MODULAR PROPERTIES OF THETA-FUNCTIONS AND REPRESENTATION OF NUMBERS BY POSITIVE QUADRATIC FORMS

Transcription

1 GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 4, 997, MODULAR PROPERTIES OF THETA-FUNCTIONS AND REPRESENTATION OF NUMBERS BY POSITIVE QUADRATIC FORMS T. VEPKHVADZE Abstract. By means of the theory of modular forms the formulas for a number of representations of positive integers by two positive quaternary quadratic forms of steps 36 and 60 and by all positive diagonal quadratic forms with seven variables of step 8 are obtain. Let rn; f denote a number of representations of a positive integer n by a positive definite quadratic form f with a number of variables s. It is well known that, for the case s > 4, rn; f can be represented as rn; f = ρn; f + νn; f, where ρn; f is a singular series and νn; f is a Fourier coefficient of cusp form. This can be expressed in terms of the theory of modular forms by stating that ϑτ; f = Eτ; f + Xτ, where Eτ; f is the Eisenstein series and Xτ is a cusp form. In his work [] Malyshev formulated the following problem: to define the Eisenstein series and to develop a full theory of singular series for arbitrary s. For s 3, its solution follows from Ramanathan s results []. In the present paper we work out a full solution of this problem. Moreover, convenient formulas are obtained for calculating values of the function ρn; f. Thus, if the genus of the quadratic form f contains one class, then according to Siegel s theorem [], [3], [4], ϑτ; f = Eτ; f and in that case the problem for obtaining exact formulas for rn; f is solved completely. If the genus contains more than one class, then it is necessary to find a cusp form Xτ. 99 Mathematics Subject Classification. E0. Key words and phrases. Quadratic form, number of representations, modular form, generalized theta-function X/97/ $.50/0 c 997 Plenum Publishing Corporation

2 386 T. VEPKHVADZE A large number of papers is devoted to finding such formulas. Cusp forms in these works are constructed in the form of linear combinations of products of simple theta-functions with characteristics or their derivatives see, e.g., [5], products of Jacobi theta-functions or their derivatives see, e.g., [6], and theta-functions with spherical polynomials see, e.g., [7]. All these functions are certain particular cases of linear combinations of the so-called generalized theta-functions with characteristics defined below by the formula. In the present paper, using modular properties of these functions, we have obtained by the unique method the exact formulas for a number of representations of numbers by positive quadratic forms both with an even forms of such a kind were considered earlier and an odd number of variables. Moreover, it is shown that using [8], one can reduce cumbersome calculations for obtaining formulas for a number of representations of numbers by the quadratic forms considered earlier and to obtain new formulas.. Let f = x Ax be a positive definite qadratic form, let A be an integral matrix with even diagonal elements, and the vector column x Z s, s N, s. We call u Z s a special vector with respect to the form f if Au 0 mod N, where N is a step of the form f. Moreover, let P ν = P ν x be a spherical function of the νth order ν is a positive integer corresponding to the form f see [9], p Then the generalized thetafunction with characteristics we define as follows: ϑ gh τ; p ν, f x g mod N h Ax g N p ν xe πiτx Ax N ; here and below, g and h are the special vectors with respect to the form f. In the sequel, use will be made of the following lemmas see Lemmas and 4 in [8]. Lemma. Let k be an arbitrary integral vector and l be a special vector with respect to the form f. Then the following equalities hold: ϑ g+nk,h τ; p ν, f = h Ak N ϑgh τ; p ν, f, ϑ g,h+l τ; p ν, f = ϑ gh τ; p ν, f. Lemma. Let F τ be an entire modular form of the type r, N, vl and let there exist an integer l for which vl l =. Then the function F τ is identically equal to zero if in its expansion in powers of Q = e πiτ the coefficients c n equal 0 for n r/n p N + p. In the main theorem below we formulate modular properties of linear combinations of functions.

3 MODULAR PROPERTIES OF THETA-FUNCTIONS 387 Theorem. Let f = f x = x A x,..., f j = f j x = x A j x be positive definite quadratic forms with a number of variables s, let P ν k = P ν k x be the coresponding spherical functions of order ν, k be the determinant of the matrix A k, N k the step of the form f k k =,..., j, the determinant of the matrix A of some positive definite quadratic form x Ax with a number of variables s + ν and the step N. Next, let g k and h k be special vectors with respect to f k ; moreover, N given N k, let h k be the vector with even components k =,..., j; α β L = Γ γ δ 0 N; vl = i ηγsgn δ s+ν s+ν i sgn δβ for s, = sgn δ s +ν s +ν for s; ηγ = for γ 0, = for γ < 0; s +ν sgn δβ is the Kronecker symbol, is the Jacobi symbol. Then the function Xτ = j k= k B k ϑ g k h kτ; P ν, f k 3 for arbitrary complex numbers B k is an entire modular form of the type s + ν, N, vl if and only if the conditions N k N, N k f k g k, 4N k N N k f k h k are fulfilled, and for all α and δ such that αδ mod N we have j k= B k ϑ αg k,hkτ; pk ν, f k sgn δ ν [ [ s+ν = ] j k= ] k B k ϑ g k h kτ; pk ν, f k. = This theorem has been proved in [8] for even h k. It can easily be adjusted to the case N N k h k. From this theorem we obtain the following two theorems which are analogues of Theorems 4 and from [8].

4 388 T. VEPKHVADZE Theorem. If all the conditions of Theorem are fulfilled and either ν > 0, or ν = 0 and all the g k vectors are nonzero, then the function 3 is a cusp form of the type + ν, N, vl. Theorem 3. Let f be an integral positive quadratic form with a number of variables s and let be a determinant of the form f. Then the function ϑτ; f defined by the formula ϑτ; f = + rn; fe πiτf Im τ > 0 4 n= is the entire modular form of the type s, N, νl, where νl are defined by the formulas for ν = 0. From the results of [], [3], [4] and [0] we obtain Theorem 4. Let f be a positive quadratic form with a number of variables s and let be its determinant. Then the function Eτ, z; f, determined for Re z s and Im τ > 0 by the formula Eτ, z; f = + e πis 4 s s q= H= H,q= Sfh, q q s qτ H s qτ H z, where Sfh, q is the Gaussian sum, can be continued analytically into the neighborhood of the point z = 0. Further, having defined the Eisenstein series Eτ; f by the formulas Eτ; f = Eτ, z; f z=0 for s = = Eτ, z; f z=0 for s >, we have Eτ; f = + ρn; fe πiτn n= for s =, = + ρn; fe πiτn for s > ; 5 n= here ρn; f is a singular series which is calculated as follows:

5 MODULAR PROPERTIES OF THETA-FUNCTIONS 389 If s, v = p n p w, = r ω ω is a square-free number, then p π s ρn; f = Γ s n s χ χ p / p p> L s ; ω k v p r p> s k s p k s. ω p s If s, n = α+γ v v = r ω, α n, γ, p l, p w n p >, v = p n p w = rω, v = p n p w+l = rω ω, ω, ω are squarefree numbers, p p, p> then ρn; f = s! r s n s Γ s s π s B s s p s s L ; ω p k r k s p k p r p> s p ω χ χ p p p> s p p s. ω p s The values of χ and χ p are given in [] formulas 9 3, p. 66; Lk; k ω = k ω l= l B s are Bernoulli s numbers. l l k = p p> k ω p p k ;. In this section we will obtain exact formulas for a number of representations of numbers by quaternary quadratic forms: x + x + x 3 + 5x 4 and x + x x + 5x + x 3 + x 3 x 4 + 5x 3. The first form has been considered by Lomadze in [5] who, for the construction of cusp form Xτ, used products of simple theta-functions with characteristics and of their derivatives some particular cases of the function and therefore he had to use modular forms of step 40 instead of 60.

6 390 T. VEPKHVADZE Theorem 5. Let f = x + x + x3 + 5x 4, f = 3x + 5x, f = 4x + x x + 4x, g 0 =, h 0 =, 0 0 g 5 =, h 0 =, p 0 5 = x, p = x + x. Then the equality ϑτ; f = Eτ; f ϑ g h τ; p, f + 0 ϑ g h τ; p, f 6 holds, where the functions ϑτ; f, ϑ g h τ; p, f, ϑ g h τ; p, f are defined by formulas 4 and, while the function Eτ; f by formula 5. Proof. By Theorem 3, the function ϑτ; f belongs to the space of entire modular forms of the type, 60, vl, where vl is the corresponding multiplier system. Then, according to Siegel s theorem see [], Eτ; f also belongs to this space. Using Lemma, we can check that the function Xτ = 4 5 ϑ g h τ; p, f + 0 ϑ g h τ; p, f satisfies all the conditions of Theorem. Indeed, f is the binary form of step 60 and f is the binary form of step N = 60, N = 30, g =, and h 0 = are special vectors with respect to the form f = 3x +5x, and g = and h 0 = 5 are special vectors with respect to the form f = 4x +x x +4x. h, N N h, since N = N = 60, N = 30; but 60 N, 30 N, 60 f g, 30 f g, 40 f h n, 0 f h. If αδ mod 60, then either α δ mod 3 or α δ mod 3. Because of Lemma, ϑ αg,hτ; p, f = { ϑ g h τ; p, f for α δ mod 3, ϑ g h τ; p, f for α δ mod 3.

7 Due to we have Thus = MODULAR PROPERTIES OF THETA-FUNCTIONS 39 ϑ g h τ; p, f = x g mod 60 ϑ αg,hτ; p, f = x g mod 60 3x +5x πiτ x e 60 3x +5x πiτ x e 60 = ϑ g h τ; p, f. { ϑ g h τ; p We have 5 sgn δ = sgn δ, Furthermore, we have = 3, f for α δ mod 3, ϑ g h τ; p, f for α δ mod 3. = { sgn δ for δ mod 3, sgn δ for δ mod We can easily verify that formulas 7 and 8 imply ϑ αg h τ; p, f sgn δ = ϑ g h τ; p, f. 9 Analogously, we get ϑ αg h τ; p, f sgn δ = Consequently, according to 9 and 0, the function ϑ αg h τ; p, f. 0 Xτ = 4 5 ϑ g h τ; p, f + 0 ϑ g h τ; p, f satisfies the conditions of Theorem and, due to Theorem, belongs to the space of cusp forms of the type, 60, vl. Thus, owing to Lemma, the function Ψτ = ϑτ; f Eτ; f Xτ, where Xτ is defined by, will be identically zero if all coefficients for Q n n 4 are zero in its expansion in powers of Q = e πiτ. Next, let n = α 3 β 5 β u, u, 30 =. Then by Theorem 4, Eτ; f = + ρn; fq n Q = e πiτ, n=

8 39 T. VEPKHVADZE where ρn; f = α+ + β u 5 β + + α+β +β u 5 3 β+ α+β d d =u u 3 5 d d. 3 Having calculated the values ρn; f for all n 4 by formula 3, we obtain because of : Eτ; f = + 3Q Q Q3 + 9Q 4 + 4Q 5 + 5Q Q Q8 + 39Q Q0 + 4Q Q + 4Q Q Q5 + 33Q 6 + 7Q Q8 + 8Q Q Q + 4Q Q3 + 75Q Formulas 4 and yield ϑτ; f = + 6Q + Q + 8Q 3 + 6Q 4 + 4Q 5 + 4Q Q Q 9 + 4Q 0 + 4Q + 8Q + 4Q Q 4 + Q 5 + 8Q 6 + 7Q 7 + 5Q Q Q Q + 4Q + 4Q Q ϑ g h τ; p, f = 6 3 Q + Q 3 Q 7 Q 8 + Q 0 Q Q 5 + Q 8 Q 3 + 4Q , 6 0 ϑ g h τ; p, f = 3 Q Q4 6Q 6 6Q 9 Q 0 6Q 5 0Q 6 + Q 9 + 6Q Taking into account 4 7, we can easily verify that all coefficients for Q n n 4 in the expansion of the function ψτ in powers of Q are zero. Thus identity 6 is proved. Theorem 6. Let f = x + x + x 3 + 5x 4, n = α 3 β 5 β u, u, 30 =. Then rn; f = 3 β+ β u β+ + β +β u 5 d + 5 d d d =u

9 MODULAR PROPERTIES OF THETA-FUNCTIONS x x + x for n mod 4, n=x +x x +4x x = u α+ + β 3 β+ α+β u 3 5 β+ + α+β +β u 5 d + 5 d + 3 d d =u x x + x n=x +xx+4x x 3n=x +5x x x mod 3 x otherwise. Proof. Equating coefficients of the same powers Q in both parts of identity 6, we get rn; f = ρn; f ν n + 3 ν n, 8 where ν n, ν n denote respectively the coefficients for Q in the expansions of the functions in powers of Q. From we have 0 ϑ g h τ; p, f, 5 ϑ g h τ; p, f 0 ϑ g h τ; p, f = 3x + e πiτ3x + +53x + 3, x,x = i.e., ν n = x. 9 3n=x +5x x x mod 3 It follows from 4 that = x,x = 5 ϑ g h τ; p, f = x x + + x e πiτ[x + +x +x +4x ],

10 394 T. VEPKHVADZE i.e., ν n = x x + x. 0 n=x +x x +4x x From formulas 8, 3, 9, 0 we obtain the desired exspression for rn; f. Theorem 7. Let f = x + x x + 5x + x3 + x 3 x 4 + 5x4, f = 8 0 3x + 9x, g =, h =, p 6 0 = x. Then ϑτ; f = Eτ; f 9 ϑ ghτ; p, f, where the functions ϑτ; f, Eτ; f and ϑ gh τ; p, f are defined respectively by the formulas 4, 5 and. Proof. Let n = α 3 β u, u, 6 =. Then by Theorem 4, Eτ; f = + n= ρn; fqn Q = e πiτ, where ρn; f = 3 β µ u µ for α > 0, β > 0, = 43 β µ u µ for α = 0, β > 0, = 4 µ u µ for α > 0, β > 0, = 4 µ for n, 6 =. 3 µ u Formulas 5 and imply Eτ; f = Q +..., ϑτ; f = + 4Q +..., 9 ϑ 9hτ; p, f = 4 3 Q By Theorem 3, the function ϑr; f belongs to the space of entire modular forms of the type, 36,. Then by Siegel s theorem see [], Eτ; f also belongs to this space. Using Lemma, we can easily verify that the function ϑ gh τ; p, f satisfies all the conditions of Theorem. Therefore by Theorem, it belongs to the space of cusp forms of the type, 36,. It is well known that this space is one-dimensional see []. Therefore from we obtain the above assertion. From Theorem 7 immediately follows

11 MODULAR PROPERTIES OF THETA-FUNCTIONS 395 Theorem 8. Let n = α 3 β u, u, 6 =, f = x + x x + 5x + x 3 + x 3 x 4 + 5x 4. Then rn; f = 3 β µ u µ for α > 0, β > 0, = 43 β µ u µ for α = 0, β > 0, = 4 µ u µ for α > 0, β = 0, = 4 µ 3 3 µ u Remark to Theorem 8. Let 4n=3x +x x mod x mod 6 x for n, 6 =. νn = x. 4n=3x +x x mod x mod 6 It can be easily shown that νn = x + x. 4n=3x +x x mod x mod 6 Further, arguing as in [] p. 33, we can easily show that νn n = νn νn if n, n = ; νp β = 0 k< p k T rπ β k p + δ β β p β, where πp is the Frobenius endomorphism of a curve y = x 3 + reduced in modulo p, δr is equal to one or to zero according to whether the number r is an integer or not. In particular, if n = p is a prime number, then p x 3 + νp =, p x=0 where x 3 + p is the Legendre symbol.

12 396 T. VEPKHVADZE 3. In this section we obtain formulas for a number of representations of numbers by quadratic forms with seven variables f = s xj + j= 7 j=s+ x j s 6. 3 The cases s = 0 and s = are considered earlier see, e.g., [3], Vol. II, pp. 305, 309, 335 and Vol. III, p. 37. In these cases the corresponding forms belong to one-class genera. The case s = 3 was considered in [6]. Theorem 9. Let f be of the kind 3, f = x + x + x 3, f = x + 4 x + x 3, g = 4, h =, p = x x, g = 4 4 4, h = Then the following equality holds, where ϑτ; f = Eτ; f + Xτ; f, 4 Xτ; f = 3 ϑ g h τ; p, f for s =, 4, = 3s ϑ g h τ; p, f for s = 3, 5, = 0 otherwise. 5 Proof. By Theorem 3, the functions ϑτ; f belong to two different spaces of modular forms 7, 8, vl, where vl is a system of multipliers corresponding to the form f. This system is the same for all s with the same evenness. Then, according to Siegel s theorem see [], the functions Eτ; f also belong to appropriate spaces of modular forms. It can be easily verified that functions 5 satisfy all the conditions of Theorem and, by Theorem, they belong to two different spaces of cusp forms depending on s. Let n = α u u, α 0, s n = rsω s, u = r ω s =,,..., 6 and let ω and ω s be square-free numbers. Then by Theorem 4 we have where Eτ; f = + ρn; fq n Q = e πiτ, 6 n= ρn; f = 5α +9 s π 3 ω 5 L3; ωs χ µ 5 p µ µ r ωs p 3. p

13 MODULAR PROPERTIES OF THETA-FUNCTIONS 397 By Lemma 7 from [4] we have L3; = π3 3π3, L3; = 3 64 ; 7 { L3; ω = π3 h h ω 6h + 3ω + 6ω 5 ω h ω ω ω 4 4 <h< ω h } + 6 hh ω, if ω mod 4, ω >, ω = π3 ω 5 = π3 3ω 5 + 4ω 8 = π3 3ω ω 4 <h ω h ω { h ω 6 ω 3ω 6 <h< 6 3ω 6 <h ω 4 {3ω ω 3ω 6 <h 6 h hω h, if ω 3 mod 4, ω h 3ω 56h ω ω + h ω 8h + 3ω h hω h ω h ω 6 3ω 6 <h ω 4 h ω }, if ω mod 8, ω >, h h h ω ω ω 3ω 6 <h< ω + 6 h + 8ω hω 4h ω 3ω 6 <h ω 4 h ω 4h ω }, if ω 6 mod 8. 8 Using formulas 33 of [9], after calculation of values χ, we obtain χ = for s, α = 0, or for s, α = 0, u mod 4, or s, α =, = + u 6 s 5, for s, α = 0, u 3 mod 4, = + s 3 5α 63 3 for s, α, u mod 4, = + s 3 5α + u 8 5α 3 3 α, u 3 mod 4 = + s 3 5α for s, for s, α, α >,

14 398 T. VEPKHVADZE = + s 5α 63 for s, α, α > 0, 3 = + s 5α 5 63 for s, α, u mod 4, 3 = + s 5α 5 + u 8 5α for s, α, u 3 mod 4. 9 By we have ϑ g h τ; p, f = 6 x,x,x 3= x +x +x 3 x + x + e πiτ[x + +x + +x 3 ], 30 ϑ g h τ; p, f = 6 x +x x + x + x,x,x 3= e πiτ[x + +x + +x 3 + ] 4. 3 Taking then into account 6 3 and arguing as in the proof Theorem 5, we obtain the above assertion. From Theorem 9 we have Theorem 0. Let n = α u u, α 0, s n = r ω s, u = r ω, and let ω and ω s be square-free numbers, s =,,..., 6, f = s x j + j= 7 j=s+ x j. Then rn; f = 5α s +9 ω 5 π 3 L3; ω s χ µ r µ 5 p µ ωs p p 3 + νn; f,

15 MODULAR PROPERTIES OF THETA-FUNCTIONS 399 where νn; f = 0 for s =, 6, = x x +x 3 x x for s =, 4, n=x +x +x 3 x, x = s 4n=x +x +x 3 x, x, x 3 x x x x for s = 3, 5. The values L3; ω 3 and χ can be calculated by formulas 7 9. References. A. B. Malyshev, Formulas for a number of representations of numbers by positive quadratic forms. Russian Actual problems of analytic number theory Russian 9 37, Minsk, K. G. Ramanathan, On the analytic theory of quadratic forms. Acta Arithmetica 97, T. V. Vepkhvadze, To one Siegel s formula. Russian Acta Arithmetica 4098, T. V. Vepkhvadze, To the analytic theory of quadratic forms. Russian Trudy Tbilis. Mat. Inst. Razmadze 7983, G. A. Lomadze, On a number of representations of numbers by quadratic forms with four variables. Russian Trudy Tbilis. Mat. Inst. Razmadze 4097, H. Peterson, Modulfunktionen und quadratische Formen, Springer- Verlag, Berlin, G. A. Lomadze, On cusp forms of a prime step and of basic type II. Russian Trudy Tbilis. Mat. Inst. Razmadze 57977, T. V. Vepkhvadze, Generalized theta-functions with characteristics and representation of numbers by quadratic forms. Russian Acta Arithmetica 53990, C. Shimura, On modular forms of half-integral weight. Ann. of Math , R. I. Beridze, On summation of a singular Hardy-Littlewood series. Russian Soobshch. Acad. Nauk Gruz. SSR 38965, No. 3, A. V. Malyshev, On the representation of integers by positive quadratic forms. Russian Trudy Mat. Inst. Steklov. 6596,.

16 400 T. VEPKHVADZE. A. N. Andrianov, Representation of numbers by some quadratic forms in connection with the theory of elliptic curves. Russian Izv. Acad. Nauk SSSR, Ser. Mat. 9965, No., L. E. Dickson, History of the theory of numbers, Vols. II and III. New York, H. Steefkerk, Over het aantal oplossingen der diophantische vergelijking U = s i= Ax i + Bx i + C i. Dissertation, Amsterdam, 943. Received Author s address: Faculty of Physics I. Javakhishvili Tbilisi State University, I. Chavchavadze Ave., Tbilisi Georgia