 AMANDAFOLSOM Θ (Φ(τ/ℓ)) converges, hence has no zeros or poles in . ℓ H Acknowledgements The author would like to thank O¨zlem Imamoglu for her comments on this paper. References 1. Borcherds, R.E.: Automorphic forms on Os+2,2(R) and infinite products. Invent. Math. 120 No. 1, 161-213 (1995) 2. Bruinier, J.H., Kohnen, W., and Ono, K.: The arithmetic of the values of modular functions and thedivisors of modular forms. Compositio Math. 140, 552-566 (2004) 3. Kohnen, W.: On a certain class of modular functions, Proceedings of the AMS Vol. 133, No. 1, 65-70 (2005) 4. Kubert D.S., Lang, S.: Modular Units. Grundlehren der Mathematischen Wis- senschaften [Fundamental Principles of Mathematical Science], 244. Springer-Verlag, New York-Berlin (1981) 5. Ono, K.: Hecke Operators and the q−expansion of modular forms. Number Theory, CRM Proc. Lecture Notes36, AMS,Providence, 229-235 (2004)