A Remark on the Behavior of Theta Series of Degree n under Modular Transformations

Transcription

1 IMRN International Mathematics Research Notices 21 No. 7 A Remark on the Behavior of Theta Series of Degree n under Modular Transformations Olav K. Richter 1 Introduction A. Andrianov and G. Maloletkin [3] [4] and Andrianov [1] [2] investigate transformation roerties of theta series corresondingto quadratic forms. Let F be a symmetric integral matrix of rank m with even diagonal entries and let q be the level of F; that is qf 1 is integral and qf 1 has even diagonal entries. Suose that F is of tye k l and let H be a majorant of F; that is HF 1 H = F and t H = H>. For Z in the Siegel uer half-lane H n = {Z M nn C Z = t Z and ImZ > } and for ζ + ζ M mn C with m>n Andrianov and Maloletkin [4] define the theta series θ κλ Z = N M mnz det t NFζ + κ det t NFζ λ e {σ F[N] ReZ+iH[N] ImZ } 1 where κ λ are nonnegative integers e{x} = exπix U[V] = t VUV for any matrices U and V and σw is the trace of the matrix W. Note that θ κλ Z is identically zero if both n and κ + λ are odd. If F is ositive definite and if F[ζ + ] = Andrianov and Maloletkin [3] show that θ κ Fζ + Z =θ κ Z is a Siegel modular form on Γ n q where Γ n = S n Z and Γ n q = { AB CD Γ n C mod q }. Andrianov and Maloletkin [4] assume that Fζ + = Hζ + Fζ = Hζ and F[ζ + ]=H[ζ ]=. They then determine the behavior of θ κλ Z under modular transformations. We determine the behavior of θ κλ Z Received 17 August 2. Communicated by Dennis Hejhal.

2 372 Olav Richter in the more general situation when F[ζ + ] H[ζ ]. In this case θ κλ Z is not modular but can be used to construct a function Θ FHζ+ζ Z X which satisfies a transformation roerty that generalizes the transformation law of Jacobi-like forms introduced by D. Zagier [1] and P. Cohen Y. Manin and Zagier [6]. Furthermore we see that functions that share the transformation roerty of Θ FHζ+ζ Z X rovide a method to construct Siegel modular forms. Define for fixed F H ζ + and ζ a function of Z H n and X M nn C by Θ FHζ+ζ Z X = κ λ κ+λ 2 θ 2κ2λ FHζ +ζ Z det2πix κ det2πix λ. 2 n! 2κ!2λ! Note that Θ FHζ+ζ Z X does not vanish identically. Our main result is the followingtheorem. Theorem 1. Suose Fζ + = Hζ + and Fζ = Hζ. Let M = AB n CD Γ q. Choose T integral and symmetric such that for D = CT + D det D = ± for an odd rime. Then Θ FHζ+ζ AZ + BCZ + D 1 CZ + D 2 X { det F[ζ+ ]CX = φm Z ex detcz + D + det H[ζ ]CX } Θ FHζ+ζ Z X detcz + D 3 where φm Z =χ F M detcz + D k/2 detcz + D l/2 and where χ F M is an eighth root of unity. More recisely φm Z =ε m { } 2 m c m det F k ls e 4 detc m/2{ det [ ic 1 CZ + D ]} k/2{det [ ic 1 CZ + D ]} l/2 where ε = 1 for 1 mod 4 ε = i for 3 mod 4 is the Legendre symbol c is any diagonal element of D 1 C with c =1 and s is the signature of D 1 C. If C is singular then C 1 is interreted as t DC t D 1 where C t D 1 is the Moore- Penrose generalized inverse and the determinants are interreted as the roduct of the nonzero eigenvalues. Furthermore detc 1/2 is ositive and {det[ ic 1 CZ + D]} 1/2 and {det[ic 1 CZ + D]} 1/2 are given by analytic continuation from the rincial value when Z = C 1 D + iy.

3 Theta Series of Degree n The symlectic theta function The symlectic grou { S n R = M = AB CD M M 2n2n R such that J[M] =J = I n } I n where I n is the n n-identity matrix acts on the Siegel uer half-lane H n = { Z M nn C Z = t Z and ImZ > }. The action of M on Z is given by M Z =AZ + BCZ + D 1. Let Γ n = S n Z. The theta subgrou Γ n = { AB CD Γ n A t B C t D have even diagonal entries } acts on the symlectic theta function u Z = v m Z n e { Z[m + v] 2 t mu t vu } 4 where u and v are column vectors in C n. It is well known see e.g. [7] that for A B M = C D u M Z M v in Γ n = χm [ detcz + D ] 1/2 Z u 5 v where χm is an eighth root of unity which deends uon the chosen square root of detcz + D but which is otherwise indeendent of Z u and v. It is also known that χm can be exressed in terms of Gaussian sums. H. Stark [8] determines χm in the imortant secial case when both C and D are nonsingular and when D 1 is integral for some odd rime. R. Styer [9] extends Stark s results and includes the case where C is singular. We use the following theorem of [9] to comute the exlicit theta multilier of Θ FHζ+ζ Z X. Theorem 2 Stark Styer. Suose M = AB n CD is in Γ where D 1 exists. Suose further that D 1 is integral and that det D = ± h for some odd rime. Then

4 374 Olav Richter χm [ detcz + D ] 1/2 2 = ε h h det [ D 1 C ] h e { } s detc 1/2 { [ det ic 1 CZ + D ]} 1/2 4 6 where ε = 1 for 1 mod 4 ε = i for 3 mod 4 is the Legendre symbol D 1 C h is any h h-rincial submatrix of D 1 C which is nonsingular mod and s is the signature the number of ositive eigenvalues minus the number of negative eigenvalues of D 1 C.IfCis singular then C 1 is interreted as t DC t D 1 where C t D 1 is the Moore-Penrose generalized inverse see [5] and the determinants are interreted as the roduct of the nonzero eigenvalues. Furthermore detc 1/2 is ositive and {det[ ic 1 CZ + D]} 1/2 is given by analytic continuation from the rincial value when Z = C 1 D + iy. 3 Proof of Theorem 1 Now we turn to the roof of Theorem 1. Let F be a symmetric integral matrix of rank m with even diagonal entries and let q be the level of F; that is qf 1 is integral and qf 1 has even diagonal entries. Suose that F is of tye k l and let H be a majorant of F; that is HF 1 H = F and t H = H>. Andrianov and Maloletkin [4] regard θ FH Z =θ Z as a symlectic theta function and then aly 5. Let U V =u ij V denote the Kronecker roduct of two matrices U and V. ForZ = X + iy H n set Z = X F + iy H H nm. One verifies that θ FH Z = Z. Furthermore if M = AB n CD Γ q then M = Ã B = C D A Im B F C F 1 D I m Γ nm and Andrianov and Maloletkin [4] show that θ FH M Z =χ M det C Z + D 1/2 Z = φm ZθFH Z 7 where φm Z =χ F M detcz + D k/2 detcz + D l/2 and χ F M is an eighth root of unity. Unfortunately Andrianov and Maloletkin [4] can determine χ F M only when m is even. In the secial case where F is ositive definite F = H Styer [9] uses Theorem 2 to determine χ F M for all m. It is easy to see that Styer s method can also be alied to determine χ F M for all m even if F is indefinite. Styer [9] shows that if AB CD Γ n then there exists a symmetric integral matrix T such that detct + D =± for some arbitrarily large rime. As in Styer [9] we set Z = Z T M = M I n T I n and we observe that θ FH M Z =χ M det C Z + D 1/2 θfh Z. 8

5 Theta Series of Degree n 375 We aly Theorem 2 and find that { φm Z =ε m 2 m c m } det F k ls e 4 detc m/2 { [ det ic 1 CZ + D ]} k/2{ [ det ic 1 CZ + D ]} 9 l/2 where ε = 1 for 1 mod 4 ε = i for 3 mod 4 is the Legendre symbol c is any diagonal element of D 1 C with c =1 and s is the signature of D 1 C. If C is singular then C 1 is interreted as t DC t D 1 where C t D 1 is the Moore- Penrose generalized inverse and the determinants are interreted as the roduct of the nonzero eigenvalues. Furthermore detc 1/2 is ositive and {det[ ic 1 CZ + D]} 1/2 and {det[ic 1 CZ + D]} 1/2 are given by analytic continuation from the rincial value when Z = C 1 D + iy. Note that if m is even and if det D = ± formula 9 matches the result from Andrianov and Maloletkin [4] and we have 1 φm Z =sgndet D k l/2 m/2 det F detcz + D k l/2 detcz + D l. det D 1 To rove Theorem 1 we roceed as in Andrianov and Maloletkin [4] and we differentiate 7. For this urose we state [3 Lemma 3] in a slightly more general form. Lemma 1. Let 1 n<m. Let P R M nn C t P = P and Q η M mn C. Denote by ξ =ξ αβ an m n-variable matrix and by = / ξ αβ the corresondingmatrix of differentiation oerators. Set L = L η = det t η. Then for ν 1 we have L ν e { σ P t ξξ + 2 t Qξ + R } = f PQην ξ e { σ P t ξξ + 2 t Qξ + R } 11 where f PQην ξ = [ν/2] j= j n! det 2πiP t ηη j det 2πi P t ξ + t Q η ν 2j ν!. 2 j! ν 2j! Remark. If in addition t ηη = then f PQην ξ =det2πip t ξ + t Qη ν and our lemma simlifies to [3 Lemma 3]. Proof. Andrianov and Maloletkin [3] oint out that

6 376 Olav Richter L e { σ P t ξξ + 2 t Qξ + R } = det 2πi P t ξ + t Q η e { σ P t ξξ + 2 t Qξ + R } and that for indices β γ µ and ι Hence m η αβ ξ αγ α=1 m ξp + Q xµ η xι = P γµ t ηη βι. x=1 L det 2πi P t ξ + t Q η = det 2πiP L det t ξη. The oerator L can be written as follows: L = n sgnσ σ S n γ j=1 η γjj ξ γjσj where the second summation is over all mas γ from {1...n} to {1...m}. We find that and therefore L det t ξη = n sgnσ sgnσ τ η γjj η γ τjj σ S n γ τ S n j=1 = n! n sgnτ η γjj η γjτj τ S n γ j=1 = n! det t ηη L det 2πiP t ξ + t Qη = n! det 2πiP t ηη. Induction on ν then gives the desired result. Now we are ready to determine the behavior of θ κλ Z under modular transformations. Theorem 3. Suose Fζ + = Hζ + and Fζ = Hζ. Let M = AB n CD Γ q. Then θ κλ M Z [κ/2] = φm Z j= [λ/2] g= detcz + Dκ j κ 2j! where φm Z is given by 9. κ!λ! n! 2 F[ζ+ ]C j+g det 2πi j! j detcz + D λ g θ κ 2jλ 2g FHζ λ 2g! +ζ Z H[ζ ]C det 2πi g! g 12

7 Theta Series of Degree n 377 Remark. In the secial case when F[ζ + ]=H[ζ ]= our theorem reduces to [4 Theorem 2]. Proof. Let ξ = ξ + ξ Mmn C where ξ + M kn C and ξ M ln C and let + and be the corresondingmatrices of differential oerators. Set η + = S 1 ζ + and η = S 1 ζ where S GL n R such that F[S] =E kl and H[S] =I m and where E kl = Ik Il. Note that η+ = η + and η = η for η+ M kn and η M ln since F H[S] η + = 2 Il η+ = and F + H[S] η + = 2 I k η =. Thus [4 4.4] can be stated as follows: { e σ P t + ξ + ξ t } Q + ξ + + R + e {σ P t ξ ξ + 2 t } Q ξ + R where N = φm Z { e σ P + t ξ + ξ t Q +ξ + + R } + N { e σ P t ξ ξ + 2 t Q ξ + R } P + = ZCZ + D 1 D P = Z CZ + D 1 D P + = Z P = Z Q + = t ZCZ + D 1t N t S 1 I k Q = t ZCZ + D 1t N t S 1 Il Q + = t Z t N t S 1 I k Q = t Z t N t S 1 Il R + = Re M Z F[N] R = i Im M Z H[N] R + = ReZF[N] R = i ImZH[N] and where φm Z is defined in 7 and given exlicitly in 9. We aly Lemma 1 with L = L + L = det t η + + κ det t η λ and we set ξ + = ξ =. Then 12 follows from observingthat t η + η + = t η + η + = F[ζ + ] and t η η = t η η = H[ζ ]. The transformation formula 3 is an immediate consequence of Theorem 3: Θ FHζ+ζ M Z CZ + D 2 X = φm Z κ j det F[ζ+ ]CX j κ 2 det2πix κ j detcz + D n! κ j= 2κ 2j! j! det λ λ g 2 det 2πiX H[ζ ]CX g λ g det CZ + D θ 2κ 2j2λ 2g n! λ g= 2λ 2g! FHζ g! +ζ Z { det F[ζ+ ]CX = φm Z ex detcz + D + det H[ζ ]CX } Θ FHζ+ζ Z X. detcz + D

8 378 Olav Richter 4 Conclusion It would be very interestingto find other examles that satisfy the transformation roerty 3. We now exlain how functions that satisfy a secial case of 3 allow us to construct Siegel modular forms. For j = 1 2 let f j Z X be holomorhic functions on H n M nn C of the form f j Z X = ν f j ν Z det2πix ν. 13 Suose that for M = AB CD Γ n { } f j M Z CZ + D 2 X = χ j M detcz + D k j detcx ex detcz + D f j Z X 14 for some nonnegative integers k j and characters χ j. Notice that when n = 1 f j are Jacobi-like forms in the sense of Zagier [1] and Cohen Manin and Zagier [6]. Then { } 1 FZ X =f 1 Z Xf 2 Z e X = F ν Z det2πix ν n ν where F ν Z is a Siegel modular form of weight k 1 + k 2 + 2ν and character χ 1 χ 2. Hence as an alication of Theorem 3 we have the followingcorollary. Corollary 1. Suose F is ositive definite F = H such that detf[ζ + ] = 1. Set Θ Fζ+ Z X = κ 2 θ 2κ FHζ +ζ Z det2πix κ 15 n! 2κ! κ and { } 1 FZ X =Θ Fζ+ Z XΘ Fζ+ Z e X = F ν Z det2πix ν. 16 n ν Then F ν Z = ν 2 ν n! κ= 1 κ θ2κ FHζ +ζ Z 2κ! θ 2ν 2κ Z 2ν 2κ! 17 is a Siegel modular form on Γ n q of weight m + 2ν and character χ F M 2 with χ F M as in Theorem 1.

9 Theta Series of Degree n 379 References [1] A. N. Andrianov Sherical theta-series Math. USSR-Sb [2] Symmetries of harmonic theta-functions of integral quadratic forms Russian Math. Surveys [3] A. N. Andrianov and G. N. Maloletkin Behavior of theta series of degree N under modular substitutions Math. USSR-Izvestija [4] Behavior of theta series of genus n of indefinite quadratic forms under modular substitutions Proc. Steklov. Inst. Math [5] A. Ben-Israel and T. N. E. Greville Generalized Inverses: Theory and Alications Wiley- Interscience New York [6] P. B. CohenY. Manin and D. Zagier Automorhic Pseudodifferential Oerators Progr. Nonlinear Differential Equations Al. 26 Birkhäuser Boston [7] M. Eichler Introduction to the Theory of Algebraic Numbers and Functions Pure Al. Math. 23 Academic Press New York [8] H. M. Stark On the transformation formula for the symlectic theta function and alications J. Fac. Sci. Univ. Tokyo Sect. 1A Math [9] R. Styer Prime determinant matrices and the symlectic theta function Amer. J. Math [1] D. Zagier Modular forms and differential oerators Proc. Indian Acad. Sci. Math. Sci Deartment of Mathematics University of California Santa Cruz California 9564 USA; richter@math.ucsc.edu