Anatoli Andrianov (St. Petersburg) and Fedor Andrianov (Los Angeles) Introduction

Transcription

1 ACTION OF HECKE OPERATORS ON PRODUCTS OF IGUSA THETA CONSTANTS WITH RATIONAL CHARACTERISTICS Anatoli Andrianov St. Petersburg) and Fedor Andrianov Los Angeles) Abstract. We derive explicit formulas for the images of products of Igusa theta constants with rational characteristics under the action of regular Hecke operators. In particular, we prove that when the class number of the sum of 2k squares is one, images of products of 2k Igusa theta constants with rational characteristics under the action of regular Hecke operators are, in general, linear combinations of similar products with explicitly given coefficients. Introduction In 1980 H.Yoshida [Yo80)], p.243) proposed a two-dimensional analog of the famous Shimura-Taniyama relation between Hasse zeta functions of elliptic curves over the field of rational numbers and Hecke zeta functions of elliptic modular forms. In confirmation of the Yoshida conjecture, R.Salvati-Manni and J.Top have considered in [SM-T93)] a number of products of four Igusa theta constants with rational characteristics which in part hypothetically) are eigenfunctions of all regular Hecke operators with Andrianov zeta function coinciding up to a finite number of Euler factors) with Hasse zeta function of appropriate two-dimensional Abelian variety. In this paper we begin to study transformation properties of products of even number of Igusa theta constants with rational characteristics considered as Siegel modular forms. More specifically, we investigate their transformations under modular substitutions and under the action of regular Hecke operators. For this we interpret products of theta constants as multiple theta functions of sums of squares 2000 Mathematics Subject Classification. Primary 11F27, 11F46, 11F60, 11F66. Key words and phrases. Igusa theta constants, Hecke operators, Siegel modular forms, theta functions of integral quadratic forms, zeta functions of modular forms. The first author was supported in part by Russian Fund of Fundamental Research Grant #

2 2 A.N. ANDRIANOV AND F.A. ANDRIANOV and then apply explicit transformation formulas for the action of modular substitutions and regular Hecke operators on theta functions of integral quadratic forms obtained in [An95)] and [An96)], respectively). In particular, if the class number of the sum of 2k squares is equal to one, we obtain explicit transformation formulas expressing images of the products of 2k theta constants with rational characteristics under the action of regular Hecke operators in the form of linear combinations of similar products Theorem 4.1). In forthcoming works we hope to extend our study in order to investigate the question of construction of common eigenfunctions for all regular Hecke operators on spaces spanned by products of theta constants with rational characteristics and, posibly, compute corresponding zeta functions. Notation. We reserve the letters N, Z, Q, R, and C for the set of positive rational integers, the ring of rational integers, the field of rational numbers, the field of real numbers, and the field of complex numbers, respectively. A m n is the set of all m n-matrices with entries in a set A, A n = A n 1, and A n = A 1 n. If M is a matrix, t M always denotes the transpose of M. If entries of M belong to C, then M is the matrix with complex conjugate entries. For a symmetric matrix Q we write Q[M] = t MQM if the product on the right is defined. 1 g is the unit matrix of order g, and 0.1) J g = 0 1g 1 g 0 ). We denote by E m = { Q = q αβ ) Z m m t Q = Q, q 11, q 22,..., q mm 2Z the set of all even matrices of order m, i.e. the set of matrices of integral quadratic forms in m variables qx) = 1 t 2 XQX, where t X = x 1,..., x m ). We recall that the level q of an invertible matrix Q E m and that of the corresponding form) is the least positive integer satisfying qq 1 E m. 1. Products of theta constants as theta functions for sums of squares Let m = m, m ) C 2g with m, m C g be a complex 2g-row vector, and Z belongs to the upper half-plane of genus g: H g = { Z = X + iy C g g t Z = Z, Y > 0.

3 PRODUCTS OF THETA CONSTANTS 3 The Igusa theta constant with characteristic m is defined by θ m Z) = n Z g exp πi { n + m )Z t n + m ) + 2n + m ) t m ). If product AB of two matrices A and B is defined and is a square matrix, then clearly the product BA is also defined and is a square matrix with the same trace as that of AB. It follows that the theta constant can be rewritten in the form 1.1) θ m Z) = n Z g e { Z t n + m )n + m ) + 2 t m n + m ), where for a square matrix A we set 1.2) e{a = expπi σa)), and σa) denotes the trace of A. Hence, it is easy to see that the product of r theta constants with characteristics m 1,..., m r can be written in the form 1.3) r θ mj Z) j=1 = n 1,...,n r Z g e { r Z t n j + m j)n j + m j) + 2 t m j n j + m j) ) j=1 = e { Z t N + M )N + M ) + 2 t M N + M ), N Z r g where we set n 1 N =., M = n r m 1. m r, and M = A product of the form 1.3) will be called the theta product of genus g with the characteristic matrix M = M, M ). The following criterion for the vanishing of theta products is a direct consequence of the Igusa result on vanishing of theta constants [Ig72)], Theorem 1, p. 174). m 1. m r.

4 4 A.N. ANDRIANOV AND F.A. ANDRIANOV Lemma 1.1. The theta product with the characteristic matrix M = M, M ) C r 2g is identically equal to 0, if and only if there is a row m j = m j, m j ) of M satisfying m j Z 2g, 2m j t m j / Z. The main purpose of this paper is to study the action of Hecke operators on the products of theta constants with rational characteristics. The action of Hecke operators on theta functions of integral positive definite quadratic forms of the form 1.4) ΘV, Z; Q) = e { Z Q[N V 2 ] + 2 tv 1 QN t V 1 QV 2, N Z r g where V = V 1, V 2 ) with V 1, V 2 C r g is the characteristic matrix of the theta function, Z H g, and where Q is an even positive definite matrix of order r, i.e. the matrix of an integral positive definite quadratic form, was considered in [An96)]. The series 1.5) converges absolutely and uniformly on compact subsets of the complex space C r 2g H g and therefore determines a complex-analytic function on the space. The formula 1.3) implies that the product of theta constants can be expressed through the theta function 1.4) of the quadratic form q r X) = x x 2 r with the matrix Q r = 2 1 r by the formula r e{ t M M θ mj Z) = { 1 e 2 Z2 1 r)[n + M ] + 2 t 1 2 M )2 1 r )N + t 1 2 M )2 1 r )M N Z r g = Θ 1 2 M, M ), 1 ) 2 Z; Q r, which we shall use in the form r 1.5) θz, M) = θ mj Z) = δm) Θ V M), 1 ) 2 Z; Q r, j=1 where, for M = M, M ), we set j=1 1.6) δm) = e{ t M M and ) V M) = V 1 M), V 2 M)) = 2 M, M 1g 0 = M J g g The following property of function δm) on rational M is an easy consequence of definition: 1.8) δm + M 1 ) = δm) if dm Z r 2g and M 1 2d Z r 2g with d N.

5 PRODUCTS OF THETA CONSTANTS 5 2. Modular transformations of theta functions and theta products Here we shall recall basic transformation formulas of the theta functions 1.4) of integral positive definite quadratic forms and then specialize the formulas to the case of sums of squares. First of all, note that for any matrix U GL r Z) we have UZ r g = Z r g and so, by replacing N by UN in 1.5), we get the identity 2.1) ΘV, Z; Q) = e { Z Q[U][N U 1 V 2 ] + 2 tu 1 V 1 )Q[U]N t U 1 V 1 )Q[U]U 1 V 2 N Z r g = ΘU 1 V, Z, Q[U]). In the case Q = Q r it implies the formula 2.2) θz, UM) = θz, M) for every U E r, where 2.3) E r = EQ r ) = { U GL r Z) t UU = 1 r is the group of units of Q r. In fact, by 1.5) and 2.1), we have θz, UM) = e{ t UM )UM Θ V UM), 1 ) 2 Z; Q r = e{ t M M Θ V M), 1 ) 2 Z; Q r[u 1 ] = θz, M). Next we mention that the function ΘV, Z; Q) is quasiperiodic with respect to the lattice Q 1 Z r g, Z r g) C r 2g. More exactly, for every T 1 Q 1 Z r g and T 2 Z r g, we have ΘV 1 + T 1, V 2 + T 2 ), Z; Q) = e { Z Q[N V 2 T 2 ] + 2 tv 1 + T 1 )QN T 2 + T 2 ) t V 1 + T 1 )QV 2 + T 2 ) N Z r g = e { t V 1 QT 2 t T 1 QV 2 + t T 1 QT 2 ΘV1, V 2 ), Z; Q), since the matrix N = N T 2 ranges over the set Z r g along with N, and the matrices t T 1 QN are integral for all N Z r g. In particular, using the above identity with

6 6 A.N. ANDRIANOV AND F.A. ANDRIANOV Q = Q r, and the formula 1.7), we conclude that for each integral r 2g-matrix S = S, S ) we have 2.4) θz, M + S) = e { t M + S )M + S ) Θ V M) + V S), 12 ) Z, Q = e { t M + S )M + S ) t M S + t S M t S S Θ V M), 1 ) 2 Z, Q r = e { 2 t S M θz, M). The following formulas for integral symplectic transformations of theta functions with respect to the variable Z are particular cases of formulas proved in [An95)], Theorems 3.1 and 4.3. Let Q be an even positive definite matrix of even order r = 2k and q the level of Q. Then the theta function ΘV, Z; Q) of genus g of the matrix Q satisfies functional equation 2.5) detcz + D) k Θ V, AZ + B)CZ + D) 1 ; Q ) A B for every matrix M = of group C D 2.6) Γ g A B 0 {M q) = = C D Z 2g 2g = χ Q M) ΘV tm 1, Z; Q) t MJ g M = J g, C 0 mod q), where J g is the matrix 0.1), with character χ of the group defined by conditions A χ Q C )) B = D { 1 if q = 1 χ Q det D) if q > 1, where χ Q is the real Dirichlet character modulo q satisfying χ Q 1) = 1) k, and 1) k ) det Q χ Q p) = p the Legendre symbol) for each odd prime number p not dividing q. Specialization of these formulas to the case of sums of squares allows us to prove the following lemma.

7 PRODUCTS OF THETA CONSTANTS 7 Lemma 2.1. The product 1.5) of even number r = 2k of theta constants satisfies the functional equation 2.7) detcz + D) k θaz + B)CZ + D) 1, M) A B for every matrix M = of group C D 2.8) Γ g A B 00 {M 2) = = C D where = χ r M)δM)δMM) θz, MM), ) Γ g = Γ g0 1) B C 0 mod 2), 2.9) χ r M) = χ Qr M) = χ k 4det D) with nontrivial Dirichlet character χ 4 modulo 4, and where δ is the function 1.6). Proof. By 1.5) we can write detcz + D) k θaz + B)CZ + D) 1, M) = δm) det 2C 1 ) k Θ V 2 Z) + D M), A 12 Z) + 12 ) B)2C12 Z) + D) 1 ; Q r. Since, clearly, M A B = A 1 C D = 2 B ) 2C D = ωmω 1 = Γ g 0 4) with ω = 1g 0, g and the level of Q r is equal to 4, then by the formula 2.5) for the matrix M, the last expression is equal to δm)χ Qr M ) Θ V M) t M ) 1, 1 ) 2 Z; Q r. By the relations 1.7) and by t M JM = J in the form J t M ) 1 J 1 = M, where J = J g is the matrix 0.1), we have V M) t M ) 1 = Mω 1 J t M ) 1 J 1 ωω 1 J = Mω 1 M ωω 1 J = V MM). Hence, detcz + D) k θaz + B)CZ + D) 1, M)

8 8 A.N. ANDRIANOV AND F.A. ANDRIANOV = δm)χ Qr det D)δMM)δMM) Θ V MM), 1 ) 2 Z; Q r, which together with the formula 1.5) for MM in place of M and with above formulas for χ Qr prove the lemma. Now we shall recall some definitions. The general real positive symplectic group of genus g consists of all real symplectic matrices of order 2g with positive multipliers: { 2.10) G g = GSp + g R) = M R 2g 2g t MJ g M = µm)j g, µm) > 0, where J g is the matrix 0.1). It is a real Lie group which acts as a group of analytic automorphisms on the gg + 1)/2-dimensional open complex variety H g by the rule G g A B M = : Z M Z = AZ + B)CZ + D) 1 Z H g ). C D By acting on the upper half-plane H g, the general symplectic group also operates on complex-valued functions F on H g by Petersson operators of integral weights k, 2.11) G g A B M = : F F C D k M = detcz + D) k F M Z ). The Petersson operators satisfy the rules 2.12) F k MM = F k M) k M M, M G g ). Let Ω be a subgroup of G g commensurable with the modular group of genus g, Γ g = Γ g 0 1), χ a character of Ω, that is a multiplicative homomorphism of Ω into nonzero complex numbers with kernel of finite index in Ω, and let k be an integral number. A complex-valued function F on H is called Siegel) modular form of weight k and character χ for the group Ω, if the following conditions are fulfilled: i) F is a holomorphic function in gg + 1)/2 complex variables on H g ; ii) For every matrix M Ω, the function F satisfies the functional equation 2.13) F k M = χm)f, where k is the Petersson operator of weight k; iii) If g = 1, then every function F k M with M Γ 1 is bounded on each subset of H 1 of the form H 1 ε = {x + iy H 1 y ε with ε > 0. The set M k Ω, χ) of all modular forms of weight k and character χ for the group Ω is clearly a linear space over the field C. Each space M k Ω, χ) has finite dimension over C.

9 PRODUCTS OF THETA CONSTANTS 9 Theorem 2.2. The product of even number r = 2k of theta constants with rational characteristic matrix M 1 d Zr 2g, where d N, satisfies the functional equation 2.14) θz, M) k M = χ M M)θZ, M) for every M Γ g d) Γ g 00 2), where Γ g d) = { M Z 2g 2g t MJ g M = J g, M 1 2g mod d), is the principal congruence subgroup of level d of the modular group Γ g = Γ g 1), Γ g 00 2) is the group 2.8), and where 2.15) χ M M) = χ r M)e{SM) t MM with B ) SM) = t B A t B t D A t D D 1 g C t B C t D ) ) A B M = ; C D the matrix SM) is symmetric. If the product θz, M) is not identically zero, then the function χ M : M χ M M) is a character of the group Γ g d) Γ g 00 2) coinciding on the subgroup Γ g 2d 2 ) with the character χ r ; the theta product is a modular form of weight k and character χ M for the group Γ g d) Γ g 00 2). Proof. Since M Γ g 00 2), by the definition of the Petersson operators of weight k, we can rewrite the functional equation 2.7) in the form θz, M) k M = χ r M)δM)δMM) θz, MM). A B Since M = Γ C D g d) and the matrix dm is integral, we conclude that the matrix T = T, T ) = MM M = MM 1 2g ) = M, M A 1g B ) C D 1 g = M A 1 g ) + M C, M B + M D 1 g )), where M = M, M ), is also integral. Therefore, by 2.4),we get θz, MM) = θz, M + T ) = e{2 t T M θz, M). The formula 2.14) follows with χ M M) = χ r M)δM)δMM)e{2 t M B + M D 1 g ))M.

10 10 A.N. ANDRIANOV AND F.A. ANDRIANOV In order to prove the formula 2.15), it is sufficient to show that 2.17) δm)δmm)e{2 t M B + M D 1 g ))M = e{sm) t MM A B with the matrix SM) defined by 2.16). Let us set M = 1 2g + C D, then, by the 1.6), we have ) δmm) = δ M, M 1g + A ) B C 1 g + D whence δm)δmm)e{2 t M B + M D 1 g ))M = e{ t M + M B + M D ))M + M A + M C )). = e{ t M M t M + M B + M D ))M + M A + M C )) e{2 t M B + M D )M = e{ t M B + M D )M t M M A + M C ) e{ t M B + M D )M A + M C ). By standard properties of traces of square matrices, the last expression can be rewritten in the form { = e{ t M M B + M D ) t M M A + M C )e t B A M D )M C = e { = e t B A MM D C { B 1g A t A 1g MM D 1 g C C t B, t D ) t A MM { B + = e t B A t B t D A t D D 1 g C t B C t D C ) t B, t D 1 g ) t MM t MM, which proves the formulas 2.15) 2.16) and 2.17). In order to prove that the matrix SM) is symmetric, ) we note that the g g blocks A, B, C, and D of every A B matrix M = complying with C D t MJ g M = J g satisfy the relations 2.18) A t B = B t A, C t D = D t C, and A t D B t C = 1 g

11 PRODUCTS OF THETA CONSTANTS 11 see, e.g.,[an87)], p. 6). It follows that the matrix SM) can be written in the form B ) SM) = t B A t B t D A t D D D t A C t D and is symmetric. If θz, M) is not identically zero, it follows from 2.12) and 2.14) that the function χ M : M χ M M) is multiplicative. If M Γ g 2d 2 ), then the matrix SM) is, clearly, divisible by 2d 2. Therefore, the matrix SM) t MM is integral and divisible by 2, and so e{sm) t MM = 1. The rest for g > 1 follows from the definition of modular forms; condition iii) for g = 1 follows from [Ig72)], Corollary, p Action of Hecke operators on theta functions Here we shall recall basic notions and facts related to Hecke operators and specialize the transformation formulas of general harmonic theta functions under the action of regular Hecke operators obtained in [An96)] to the case of theta functions 1.4). For details and proofs see the book [An87)], Chapters 3 and 4, and [An96)], 5. Let be a multiplicative semigroup and Ω a subgroup of such that every double coset ΩMΩ of modulo Ω is a finite union of left cosets ΩM. Let us consider the vector space over a field, say, the field C of complex numbers, consisting of all formal finite linear combinations with coefficients in C of symbols ΩM) with M which are in one-to-one correspondence with left cosets ΩM of the set modulo Ω. The group Ω naturally acts on this space by right multiplication defined on the symbols ΩM) by We denote by ΩM)ω = ΩMω), HΩ, ) = HS C Ω, ) M, ω Ω). the subspace of all Ω invariant elements. The multiplication of elements of HΩ, ) given by the formula α a α ΩM α )) β b β ΩN β )) = α,β a α b β ΩM α N β ) does not depend on the choice of representatives M α ΩM α and N β ΩN β, and turns HΩ, ) into an associative algebra over C with the unity element Ω1 Ω ), called the Hecke Shimura ring or HS ring of relative to Ω over C). Elements 3.1) M) = M) Ω = ΩM i ), M ) M i Ω\ΩMΩ

12 12 A.N. ANDRIANOV AND F.A. ANDRIANOV are in one-to-one correspondence with double cosets of modulo Ω, belong to HΩ, ), and form a basis of the ring over C. For brevity, the symbols ΩM) and M) will be referred as left and double classes of modulo Ω), respectively. We consider the semigroup Σ g = G g Z 2g 2g = { M Z 2g 2g t MJ g M = µm)j g, µm) > 0 and its subsemigroups Σ g q = {M Σ g gcdµm), q) = 1 Σ g A B 0 {M q) = = Σ g q C 0 mod q), C D 3.2) Σ g A B 00 {M q) = = Σ g q B C 0 mod q), C D { Σ g q) = M Σ g µm)1g 0 q M mod q) 0 1 g with q N. According to [An87)], Lemma and similarly treated the case of groups Γ g 00 q), for every q, q N with q q the semigroups satisfy the relations 3.3) Σ g 0 q) Σ g q = Γ g 0 q)σg q ) = Σ g q )Γ g 0 q), and 3.4) Σ g 00 q) Σ g q = Γ g 00 q)σg q ) = Σ g q )Γ g 00 q), where 3.5) Γ g A B 00 {M q) = = C D ) Γ g = Γ g0 1) B C 0 mod q), If Ω is a subgroup of finite index of the modular group Γ g, then the pair Ω, Σ g ) satisfies the conditions of the definition of HS rings, as well as each pair Ω, ) with Ω Σ g, and we can define corresponding Hecke Shimura rings HΩ, ). We shall say that a group Ω satisfying Γ g q) Ω Γ g is q-symmetric if ΩΣ g q) = Σ g q)ω. For such group Ω the HS ring H reg Ω) = HΩ, RΩ)) with RΩ) = ΩΣ g q) = Σ g q)ω is called the regular HS ring of the group Ω of level q. According to Theorem of [An87)], all regular HS rings for given genus g and level q are isomorphic

13 PRODUCTS OF THETA CONSTANTS 13 to each other. In particular, by 3.3) and 3.4) with q = q, the groups Γ g 0 q) and Γ g 00 q) are q-symmetric, and the regular HS rings of the groups Γg 0 q), Γg 00 q), and Γ g q), 3.6) H g 0 q) = H regγ g 0 q)) = HΓg 0 q), Σg 0 q)), H g 00 q) = H regγ g 00 q)) = HΓg 00 q), Σg 00 q)), H g q) = H reg Γ g q)) = HΓ g q), Σ g q)) are naturally isomorphic. For example, the isomorphism of the first and the third rings can be defined as follows. Let T = α a α Γ g 0 q)m α) H g 0 q), where the left classes Γ g 0 q)m α) with a α 0 are pairwise distinct. Using 3.3), without loss of generality one may assume that all representatives M α of the left cosets Γ g 0 q)m α belong to Σ g q). Then, obviously, we have 3.7) T = α a α Γ g q)m α ) H g q), and the map T T is a homomorphic embedding of the ring H g 0 q) into Hg q). In fact, it is a ring isomorphism, and the inverse isomorphism is determined by the condition H g q) b β Γ g q)n β ) b β Γ g 0 q)n β). β β Note also that the map 3.8) M M = ω 1 Mω with ω = ω g 1g 0 q) = 0 q 1 g defines an isomorphism of the pair Γ g 0 q2 ) Σ g 0 q2 ) with the pair Γ g 00 q) Σg 00 q), which induces the isomorphism 3.9) ω = ω g q) : H g 0 q2 ) H g 00 q2 ). The isomorphisms of the rings allows one to transfer various constructions from one of the rings to another. For example, Zharkovskaya homomorphisms Ψ g,n = Ψ g,n k,χ of the ring Hg 0 q) to Hn 0 q), where g > n 1, k is an integer, and χ is a Dirichlet character modulo q satisfying χ 1) = 1) k, can be naturally carried

14 14 A.N. ANDRIANOV AND F.A. ANDRIANOV out to the corresponding HS rings of principal congruence subgroups, so that we get the homomorphisms Ψ g,n = Ψ g,n k,χ : Hg q) H n q), g > n 1) satisfying the commutative diagram H g 0 q) Ψ g,r H n 0 q) η H g q) Ψ g,r η H n q) We remind that the Zharkovskaya map from genus g to genus n, 3.10) Ψ g,n = Ψ g,n k,χ : Hg 0 q) Hn 0 q), can be defined in the following way. Let T = α a αγ g 0 q)m α) H g 0 q). One can assume that each representative M α Γ g 0 q)\σg 0 q) is chosen in the form Aα B M α = α D with D 0 D α = α α 0 D α and D α Z n n. A If A α = α B and B α = α with r r-blocks A α and B α, then and we put M A α = α B α 0 D α Σ r 0q), 3.11) Ψ g,n k,χ T ) = α a α det D α k χ 1 det D α )Γ n 0 q)m α). Hecke Shimura rings act on modular forms and on theta functions by means of linear representation given by Hecke operators. First, let us consider the space F = Fr, g) of all real analytical functions F = F V, Z) : C r 2g H g C with even r = 2k and define action of the semigroup Σ g 0 q), where q is the level of an even positive definite matrix Q of order r, on F by 3.12) Σ g A B 0 q) M = C D ) : F F M = jm, Z) 1 F V tm, M Z ),

15 here PRODUCTS OF THETA CONSTANTS 15 jm, Z) = j Q M, Z) = χ Q M) detcz + D) k, χ Q is defined in previous section character assiciated with the matrix Q, and M Z = AZ + B)CZ + D) 1. It is easy to see that the function jm, Z) satisfies the relations jm, M Z ) jm, Z) = jmm, Z) for every M, M Σ g 0 q) and Z Hg. Hence, 3.13) F M M = F MM F F, M, M Σ g 0 q)). This property of the operators M allows one to define the standard representation of the Hecke Shimura ring H g 0 q) = HΓg 0 q), Σg 0 q)) on the subspace FΓ g 0 {F q)) = F F γ = F, γ Γ g 0 q) of all Γ g 0 q)-invariant functions of F. Namely, if 3.14) T = α a α Γ g 0 q)m α) H g 0 q) is an element of the ring H g 0 q) and F FΓg 0 q)), then the function 3.15) F T = α a α F M α does not depend on the choice of representatives M α Γ g 0 q)m α and again belongs to FΓ g 0 q)). The operators T are clearly linear. The map T T is linear and, as follows from 2.12) and the definition of multiplication in HS rings, it satisfies T T = T T. Thus, we get a linear representation of the ring H g 0 q) on the space FΓ g 0 q)). The operators T are called Hecke operators. By 2.5), the theta function ΘV, Z; Q) with an even positive definite matrix Q of even order r = 2k and level q, considered as a function of V and Z, belongs to the space FΓ g 0 q)). By the above, its image 3.16) ΘV, Z; Q) T = α a α jm α, Z) 1 ΘV tm α, M α Z ; Q) under the action of Hecke operator corresponding to the element 3.13) does not depend on the choice of representatives M α Γ g 0 q)m α and again belongs to FΓ g 0 q)). Formulas proved in [An96)], Theorem 4.1 express the image of a theta

16 16 A.N. ANDRIANOV AND F.A. ANDRIANOV function under the action of Hecke operators as a linear combination with constant coefficients of similar theta functions. In order to formulate the theorem we have to remind two related reductions. The first reduction relates to each of HS-rings HΩ, Σ) with Ω Γ g and Σ Σ g. By the definition of the semigroup Σ g, each matrix M Σ g satisfies the relation t MJ g M = µm)j g, where µm) is a positive integer called the multiplier of M. The multipliers satisfy the rules µmm ) = µm)µm ) M, M Σ g ), and µm) = 1 if M Γ g. It follows that the function µ takes the same value on each left and each double coset of Σ modulo Ω, so one can speak of the multipliers of the cosets. We say that a nonzero formal finite linear combination T of left or double cosets of Σ modulo Ω is homogeneous of multiplier µt ) = µ, if all of the entering cosets have the same multiplier µ. It is clear that every finite linear combination of the cosets is a sum of homogeneous components having different multipliers, and the components are uniquely determined. In particular, this allows one to reduce the consideration of arbitrary Hecke operators T to the case of homogeneous T. Another reduction is related to special choice of representatives in left cosets ΩM when Γ g 0 q) Ω and in Σ Σg 0 q). By Lemma in [An87)], each of the left cosets contains a representative of the form A B 3.17) M = with A, B, D Z g 0 D g, t AD = µm)1 g, t BD = t DB. Such representatives are often convenient for computations with Hecke operators and will be referred as triangular representatives. The following particular case of Theorem 4.1 [An96)] expresses the images of theta-series 1.4) under the action of regular Hecke operators in the form of linear combinations of similar theta-series with coefficients explicitly given in terms of certain trigonometric sums. Theorem 3.1. Let Q be an even positive definite matrix of even order r = 2k and level q. Let T = a α Γ g 0 q)m α) α be a homogeneous element of the Hecke Shimura ring H g 0 q) with µt ) = µ. Let us assume that in the case g < r the element T belongs to the image of the ring H0q) r under the Zharkovskaya map 3.11) from genus r to genus g with k = r/2 and the character χ = χ Q defined in 2, 3.18) T = Ψ r,g r/2,χ Q T ) with T H r 0q) g < r).

17 PRODUCTS OF THETA CONSTANTS 17 Then the image of a theta-series 1.5) under the Hecke operator T can be written in the form 3.19) ΘV, Z; Q) T = where Q, µ) = { D Z r r D Q,µ)/Λ ID, Q, Ψ g,r T ))Θ µd 1 V, Z; µ 1 Q[D] ), det D = ±µ r/2, µ 1 Q[D] E r, Λ = Λ r = GL r Z), 3.20) Ψ g,r T ) = Ψ g,r r/2,χ Q T ) = Ψ g,r r/2,χ Q T ) T ID, Q, T ), for written in the triangular form elements T = β are trigonometric sums defined by 3.21) ID, Q, T ) = if g > r if r = g T Ψ r,g r/2,χ Q ) 1 T ) if g < r, ) b β Γ r Aβ B β 0q) H 0 D 0q) r t A β D β = µ1 r ), β β; D t D β 0 mod µ) with e{ being the exponent 1.2). b β det D β r/2 χ 1 Q det D β )e{µ 2 Q[D] t D β B β, As to the condition 3.18), it follows from [An87), Proposition ] that it can be replaced by a more explicit condition of the following lemma. Lemma 3.2. In the notation of the theorem, a nonzero homogeneous element T H g 0 q) with multiplier µt ) = µ belongs to the image of the ring Hr 0q) for r > g under the Zharkovskaya map 3.10) from genus r to genus g with k = r/2 and χ = χ Q if and only if g k, or g < k and χ Q p) = 1 for each prime divisor p of µ entering the prime numbers factorization of µ in an odd degree. It is sometimes convenient to rewrite the above transformation formulas in somewhat different form. In order to do this we have to make several preliminary remarks. We shall say that an even matrix Q is similar to a nonsingular matrix Q E r, Q Q, if it can be written in the form Q = µ 1 Q[D] with D Q, µ),

18 18 A.N. ANDRIANOV AND F.A. ANDRIANOV where µ is coprime with the level q of Q. It easily follows from the definition that the relation of proper similarity is symmetric. Besides, it is clearly reflexive and transitive. Thus, the set of all nonsingular matrices of E r is disjoint union of s-classes { s{q = Q E r Q Q Q E r, det Q 0). It is easy to see that all matrices of the s-class of a matrix Q have the same signature, determinant, level, and divisor as those of Q, respectively. Further, we recall that two matrices Q, Q of E r are said to be equivalent, Q Q, if Q = Q[U] with U Λ r = GL r Z). All matrices properly equivalent to a matrix Q form the e-class of Q, e{q = { Q = Q[U] U Λ. According to the reduction theory of integral quadratic forms, the s-class of every nonsingular matrix Q E r is a finite union of the e-classes: s{q = hq) j=1 e{q j. The quantity hq) is called the class number of Q. Now we can reformulate Theorem 3.1 in the following form: Theorem 3.3. With the notation and under assumptions of Theorem 3.1, the formula 3.19) can be rewritten in the form 3.22) ΘV, Z; Q) T = h + Q) j=1 D RQ,µQ j )/EQ j ) ID, Q, Ψ g,r T ))ΘµD 1 V, Z, Q j ), where Q 1,..., Q hq) is a system of representatives of all different e-classes contained in the s-class of Q, RQ, Q ) = { M Z r r Q[M] = Q Q, Q E r ) is the set of all integral representations of Q by Q, and denotes the group of all units of Q. EQ ) = RQ, Q ) Q E r, det Q 0)

19 PRODUCTS OF THETA CONSTANTS 19 In particular, if the class number hq) of Q is equal to 1, then the formula 3.22) takes the form 3.23) ΘV, Z; Q) T = ID, Q, Ψ g,r T ))ΘµD 1 V, Z, Q). D RQ,µQ)/EQ) Proof. It is an easy consequence of the above definitions that the sums 3.21) are independent of the choice of representatives in the decomposition of T and satisfy 3.24) IUDU, Q, T ) = ID, Q[U], T ) for all U, U GL m Z) see [An96), Lemma 3.1]). From 2.1) and 3.24) we conclude that the term of the sum on the right of 3.19) corresponding to a matrix D Q, µ) depends only on the coset DΛ. On the other hand, each of the matrices µ 1 Q[D] with D Q, µ) is similar to Q and so is equivalent to one of the matrices Q 1,..., Q hq), say, µ 1 Q[D] Q j. This means that µ 1 Q[D][U] = µ 1 Q[DU] = Q j with U Λ. By replacing D by DU, we can assume that D RQ, Q j ). If D = DU is another such matrix, then, clearly, Q j [U ] = Q j, whence U EQ j ). This proves the formulas. Note that the summs 3.21) were explicitely computed in [An91)] and [An93)] for certain generators of the rings H r 0q) including, in particular, all generators of the rings H 1 0q) and H 2 0q). 4. Action of Hecke operators on theta products Here we shall consider the action of regular Hecke operators on products of even number r = 2k of theta constants with rational characteristics considered as modular forms, assuming that the class number hq r ) of the sum of r squares is equal to 1. Let F belong to the space M k Ω) = M k Ω, 1) of modular forms of integral weight k and the trivial character χ = 1 for a subgroup Ω of finite index in the modular group Γ g, and let T = i a α ΩM α ) HΩ, Σ g ). Then, as it easily follows from definitions of modular forms and HS rings, and the properties 2.12), 2.13) of the Petersson operators 2.11), the function 4.1) F T = F k T = α a α F k M α

20 20 A.N. ANDRIANOV AND F.A. ANDRIANOV does not depend on the choice of representatives M α ΩM α and again belongs to the space M k Ω). These operators are called Hecke operators of weight k for the group Ω). The Hecke operators corresponding to elements of regular HS rings are called regular. It follows from the definition of multiplication in HS rings and from 2.12) that the map T T is a linear representation of the ring HΩ) on the space M k Ω). According to Theorem 2.2, if r = 2k, M 1 d Zr 2g, then the theta product θz, M) is a modular form of weight k and character χ r for the group Γ g 2d 2 ). Assuming in addition that d is even, we have the inclusion 4.2) θz, M) M k Γ g 2d 2 )). In this case the action 4.1) of the ring H g 2d 2 ) on the modular form θz, M) is correctly defined. In order to compute the corresponding images we assume also that hq r ) = 1 and use the formulas 3.23) for the image ΘV, Z ) T = ΘV, Z, Q r ) T, where V = V M), Z = 1 2Z, and 4.3) T = α a α Γ g 0 4)M α) H g 0 4), is a homogeneous element of multiplier µt ) = µ coprime with d. Under the assumptions of Theorem 3.3, by 3.23), we get the formula 4.4) ΘV, Z ) T = A If M α = α B α C α D α can be rewritten in the form D RQ r,µq r )/EQ r ) ID, Q r, Ψ g,r T ))ΘµD 1 V, Z ). Σ g 0 4), then by 3.12) and 2.15) the left-hand side of 4.4) ΘV, Z ) T = α a α j Qr M α, Z ) 1 ΘV tm α, M α Z ; Q). Let us set Aα B M α = α C α D α = ω 1 M αω = A α 2B α 1 2 C α D α ) ω 1 Σ g 0 4)ω Σg 00 2), where ω = ω g 1g 0 2) =, see 3.8). Then it follows directly from definitions that g j Qr M α, Z ) = χ r M α ) detc α Z + D α ) k,

21 where χ r is the character 2.15), also PRODUCTS OF THETA CONSTANTS 21 V tm α = V M) tm α = Mω 1 J g tm α = MµM 1 α ω 1 J g = V M M α), where M α = µm 1 α, and we have used the relations 1.7) along with the fact that t M J g M = µj g, and finally, M α Z = 1 2 M α Z. Hence, the left-hand side of 4.4) can be written in the form a α χ r M α ) 1 δmm α) detc α Z + D α ) k δmm α)θv MM α), 1 2 M α Z ) α = α a α χ r M α ) 1 δmm α)θz, MM α) k M α, where we have used the relation 1.5) and definition 2.11) of Petersson operators. On the other hand, since,clearly, µd 1 V M) = V µd 1 M), the right-hand side of 4.4) can be rewritten in the form D RQ r,µq r )/EQ r ) = ID, Q r, Ψ g,r T ))δµd 1 M)δµD 1 M)ΘV µd 1 M), 1 2 Z) D RQ r,µq r )/EQ r ) = D EQ r )\RQ r,µq r ) ID, Q r, Ψ g,r T ))δµd 1 M)θZ, µd 1 M) IµD 1, Q r, Ψ g,r T ))δdm)θz, DM), since the mapping D µd 1 translates the set RQ r, µq r ) into itself and sends right cosets modulo EQ r ) to left cosets. Hence, the relation 4.4) takes the form 4.5) = a α χ r M α ) 1 δmm α)θz, MM α) k M α α D EQ r )\RQ r,µq r ) The linear combination IµD 1, Q r, Ψ g,r T ))δdm)θz, DM). ωt ) = α a α ω 1 Γ g 0 4)ω ω 1 M αω) = α a α Γ g 00 2)M α)

22 22 A.N. ANDRIANOV AND F.A. ANDRIANOV belongs to the ring H g 00 2) as the image of T under the map 3.9) for q = 2. According to 3.4) for q = 2 and q = 2d 2, we can assume without loss of generality that all representatives M α Γ g 00 2)M α belong to the semigroup Σ g 2d 2 ). Then the linear combination T = a α Γ g 2d 2 )M α ) α belongs to the ring H g 2d 2 ), it is a homogeneous element of multiplier µt ) = µ, and each such element has the this form for a homogeneous element T H g 0 4) of multiplier µt ) = µ. For µ N we denote by [µ] g and [µ] g matrices of order 2g of the form 4.6) [µ] g µ 1g 0 1g 0 = and [µ] 0 1 g =. g 0 µ 1 g Since M α Σ g 2d 2 ), we have M α [µ] g mod 2d 2 ), hence M α [µ] g mod 2d 2 ), that is M α = [µ] g + 2d 2 N with N = N α Z 2g 2g it is easy to see that the map G g M M = µm)m 1 is an antiautomorphism of the semigroup Σ g q). It follows that MM α = M[µ] g + 2dR with the integral matrix R = dmn ) = R, R ). Since the matrix dm is integral, by 1.8) we obtain δmm α) = δm[µ] g ) and by 2.4)we have θz, MM α) = e4d t R M ))θz, M[µ] g ) = θz, M[µ] g ). Therefore, since clearly χ r M α ) = 1 for M α Σ g 2d 2 ) with even d, we can rewrite the relation 4.5) in the form 4.7) δm[µ] g )θz, M[µ] g ) k T = D EQ r )\RQ r,µq r ) IµD 1, Q r, Ψ g,r T ))δdm)θz, DM). After these considerations we come to the following theorem. Theorem 4.1. Let r = 2k 2N be such that the class number hq r ) of the sum of r squares is equal to 1, g N, and M 1 d Zr 2g with d 2N. Let T = α a α Γ g 2d 2 )M α ) H g 2d 2 ) be a homogeneous element with multiplier µt ) = µ, such that for the corresponding element T = a α Γ g 0 4)ωM αω 1 )) H g 0 4) with ω = 1g g α

23 PRODUCTS OF THETA CONSTANTS 23 there exists an element of the form Ψ g,r T ) H r 04) defined by 3.20). Then the theta product θz, M) is a modular form of the space M k Γ g 2d 2 )), and its image under the Hecke operator k T is again a linear combination of theta products: 4.8) θz, M) k T = D EQ r )\RQ r,µq r ) IµD 1, Q r, Ψ g,r T ))θz, DM[ µ] g ), where I are trigonometric sums defined by 3.21), µ N is an inverse of µ modulo 2d 2, and [ µ] g is defined by 4.6). Proof. The first assertion follows from Theorem 2.2. The formula 4.8) follows from the formula 4.7), if we note that for D RQ r, µq r ), δdm) = e{ t M t DDM = e{µ t M M = δm[µ] g ) by 1.6), and replace M by M[ µ] g. As to existence of elements Ψ g,r T ), specialization of Lemma 3.2 to the case Q = Q r and χ Q = χ 4 gives us the following lemma. Lemma 4.2. Under the notation of Theorem 4.1, element Ψ g,r T ) H r 04) exists if and only if g k, or g < k and each prime divisor p of µ entering the prime numbers factorization of µ = µt ) in an odd degree satisfies the congruence p 3 mod 4).

24 24 A.N. ANDRIANOV AND F.A. ANDRIANOV References [An87)] A. N. Andrianov, Quadratic Forms and Hecke Operators., Grundlehren math. Wiss. 286, Springer-Verlag, Berlin Heidelberg..., [An91)] A. N. Andrianov, Composition of solutions of quadratic Diophantine equations, Uspekhi Mat. Nauk ), no. 2278), 3-40; English transl., Russian Math.Surveys ), no. 2, [An93)] A. N. Andrianov, Factorizations of integral representations of binary quadratic forms, Algebra i Analiz ), no. 1, ; English transl., St. Petersburg Math. J ), no. 1, [An95)] A. N. Andrianov, Symmetries of harmonic theta-functions of integral quadratic forms, Uspekhi Matem. Nauk ), no. 4304), 3-44; English transl., Russian Math. Surveys ), no. 4, [An96)] A. N. Andrianov, Harmonic theta-functions and Hecke operators, Algebra i Analiz ), no. 5, 1-31; English transl., St. Petersburg Math. J ), no. 5, [Ig72)] J.I. Igusa, Theta Functions, Grundlehren math. Wiss. 194, Springer-Verlag, Berlin Heidelberg..., [SM-T93)] R. Salvati-Manni and J. Top, Cusp forms of weight 2 for the group Γ 2 4, 8), Amer. J. Math ), no. 2, [Yo80)] H. Yoshida, Siegel s Modular Forms and the Arithmetic of Quadratic Forms, Invent. Math ), no. 3, Steklov Mathematical Institute, St.Petersburg Branch, Fontanka 27, St.Petersburg, Russia address: anandr@pdmi.ras.ru UCLA, Department of Mathematics, 6363 Math. Sciences Building, Box , Los Angeles, CA address: fedandr@math.ucla.edu