INDIVISIBILITY OF CENTRAL VALUES OF L-FUNCTIONS FOR MODULAR FORMS

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1 INIVISIBILITY OF CENTRAL VALUES OF L-FUNCTIONS FOR MOULAR FORMS MASATAKA CHIA Abstract. In this aer, we generaize works of Kohnen-Ono [7] and James-Ono [5] on indivisibiity of (agebraic art of centra critica vaues of L-functions to higher weight moduar forms. 1. Introduction In this artice, we show an indivisibiity resut on centra critica vaues of L-functions associated to quadratic twists of moduar forms using a method of Kohnen-Ono [7] and James-Ono [5]. Let f(z = a(nqn be a normaized newform of weight 2k for Γ 0 (N with trivia character. For a fundamenta discriminant with (, N=1, we define the -th quadratic twist of f by f χ = a(nχ (nq n, where χ is the quadratic character corresonding to the quadratic extension Q( /Q. Then f χ is a newform of weight 2k for Γ 0 ( 2 N. Simiary, the -th quadratic twist of the L-function L(f, s is given by L(f χ, s = a(nχ (n n s. Let E be the number fied generated by a Fourier coefficient of f and Q. Then it is known that there exists a eriod Ω f C satisfying that L(f χ k 1/2,k 0 Ω f are integers in E for a fundamenta discriminant with δ(f > 0, where δ(f ±1} is the sign defined in Ono-Skinner [10,. 655] and 0 is given by if is odd, 0 = /4 if is even. We fix such a eriod Ω f. For convenience, we denote S( = Z <, : fundamenta discriminant }, and if functions f, g on R satisfy that there is a ositive constant c such that f( c g( for sufficienty arge > 0, then we write f( g(. ate: January 19, Mathematics Subject Cassification. Primary 11F67 ; Secondary 11F37. 1

2 2 MASATAKA CHIA Theorem 1.1. Let f(z = a(nqn be a normaized newform of weight 2k for Γ 0 (N with trivia character. Then, for a but finitey many rimes λ of E, we have # S( δ(f > 0, λ and L(f χ, k k 1 } mod λ Ω f f,λ og. This resut is a refinement of resuts of Bruinier [2] and Ono-Skinner [10]. The roof is based on a generaization of a method of Kohnen-Ono [7] and James-Ono [5]. In the above theorem, we do not assume that the Fourier coefficients of f beong to Z, therefore it does not hod the surjectivity of the residua Gaois reresentation associated to f for amost a aces in genera. This makes some technica difficuty on the roof. To sove this robem, we may use a resut of Ribet [12] on the image of Gaois reresentations associated to moduar forms. This is an ingredient in our roof. In the ast section, we aso consider an indivisibiity resut on non-centra critica vaues of L-functions for higher weight moduar forms using congruences of moduar form with different weights. Acknowedgment. It is great easure to thank Professor Ken Ono for his encouragements and hefu suggestions. The author is suorted in art by the Jaan Society for the Promotion of Science Research Feowshis for Young Scientists. 2. Moduar forms of haf-integra weight We denote the sace of moduar forms of weight k + 1/2, eve N with character χ by M k+1/2 (N, χ, and the sace of cus forms of weight k + 1/2, eve N with character χ by S k+1/2 (N, χ. Then M k+1/2 (N, χ and S k+1/2 (N, χ are comex vector saces. For a moduar form of haf-integra weight g(z = b(nq n M k+1/2 (N, χ, n=0 we define the action of Hecke oerator T 2 by T 2(g(z = b (nq n, where b (n are given by ( k ( 1 n b (n = b( 2 n + χ( k 1 b(n + χ( 2 2k 1 b(n/ 2 and b(n/ 2 are zero if 2 n. Now we give a short review of the theory of the Shimura corresondence. Let N be a ositive integer which is divisibe by four and χ a irichet character mod N. Then we define a vector sace S 0 3/2 (N, χ to be the subsace of S 3/2(N, χ generated by f(z = n=0 } ψ(nnq tm2 4cond(ψ 2 t N, χ = ψχ t and ψ( 1 = 1 and denote the orthogona comement by S 3/2 (N, χ. Then we assume g(z = b(nq n S k+1/2 (N, χ

3 if k 2, and INIVISIBILITY OF CENTRAL VALUES OF L-FUNCTIONS FOR MOULAR FORMS 3 g(z = b(nq n S 3/2(N, χ if k = 1. Let t be a square-free ositive integer. efine a number A t (n to be ( A t (n χ(n( ( ( 1 n = k t n b(tn 2. n s n s k+1 n s Then Shimura [14] roved that there is a ositive integer M such that SH t (g(z = f t (z = A t(nq n S 2k (M, χ 2. (In fact, one can rove that M = N/2. Furthermore for any t, t, the difference between SH t (g and SH t is ony constant mutie, so essentiay this corresondence is indeendent of choice of t. This corresondence between moduar forms is caed the Shimura corresondence. Moreover if g is an eigenform for a Hecke oerators T 2 with (, 2N = 1, then the image of g under the Shimura corresondence is aso an eigenform for a Hecke oerators T with (, 2N = 1 and the Hecke eigenvaue of T 2 for g coincides with the Hecke eigenvaue for T for SH t (g. We reca the foowing resut which is a usefu version of Wadsurger s formua ([17, Thèorém 1] by Ono-Skinner. This formua gives a reation between the Fourier coefficients of moduar forms of haf-integra weight and the centra vaues of twisted L-functions for moduar forms. Theorem 2.1 (Ono-Skinner [9], (2a,(2b. Let f(z = a(nqn be a normaized newform of weight 2k, eve M with trivia character. Then there is δ(f ±1}, a ositive integer N with 4M N, a irichet character χ moduo N, a eriod Ω f C and a non-zero eigenform g(z = b(nq n S k+1/2 (N, χ with the roerty that g(z mas to a twist of f under the Shimura corresondence and for a fundamenta discriminants with δ(f > 0 we have k 1/2 L(f χ, k 0 b( 0 2 α = if (, N = 1, Ω f 0 otherwise, where α and b(n are agebraic integers in some finite extension of Q. Moreover, there exists a finite set of rimes S such that if is a square-free integer for which (1 δ(f > 0, (2 (, N = 1, then we have L(f χ, k 0 k 1/2 /Ω f λ = b( 0 2 λ for λ S. 3. Some roerties of Fourier coefficients of moduar forms and Gaois reresentations In this section we generaize some resuts of Serre [13] and Swinnerton-yer [16] using a resut of Ribet [12]. These resuts shoud be we-known for seciaists. However we give a short review for them, since it does not seem to be avaiabe in the iterature. Let f = a(nqn be a normaized newform of weight 2k for Γ 0 (N with trivia character. Let E be the subfied of C generated by the Fourier coefficients a(n of f. Then E is a finite extension of Q. Let O E be the ring of integers of E. For each rime, we et O E, = O E Z Z and E = E Q Q.

4 4 MASATAKA CHIA Theorem 3.1 (eigne [3]. For each rime, there exists a continuous reresentation ρ f, : Ga(Q/Q GL 2 (O E, GL 2 (E unramified at a rimes N such that traceρ f, (Frob = a( and detρ f, (Frob = 2k 1 for a rimes N, where Frob is the arithmetic Frobenius at. For each rime, denote } A = g GL 2 (O E, det(g Z (2k 1, where Z (2k 1 is the grou of (2k 1-th owers of eements in Z. Reacing ρ f, by an isomorhic reresentation, we may assume that for amost a ρ f, sends Ga(Q/Q to A. Then Ribet roved the foowing theorem. Theorem 3.2 (Ribet [12]. Assume that f has no comex mutiication. amost a, we have ρ f, (Ga(Q/Q = A. Then for We ca the set of rimes with the roerty ρ f, (Ga(Q/Q A by the excetiona rimes for f. Let S be the set of excetiona aces for f. Let ε : Ga(Q/Q Z be the -adic cycotomic character. Then by a simiar argument with Swinnerton-yer [16], one can see that the image of (ρ f,, ε : Ga(Q/Q GL 2 (O E, Z is (g, α GL 2 (O E, Z } det(g = α 2k 1 if is not excetiona. Since A contains an eement with the form ( traceρf, (σ 1, detρ f, (σ 0 the ma (traceρ f,, ε : Ga(Q/Q O E, Z argument, one can see that the ma S(traceρ f,, ε : Ga(Q/Q S is surjective. Moreover by a ramification (O E, Z is aso surjective. Therefore we have the foowing resut which is a generaization of a resut of Serre [13, THÈORÉM 11] using Chebotarev density theorem. Theorem 3.3. Assume that f has no comex mutiication. Let t be a ositive integer and α a non-zero integer in E. Fix β O E /αo E and r (Z/tZ. Suose that α does not contain a rime divisor which divides an excetiona rime for f. Then the set of rime with the roerties a( β mod α and r mod t has ositive density. 4. Indivisibiity of Fourier coefficients of moduar forms of haf-integra weight In this section, we give a resut on moduo indivisibiity of Fourier coefficients of haf-integra weight moduar forms using a method of Kohnen-Ono [7] and James-Ono [5]. Our resut is a refinement of a resut of Bruinier [2] and Ono-Skinner [10]. To consider the indivisibiity of Fourier coefficients of haf-integra weight moduar forms, we wi use the foowing resuts. Theorem 4.1 (Sturm [15]. Let g(z = b(nq n M k (N, χ

5 INIVISIBILITY OF CENTRAL VALUES OF L-FUNCTIONS FOR MOULAR FORMS 5 be a haf-integra or integra weight moduar form for which the coefficients b(m are agebraic integers contained in a number fied E. Let v be a finite ace of E and et + if b(n 0 mod v for a n, ord v (g = min n b(n 0 mod v} otherwise Moreover ut Assume that then ord v (g = +. µ = k 12 [Γ 0(1 : Γ 0 (N] = kn 12 ord v (g > µ, N + 1. Remark 4.2 (cf. [5] Proosition 5. In [15], Sturm roved this theorem for integra weight moduar forms with trivia character, but the genera case foows by taking an aroriate ower of g. Lemma 4.3 (Shimura, [14] Section 1. Suose g(z = b(nq n S k+1/2 (N, χ is a haf integra weight cus form and is a rime. We define (U g(z, (V g(z by (U g(z = u (nq n = b(nq n, Then Let (V g(z = v (nq n = b(nq n. (U g(z, (V g(z S k+1/2 (N, χ f(z = a(nq n M k (N, χ ( 4. be an integra weight moduar form for which the coefficients a(m are agebraic integers in E. For a rime λ of E and ositive integers r, t with (r, t = 1, define T (r, t and T (λ, r, t by T (r, t = : rime a( = 0, r mod t} and T (λ, r, t = : rime a( 0 mod λ, r mod t}. For a ositive rea number, we aso denote T (r, t, = T (r, t } and T (λ, r, t, = T (λ, r, t }. For g = b(nqn S k+1/2 (N, χ O E,λ [[q]], denote s λ (g = minord λ (b(n n Z >0 }. The foowing two emmas give an estimate for indivisibiity of Fourier coefficients of moduar forms of haf integra weight.

6 6 MASATAKA CHIA Lemma 4.4. Let be a rime greater than 3. Let f(z = a(nqn be a normaized Hecke eigen newform of weight 2k, eve M with trivia character and et g(z = b(nq n S k+1/2 (N, χ be the eigenform given in Theorem 2.1. Assume that f has comex mutiication in the sense of Ribet [11] and λ be a ( rime in E above. If there exists an integer such that δ(f > 0, (, N = 1, ε = 0 and ord λ (b( = s λ (g, then ( } # S( = ε, ord λ (b( = s λ (g f, og. Proof. By dividing g by λ sλ(g, we may assume s λ (g = 0. If we ut ( n b(n if (n, N = 1 and = ε, b 0 (n = 0 otherwise, then g 0 (z = b 0 (nq n S k+1/2 (N 2, χ for a suitabe character χ. Since f has comex mutiication, so there exists a imaginary ( quadratic fied K such that for every rime satisfying 3 mod 4, (, N = 1 and K = 1 we have a( = 0, where K is the discriminant of K. Therefore, for such, using the formuae for the action of Hecke oerator T 2, we find that ( ( 1 b( 2 n + χ ( k 1 k n b(n + χ ( 2 2k 1 b(n/ 2 = 0. Hence if (r, t = 1, 4 t, r 3 mod 4, then and for any T (r, t we have #T (r, t, = # T (r, t } f og (4.1 b( 2 n = χ ( k 1 ( ( 1 k n b(n χ 2 ( 2k 1 b(n/ 2. Put κ = (k+ 1 2 [Γ 0(1:Γ 0 (N 2 ] Now, we choose (r 0, t 0 satisfying the foowing roerties: (1 N 2 t 0, (r 0, t 0 = 1, χ (r 0 = 1 and 3 ( mod 4. ( 1 k n (2 If is a rime with r 0 mod t 0, then = 1 for any 1 n κ with (n, N 2 = 1. ( K (3 For each rime r 0 mod t 0 we have = 1. ( (4 Each rime r 0 mod t 0 satisfies ( 1 χ ( 2 χ k ( = 1. λ

7 INIVISIBILITY OF CENTRAL VALUES OF L-FUNCTIONS FOR MOULAR FORMS 7 If T (r 0, t 0 is a sufficienty arge rime, for a 1 n κ u (n = b 0 ( 2 n = χ ( k 1 ( ( 1 k n b 0 (n 2k 1 χ 2 (b 0 (n/ 2 Since b 0 (n/ 2 = 0, we have u (n = χ ( k 1 b 0 (n = k 1 b 0 ( = k 1 v (n. By the reation (4.1, ( ( 1 v ( 3 = b 0 ( 2 = χ ( k 1 k b 0 (, and u ( 3 = b 0 ( 4 = 2k 1 χ ( 2 b 0 (. Therefore by the assumtion and the choice of (r 0, t 0, u ( 3 k 1 v ( 3 ( ( λ = ( 1 χ ( 2 2k 1 χ ( 2k 2 k b 0 ( = 1. λ Hence ord λ (U g 0 k 1 V g 0 < +. By Theorem 4.1 and Lemma 4.3, there exists a integer n such that ( 1 n k + 1 [Γ0 (1 : Γ 0 (N 2 ] = κ( + 1, (n, = and b 0 (n = u (n k 1 v (n = 0 mod λ. Consequenty, et sf be the square-free art of = n, then b 0 ( sf λ = 1. For convenience, et i be the rimes in T (r 0, t 0 in increasing order, and et i be the square-free art of i n i. If r < s < t and r = s = t, then r s t r. However this can ony occur for finitey many r, s and t since i < κ i ( i + 1. Therefore, the number of distinct i < is at east haf the number of T (r 0, t 0 with /κ. Therefore the emma foows from #T (r 0, t 0, f,λ / og. Lemma 4.5. Let f(z = a(nqn be a normaized Hecke eigen newform of weight 2k, eve M with trivia character. enote E = Q(a(n n 1} and et g(z = b(nq n S k+1/2 (N, χ be the eigenform given in Theorem 2.1. We fix a rime number greater than 3 and et λ be a rime in E above. Assume that f does not have comex mutiication and the image of the Gaois reresentation associated to f ρ f, : Ga(Q/Q GL 2 (O E, coincides ( with A. If there exists an integer such that δ(f > 0, (, N = 1, ε = 0 and ord λ (b( = s λ (g, then ( } # S( = ε, ord λ (b( = s λ (g f,λ og.

8 8 MASATAKA CHIA Proof. First, we may assume ord λ (g = 0. If we ut then ( n b(n if (n, N = 1 and b 0 (n = 0 otherwise, g 0 (z = b 0 (nq n S k+1/2 (N 2, χ = ε, for a suitabe character χ. If a( 0 mod λ, by the formua for the action of Hecke oerator T 2 we find that ( ( 1 b( 2 n + χ ( k 1 k n b(n + χ 2 ( 2k 1 b(n/ 2 0 mod λ. By the assumtion, is not excetiona. Hence Theorem 3.3 imies and for each T (λ, r, t #T (λ, r, t, = # T (λ, r, t } f,λ og (4.2 b( 2 n χ ( k 1 ( ( 1 k n b(n χ 2 ( 2k 1 b(n/ 2 mod λ. Let κ be the number as in the roof of Lemma [?]. Now, we choose (r 0, t 0 satisfying the foowing roerties: (1 N 2 t 0, (r 0, t 0 = 1, χ (r 0 = 1. ( ( 1 k n (2 If is a rime with r 0 mod t 0, then = 1 for any 1 n κ with (n, N 2 = 1. ( ( 1 k (3 For each rime r 0 mod t 0 we have = 1. (4 Each rime r 0 mod t 0 has the roerty that mod λ. If T (λ, r 0, t 0 is a sufficienty arge rime, for a 1 n κ with (n, N 2 = 1, one has ( ( 1 u (n = b 0 ( 2 n k 1 k n b 0 (n 2k 1 b 0 (n/ 2 = k 1 b 0 (n = k 1 v (n mod λ. By the reation (4.2, we have aso v ( 3 = b 0 ( 2 k 1 b 0 ( mod λ, u ( 3 = b 0 ( 4 2k 1 b 0 ( mod λ. Therefore by assumtion and the choice of (r 0, t 0, Hence k 1 v ( 3 u ( 3 2k 2 (1 + b 0 ( 0 mod λ. ord λ (U g 0 k 1 V g 0 < +. By Theorem 4.1 and Lemma 4.3, there exists a integer n such that 1 n (k + 1/2[Γ 0 (1 : Γ 0 (N 2 ]/12 = κ( + 1, (n, = 1

9 INIVISIBILITY OF CENTRAL VALUES OF L-FUNCTIONS FOR MOULAR FORMS 9 and b 0 (n = u (n k 1 v (n = 0 mod λ. In articuar, et sf be the square-free art of = n, then b 0 ( sq λ = 1. Now the emma foows from the same argument with the roof of the revious emma using Theorem 3.3. Proof of Theorem 1.1. Now we give the roof of Theorem 1.1. Let g(z = b(nq n S k+1/2 (N, χ be the eigenform given in Theorem 2.1 for f. By reacing f by a suitabe quadratic twist of f if necessary, we may assume that ε = δ(f, where ε is the sign of the functiona equation of L(f, s. By the resut of Friedberg and Hoffstein [4], we can take an integer such that δ(f > 0, (, 2N = 1 and b( 0. In articuar, for amost a finite aces λ of E we have b( λ = 1. Thus by Lemmas 4.4, 4.5, Theorem 2.1 and Theorem 3.3, for a but finitey many rimes λ we have } L(f χ, k k 1/2 0 # S( δ(f > 0, (, = 1 and = 1 Ω f f,λ og. λ This cometes the roof. 5. Indivisibiity for the non-centra critica vaues In this section, we consider a secia case for non-centra vaues of L-functions for moduar forms. We fix a rime greater than 7 and et f = a(nqn be a normaized Hecke eigenform of weight + 1 for SL 2 (Z. Let λ be a rime in a number fied E. We assume that the integer ring of E contains a Fourier coefficients of f and choose a eriod Ω ± f as in Ash-Stevens [1, Theorem 4.5]. Then for any irichet character χ, the quotient τ(χ 1 L(f χ, 1 (2πiΩ ± is an integer in E λ (χ where τ is the Gauss sum and ± = χ( 1. f Theorem 5.1. Let λ be a rime in E above. We assume the foowing conditions. (1 There exists a unique eigenform F of weight 2 for Γ 0 ( such that F f mod λ. (2 is not excetiona. (3 There exists an square-free negative integer d 0 such that (d 0, 2 = 1, χ d0 ( = ε(f, where ε(f is the sign of functiona equation of L(F, s and Then we have # S( L(f χ d0, 1 d 0 (2πiΩ ± f L(f χ, 1 (2πiΩ ± f 0 mod λ. 0 } mod λ f,λ og.

10 10 MASATAKA CHIA For the roof, we reca a resut of Ash and Stevens. Theorem 5.2 (Ash-Stevens, [1]. Let k be a ositive integer ess than + 2 and f = a(nqn S k (Γ 0 (1 an eigenform satisfying the assumtions of Theorem 5.1. We fix a rime λ above in a number fied E which contains a Fourier coefficients of f. Assume that (1 There exists a rime q satisfying a(q q k mod λ. (2 There exists an unique eigenform F S 2 (Γ 1 ( such that f F mod λ. Then there exists a comex number Ω ± F such that for any irichet character χ satisfying (cond χ, = 1, we have τ(χ 1 L(f χ, 1 (2πiΩ ± f τ(χ 1 L(F χ, 1 (2πiΩ ± F mod λ. Now we rove Theorem 5.1. By the Kohnen-Zagier formua [6], there exists an eigenform g(z = b(nq n S 3/2 (Γ 0 (4 such that for any negative square-free integer satisfying b( 2 = 2 π g, g F, F L(F χ, 1, ( = ε(f, where, is the Petersson inner roduct. We can normaize g by the reation F,F = g,g Ω± f. Taking a inear combination of twists of g, one may assume b( = 0 if ( ε(f and < 0. From the assumtions of the theorem, is not excetiona. This imies the existence of a rime q satisfying a(q q k mod λ, therefore the assumtions of Theorem 5.1 imies the assumtions of Theorem 5.2. Since τ(χ 1 = ±1/, one can see that L(f χ, 1 (2πiΩ ± f L(F χ, 1 (2πiΩ ± F with a λ-adic unit c. By the assumtion (3, we have ord λ ( L(f χ d0, 1 d 0 (2πiΩ ± f = b( 2 c mod λ therefore ord λ (b(d 0 = minord λ (b(n n : square-free, χ d0 ( = ε(f}. Hence Lemma 4.5 imies thus we have = 0, # S( χ ( = ε(f, ord λ (b( = s } f,λ og, # S( This cometes the roof. L(f χ, 1 (2πiΩ ± f 0 } mod λ f,λ og. Remark 5.3. Lemma 4.5 states ony for g given in Theorem 2.1, but one can show the simiar resut for any eigenform g S k+1/2 (N, χ if k 2 (S 3 (N, χ if k = 1 2 corresonding to some eigenform f S 2k (Γ 0 (M under the Shimura corresondence.

11 INIVISIBILITY OF CENTRAL VALUES OF L-FUNCTIONS FOR MOULAR FORMS 11 Exame 5.4. Let and F = q f = = q (1 q n 24 S 12 (Γ 0 (1 (1 q n 2 (1 q 11n 2 S 2 (Γ 0 (11. Then it is we-known that f F mod 11, dims 2 (Γ 0 (11 = 1 and the mod 11 Gaois reresentation associated to f is surjective. Moreover one can check that L( χ 3, 1 Ω + χ 3 = mod 11 by using MAGMA. So the assumtions of Theorem 5.1 are satisfied for f =. Hence we have # S( L( χ, 1 } (2πiΩ ± 0 mod 11 og. References [1] A. Ash and G. Stevens, Moduar forms in characteristic and secia vaues of their L-functions, uke Math. 53 (1986, [2] J. H. Bruinier, Non-vanishing of moduo of Fourier coefficients of haf-integra weight moduar forms, uke Math. J. 98 (1999, [3] P. eigne, Formes moduaires et rerésentations -adiques, Sém. Bourbaki, éx. 355, Lect. Notes in Math. 179 (1969, [4] S. Friedberg, J. Hoffstein, Nonvanishing theorems for automorhic L-functions on GL(2, Ann. Math. 142 (1995, [5] K. James and K. Ono, Semer grous of quadratic twists of eitic curves, Math. Ann. 314 (1999, [6] W. Kohnen, Fourier coefficients of moduar forms of haf-integra weight, Math. Ann. 271 (1985, [7] W. Kohnen and K. Ono, Indivisibiity of cass numbers of imaginary quadratic fieds and orders of Tate-Shafarevich grous of eitic curves with comex mutiication, Invent. Math. 135 (1999, [8] B. Mazur, J. Tate, J. Taitebaum, On -adic anaogues of the conjectures of Birch and Swinnerton- yer, Invent. Math. 84 (1986, [9] K. Ono and C. Skinner, Non-vanishing of quadratic twists of L-functions, Invent. Math. 134 (1998, [10] K. Ono and C. Skinner, Fourier coefficients of haf-integra weight moduar forms mod, Ann. of Math. 147 (1998, [11] K. Ribet, Gaois reresentations attached to eigenforms with Nebentyus, Lect. Notes in Math. 601 (1977, [12] K. Ribet, On -adic reresentations attached to moduar forms. II, Gasgow Math. J. 27 (1985, [13] J.-P. Serre, Congruences et formes moduaires (d ares H. P. F. Swinnerton-yer, Lecture Notes in Math. 317 (1973, [14] G. Shimura, On moduar forms of haf-integra weight, Ann. Math. 197 (1973, [15] J. Sturm, On the congruence of moduar forms, Sringer Lect. Notes Math (1984, [16] H. P. F. Swinnerton-yer, On -adic reresentations and congruences for coefficients of moduar forms, Lecture Note in Math. 350 (1973, [17] J. L. Wadsurger, Sur es coefficients de Fourier des formes moduaires de oids demi-entier, J. Math. Pures et A. 60 (1981, E-mai address: chida@math.kyoto-u.ac.j

12 12 MASATAKA CHIA Graduate Schoo of Mathematics, Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto, Jaan,