Y. CHOIE AND Y. TAGUCHI

Transcription

1 A SIMPLE PROOF OF THE MODULAR IDENTITY FOR THETA SERIES Y. CHOIE AND Y. TAGUCHI Abstract. We characterie the function spanned by theta series. As an application we derive a simple proof of the modular identity of the theta series. 1. Introduction The standard proof of the theta modular transformation formula uses the Poisson summation formula. In [1] using the elliptic shift relations a simple proof of the theta modular transformation formula was given for the theta function θ(, ) = r Z eπir +πir. In this paper we show that a similar method can be employed to derive the theta modular transformation formula for more general theta series by taking into account the heat kernel property in addition to the elliptic shift relations. These properties characterie the functions spanned by certain theta functions (Section ). This is applied in Section 3 to derive the modular identity for the theta series.. Theta series Let H C be the complex upper half plane. The following theta series has been studied in connection with Jacobi forms ([], 5); for each m Z (m 1) and each µ (mod m), (.1) θ m,µ (, ) = r Z,r µ (mod m) q r 4m ξ r, q = e πi, ξ = e πi. This series converges uniformly in C and H and hence defines a holomorphic function on H C. One easily checks that θ m,µ (, + 1) = 000 Mathematics Subject Classification: Primary 14K5, Secondary 11F50, 11F03. This work was partially supported by KOSEF R

2 Y. CHOIE AND Y. TAGUCHI θ m,µ (, ) and θ m,µ (, + ) = q m ξ m θ m,µ (, ). Further one can check that L m (θ m,µ (, )) = 0, where L m is the heat operator defined by L m = 8πim. Since these conditions are C-linear, any C-linear combination of the θ m,µ (, ) s satisfies these three conditions. Conversely, we prove the following result: Theorem.1. Let f(, ) be a holomorphic function defined over H C satisfying the following three conditions; (1) (shift) f(, + 1) = f(, ), () (elliptic property) f(, + ) = e πim(+) f(, ), (3) (heat kernel) L m (f(, )) = 0. Then f(, ) belongs to the vector space spanned by {θ m,µ (, ) 0 µ (m 1)} over C. Proof Since f is periodic with respect to, f has the following Fourier expansion in ξ = e πi : f(, ) = r Z φ r ()ξ r. Secondly, since L m (f) = 0, we note that 4m dφ r() d = πir φ r (), r πi so that φ r () = c r e 4m f(, ) = r Z c rq r 4m ξ r. f(, + ) = r Z = c r q r 4m, where c r is a constant depending on r. So, On the other hand, since c r q r 4m (ξq) r = r Z c r q (r+m) 4m q m ξ (r+m) m = q m ξ m r Z c r m q r 4m ξ r, the elliptic property f(, + ) = q m ξ m f(, ) implies that c r = c r+m for every r Z. Now it follows that f(, ) = m 1 µ=0 c µ θ m,µ (, ).

3 A SIMPLE PROOF OF THE MODULAR IDENTITY FOR THETA SERIES 3 Remark.. (1) The case of m = 1 was studied in [1]. In that case, only the first two conditions of Theorem.1 (that is, the shift and elliptic properties) were needed to show that f(, ) is a constant multiple of the theta series θ 1/,0 (, ) = r Z eπir +πir. This is because such an f(, ) has only one ero, +1, modulo the period lattice Z + Z, which is already a strong enough restriction. This doesn t hold anymore in our case. () The referee pointed out that characteriation of theta series θ(, ) = r Z +πi eπir, i.e. the case when m = 1, has been also studied in [3](see Proposition11.1 in page 53). Theorem.3. For each fixed, let f(, ) be a holomorphic function satisfying the two relations (1) and () in Theorem.1. Then (1) f vanishes identically or it has m roots (counting with multiplicity) modulo the lattice Λ() generated by and 1, i.e., Λ() = Z + Z. () The sum of all the roots j modulo Λ() of f satisfies m 1 m( + 1) (mod Λ()). Proof Suppose that f does not vanish identically. The shift and elliptic relations imply that, for each, f(, + 1) f(, + 1) = f(, ) f(, ), f(, + ) f(, + ) = f(, ) f(, ) 4πim. Let F = {c + a + b a, b [0, 1]}, for c C, be a closed fundamental domain of the lattice Λ(). By choosing c properly, we may assume that f has no roots on F. Then the number and the sum of roots of f on F can be computed respectively by the integrals 1 f(, ) πi f(, ) d and 1 πi F F f(, ) d. f(, ) From the shift relations for f one can check that the first integral evaluates to m, so f has exactly m roots on F. The second integral evaluates m 1 m( + 1) (mod Λ()). Now the proof is complete.

4 4 Y. CHOIE AND Y. TAGUCHI Remark.4. The referee pointed out that the related result of Theorem.3 is also contained in Lemma4.1 in [3]. Lemma 3.1. Let F (, ) := 3. The modular identity e πim θm,µ ( 1, ). Then F (, ) belongs to the vector space spanned by {θ m,ν (, ) 0 ν (m 1)}, so there exist complex numbers c 0,µ,..., c m 1,µ such that e πim θm,µ ( 1, ) = c 0,µθ m,0 (, ) + c 1,µ θ m,1 (, ) c m 1,µ θ m,m 1 (, ). Proof We check the three conditions of Theorem.1 for F (, ). This can be done by straightforward calculations, and we omit the details. For example, the condition (1) can be checked as follows by looking at the Fourier expansion: (+1) F (, + 1) = e πim e πir 4m e πir +1 r Z,r µ (mod m) = e πim e πi(r m) 4m e πi(r m) = F (, ). r Z,r µ (mod m) In what follows, we put ζ m = e πi m. Lemma 3.. (1) θ m,µ (, + ) = m q 1 4m ξ 1 θ m,µ+1 (, ) for each µ, 0 µ (m 1). Here we put θ m,m (, ) = θ m,0 (, ). () θ m,µ (, + 1 m ) = ζµ mθ m,µ (, ). Proof (1) We have θ m,µ (, + m ) = q r 4m ξ r q r m r µ = q 1 4m ξ 1 r µ+1 = r µ q (r+1) 4m q 1 4m ξ (r+1) 1 q r 4m ξ r = q 1 4m ξ 1 θ m,µ+1 (, ). () can be check even more easily and the proof is omitted.

5 A SIMPLE PROOF OF THE MODULAR IDENTITY FOR THETA SERIES 5 Finally we can derive the following well-known modular transformation formula: Theorem 3.3. We have θ m,µ ( 1, ) e πim = θ m,0 (, ) + ζ µ mθ m,1 (, ) ζ (m 1)µ m θ m,m 1 (, ) Proof We prove this result by three steps. (Step 1) (3.1) First note that, from Lemma 3.1, we can let θ m,µ ( 1, ) e πim = c 0,µ θ m,0 (, )+c 1,µ θ m,1 (, )+...+c m 1,µ θ m,m 1 (, ). Replacing by + /m in this equation, we have θ m,µ ( 1, + 1 m ) (+ e πim m ) = c 0,µ θ m,0 (, + m )+...+c m 1,µθ m,m 1 (, + m ). Using Lemma 3. and multiplying q 1 4m ξ with the both sides, we have (3.) ζ mθ µ m,µ ( 1, ) e πim = (c m 1,µ θ m,0 (, )+c 0,µ θ m,1 (, )+...+c m,µ θ m,m 1 (, )). By comparing the two equations (3.1) and (3.), we conclude that (3.3) e πim θm,µ ( 1, m 1 ) = c 0,µ ζ λµ m θ m,λ (, ). (Step ) Next, we need to determine the constant c 0,µ in Equation (3.3) for each µ, 0 µ (m 1). We shall use (3.3) twice to deduce that c 0,µ = 1. By replacing by and then by 1 in (3.3), we have Since m 1 1/ θ m,µ (, ) = c 0,µ e πim ζ λµ m θ m,λ ( 1, ). θ m,λ (, ) = θ m,m λ (, )

6 6 Y. CHOIE AND Y. TAGUCHI for each λ, we have m 1 1/ θ m,µ (, ) = c 0,µ e πim ζmθ λµ m,λ ( 1, ). Applying (3.3) again to the θ m,λ ( 1, ) on the right-hand side, we derive θ m,µ (, ) = c m 1 0,µ ( m ν=0 m 1 c 0,λ ζ λ(µ ν) m )θ m,ν (, ). Since the θ m,ν (, ) s are linearly independent over C (as can be seen from the Fourier series expansion), this implies that, for each µ and ν, m 1 c 0,µ c 0,λ ζ λ(µ ν) m = δ µ,ν. m Here δ µ,ν denotes the Kronecker delta function. Taking µ = ν and setting α = m 1 c 0,λ, we have c 0,µ α = 1. In particular, the c m 0,µ is independent of µ. It follows that α = mc 0,0 and c 0,0 = 1. (Step 3) The positivity of c 0,0 can be seen as follows: looking at the Fourier series expansion we note that θ m,µ (it, 0) > 0 for any real number t > 0 and for every µ. So in the functional equation m t θ m,0( i t, 0) = c 0,0(θ m,0 (it, 0) + θ m,1 (it, 0) θ m,m 1 (it, 0)), everything except c 0,0 is positive. This implies that c 0,0 > 0. By (Step ) we conclude that c 0,0 = 1. This ends our proof. Acknowledgement The second-named author thanks Professor YoungJu Choie and POSTECH for their hospitality during his stay at Pohang, Septemper, 003 February, 004. We would like to thank the referee for informing us of the related theta series results in [3]. References [1] W. Couwenberg, A simple proof of the modular identity for theta functions, Proceeding of the AMS, Vol 131, No 11, Page [] M. Eichler, D. Zagier, The theory of Jacobi forms, Progress in Math. 55, Birkhäuser, 1985

7 A SIMPLE PROOF OF THE MODULAR IDENTITY FOR THETA SERIES 7 [3] D. Mumford, Tata Lectures on Theta I, Progress in Math. 8, Birkhäuser, 1983 Department of Mathematics, Pohang University of Science and Technology, Pohang, , Korea address: yjc@postech.ac.kr Department of Mathematics, Kyushu University, Hakoaki, Fukuoka, , Japan address: taguchi@math.kyushu-u.ac.jp