Modular Forms Whose Coefficients Possess Multiplicative Properties, II

Annals of Mathematics

Modular Forms Whose Coefficients Possess Multiplicative Properties, IIAuthor(s): Morris NewmanSource: Annals of Mathematics, Second Series, Vol. 75, No. 2 (Mar., 1962), pp. 242-250Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1970172 .

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ANNALE OF IAIATHMATICS

Vol. 75, No. 2, March, 1962 Printed in Japan

MODULAR FORMS WHOSE COEFFICIENTS POSSESS MULTIPLICATIVE PROPERTIES, II

BY MORRIS NEWMAN*

(Received March, 3, 1961)

Introduction

In a recent article [6], a study was made of the coefficients of the modular forms

X (r+sq )24J1( 1 - Xn)r(1 - Xnq)

for certain integers r, s and a prime q. The method employed was to construct suitable entire modular functions on the congruence subgroup F0(q) of the modular group F, and then to determine when these functions were of total valence zero' (and so constant). In the cases considered, r and s were of the same parity so that the forms were of integral dimension. The purpose of this article is to discuss the cases when r and s are of opposite parity, when the forms are of half-integral dimension. The differ- ences arise as follows: When r and s are of the same parity, it is possible to construct appropriate entire modular functions on F0(pq) vanishing nowhere throughout the interior of the upper half-plane, where p is a prime satisfying certain congruence conditions, from which the functions on F0(q) may be obtained. This is equivalent to the existence of a Hecke operator of level p having certain modular forms as eigen functions. When r and s are of opposite parity this is not possible, but such functions may be constructed for F0(p2q), where now the prime p is unrestricted, and the functions on F0(q) obtained from these. This is equivalent to the existence of a Hecke operator of level p2 and has been discussed by Hecke in [9]. This operator also exists when r and s are of the same parity. We shall employ the "subgroup" method in what follows. The author is indebted to the referee for his critical reading of the manuscript which removed a number of errors and clarified the exposition at many points.

Notation and preliminary results

For the sake of completeness we reproduce briefly some of the material of [6]. F is the group of matrices

* The preparation of this paper was supported (in part) by the Office of Naval Research. 1 More generally, if p is the minimal valence that can occur for a non-constant automor-

phic function on a group of genus g, then p = 1 when g 0 0 and p ?2 when g>0. Thus to prove that an automorphic function is constant, it is sufficient to show that, for g = 0 it is of total valence 0, and for g > 0 of total valence at most 1.

242



MODULAR FORMS 243

[a b] a, b, c, d integers, ad-be =1 .

The congruence subgroups F0(q) are defined by c 0 (mod q). We set

B(z') = r(ZT)ys(qz.)

where

( -.) = xl/241(1 - XI,) , x = exp 2wiz is the Dedekind eta-function, an entire modular form on P of dimension - 1/2, analytic throughout the interior of the upper z half-plane and zero- free there. Here, and in what follows, products will be extended from 1 to co and sums from 0 to co, unless otherwise indicated. We set

9(z) = I(1 - Xn)r(j - xnQ)s = Ec(n)x

= (r + s), 2

t = (r + sq)/24 . Thus B(z) = xt9(z).

If f = f(r, s) is a function of the variables r, s, we shall write f* = f(s, r). As in [6] we give a transformation formula for B(z).

LEMMA 1. Suppose that

A= e a ] b e q) , c > O. (a, 6) = 1.

Then

( 1 ) B( (Az)) = {- i(qcz + d)}j ( qrc2E\) exp 2ia (bt - ct* + a B(z-) Ifa 1 4/

Here (qrC26/I a 1) is the generalized Legendre-Jacobi symbol. It was not made clear in the statement of the corresponding lemma in [6] that S was supposed to be integral there, so that the term (c/l a 1)26 appearing in (1) was not present. The proof is like that of the corresponding lemma in [6] and we omit it.

We note also that the transformation formula for the Dedekind function implies

(2) B(Sz-) = exp 2wit- B(z), S [1 1]1 (3) TqB(z) = B(-l/qz) = (-?iz-)qqr22B*(z-)

Construction of the functions

We suppose now that p and q are distinct primes. It is known [3] that if



244 MORRIS NEWMAN

W = [1 1]

then rF(p2q) is subgroup of rF(pq) of index p, and

(4) W-kPq Ok <p 1,

forms a complete set of right coset representatives for F0(pq) modulo rP(p2q). Thus if g(r) is a function on rP(p2q) then

( 5) Sk-Og( Wkpqr

is a function on ro(pq) (see [4, p. 315]). We choose

(6) g(z) - B(p) - 5 72() } { Then Theorem 1 of [2] implies

LEMMA 2. If A = t(p2 - 1) = (r + sq)(p2 - 1)/24, and A* = t*(p2 1) - (s + rq) (p2 - 1)/24 are integers, then the function g(r) defined above is an entire modular function on ro(p2q) vanishing nowhere throughout the interior of the upper r half-plane.

Thus if p>3, A and A* are integers for all integral r, s. If p = 2 then A and A* are integers provided that r + sq _ s + rq 0 0 (mod 8); and if p=3 then A and A* are integers provided that r+sq-=s +rq- 0(mod 3).

It is also known [3] that ro(pq) is a subgroup of ro(q) of index p + 1 and that

( 7 ) W-kqq 0 _ k _ p - 1 R =[q p] pop - qoq=1

forms a complete set of right coset representatives for ro(q) modulo ro(pq). Thus if h(r) is a function on ro(pq) then

EIk"h( W-kgZ) + h(Rz) is a function on ro(q) [4, p. 315]. For h(r) we choose the function defined by (5), where g(r) is defined by (6), and we obtain

LEMMA 3. Define

(8) G1(r) 2= - g( W-kz),

(9) GAZr) = EkP-g( W-kPlRz),

(10) G(r) = G1(r) + G2(t).

Then G(r) is an entire modular function on ro(q).



MODULAR FORMS 245

Since R e rP(q) and G(r) is a function on rF(q), we have that G(R-'r) = G(z). Define

(11) F,(r) = Gl(R-'z) = Ep2-g(W-k2Rl),

(12) F2(r) = G2(R' z1) = 'EL2g( W-kpgZ;) Then we have

LEMMA 4. The function G(r) defined above satisfies

(13) G(r) = F,(r) + F2(r) where both F,(r) and F2(z) are entire modular functions on ro(pq).

The valence of the functions

Since q is prime, the fundamental region of ro(q) has just the parabolic points r = i co, r =0; and since G(r) is an entire modular function on ro(q), its singularities (if any) must occur at these points, and are at most polar when measured by the proper uniformizing variables. Thus it is only necessary to know the behavior of G(r) at these points to determine its total valence throughout the fundamental region of Po(q). We study G(r) directly at r = i co in terms of the uniformizing variable x = exp 22riz, but find it more convenient to study G(r) at T = 0 by studying TqG(t) at r = i co in terms of x. The substitution t Tq'r = -1/qz permutes icc and 0, and it is known (see [5]) that TqG(Z) is an entire modular function on ro(q) if and only if G(r) is an entire modular function on FO(q).

A certain amount of routine calculation involving the transformation formulae (1), (2), (3) is unavoidable. Since similar calculations have already been performed in [6] and no new ideas are involved, it is perhaps sufficient to give only the results.

At r = i co we use the representation (13). Making use of the transformation formulae (1), (2), (3) and following the line of reasoning set down in [6], it is not difficult to obtain

LEMMA 5. The functions Fl(z), F2(z) satisfy

B r + 24k) (14) F,(r) = p-28p2-1 p2

B~p~z)+ ~ ~( k \28 Bz + k (15) F2(Z) = B(pZ) )+ PP- p-( k B( p

where



246 MORRIS NEWMAN

()2()S(lp)

To study G(r) at z = 0 we apply the transformation Tq and study TqG(72) at z = ico. Using the representation (10) we find

LEMMA 6. The funtions Gl(z-), G2(Z) satisfy (16) TqG,(z) = F (r)z)

(17) TqG2(Z) = F2*(7) Thus by (10), (13), (16) and (17) we obtain

LEMMA 7. G(r) satisfies (18) TqG(z) = G*(z)

Lemma 7 shows that it is only necessary to determine the Fourier ex- pansion of G(r) at r = ico in terms of the uniformizing variable x in order to have complete information about the singularities of G(r). For- mulae (14) and (15) allow us to do this and we find that

(19) 9(z)G(z) = p2-2e Ic(np2 + A)xn + 7pl-e I

np+A?O c(np + A)xn,,+A

-yp-eEc(n)xn + EC(f A)Xn 2s even

( (Z)G(z) = p2-28Ec(np2 + A)Xn + p /2- ( n A>)n

(20) + 70(ln, A)Xn. 2s odd.

Here

(21) = ( q)S(l-

(22) 0 = (_ 1)l/2-e2qs

From (19), (20), and Lemma 7, we obtain our first principal result concerning the valence of G(z):

THEOREM 1. At ioo, G(r) has a pole of order [A/p2] at most if A ? 0, and a pole of order -A if A < 0. At 0, G(r) has a pole of order [A*/p2] at most if A* > 0, and a pole of order -A* if A* < 0.

A word or two concerning the case p = q is in order. Lemma 2 remains true when p = q, so that the function B(q2z)/B(z) is a function on FO(q3), provided that (r + sq)(q2 -1) (s + rq)(q2 -1) 0 (mod 24). Fo(q3) is a subgroup of F0(q) of index q2 and



MODULAR FORMS 247

W-kq9 0 < O < q2_1

forms a complete set of right coset representatives for F0(q) modulo rF(q3). Hence the function

,2-i B(q2 W-kZ-)

B( W-k rZ) is an entire modular function on FO(q), which can be discussed in a man- ner similar to those treated above. We forego this discussion here.

Conclusions

Now that we have determined the valence of G(r) throughout the fundamental region of Po(q), we determine when G(z) is constant by using the fact that a non-constant automorphic function takes each value equally often in the fundamental region (cusps included). Thus for example, if G(z) is pole-free throughout the fundamental region of FO(q), then it is constant. Then Theorem 1 implies our second principal result:

THEOREM 2. Suppose that

(23) O? A< p2, 0 <A* < p2

Then G(z) is constant.2 There is no difficulty in determining all instances of (23), which are

numerous. Since we are primarily interested here in the case 2s odd, we do not discuss the case 2s even, except to say that G(z) is constant, then for all the values of r and s given in Theorem 3 of [6]. Further, we assume that neither r nor s vanishes, since the identities so obtained have already been discussed in [5]. Following the method set down in [6], we find

THEOREM 3. The values of r and s such that rs # 0, r Es(mod 2), and G(z) = c (a constant) are just those given in the following table, where with an entry (r, s) we must also include (s, r). For these values, the coefficients c(n) of p(z) satisfy

(24) c(np2 + A) - y7c(n) + p2e-2c(nl O 0

where

7n = p2e-2 - (c -)0pg3/2( n)

2 The referee points out that if q = 11 or q > 13 and 0 _ min (A, A*) < p2, max (A, A*)< 2p2, then again G(z) is constant. This follows from footnote 1 and the fact that the genus of ro(q) is given by x - 1, x, x, x + 1 for q 12x + 1, 12x + 5, 12x + 7, 12x + 11 respectively.



248 MORRIS NEWMAN

and

0 = (-1)1/2-2qs

q 2 3 5 7 11 17 19

r 2 2 3 4 4 4 2 3 2 2 2 2 2

s 1-1 2 1 3-1 1 2 1 1 1 1 1

r 5 5 5 6 6 6 4 4 3 3 4 s 2 4-2 1 3 5 1 3 2 2 1

r 6 6 7 7 7 7 4 5 4 s -1-3 2 4 6-2 -1 2 1

r 8 8 8 8 8 8 5 6 4 s 1 3 5 7 -1 -3 4 1 3

r 9 9 9 9 9 10 6 6 s 2 4 6-2-4 1 3 5

r 10 10 10 10 11 11 6 7 s 3-1 -3 -5 2.-2 -1 2

r 11 12 12 12 13 13 7 8 s -4 -1 -3 -5 -2 -4 -2 -1

r 13 14 14 14 15 s -6 -3 -5 -7 -6

Certain special cases are of particular interest. Thus for q = 2, r = 2k, s =-k, k = 1, 3, 5, 7 the function p(zT) becomes

1(1 _ Xn)2k(1 _ X2-)-k = {H1 _

}

= -E00 _00(_j) X }

= E(-1)"rk(n)x

where rk(n) is the number of representations of n as the sum of k squares, counted in the usual way. Identity (24) then becomes

rk(np ) = {1 + pk-2 - (_ 1)(P-l(k-1)/4p(k-3)/2( n rk() (25) V

- Pk2rk(-2h) k = 1,3,5,7p.

The case k=3 is given by G. Pall in [8] and a proof of (25) for k=3, 5, 7 is given by J. van Lint in [1] which makes use of the Hecke operators. See also the author's paper [7], where identities of type (25) are derived for



MODULAR FORMS 249

all positive integral k. Choosing q = 2, r =-k, s = 2k, k = 1, 3, 5, 7, the function q(z)

becomes

1(1 _ x)-k(l - X2n)2k = {ll }

= {IX(n2+n)/2}1k

= Etk(n)x ,

where tk(n) is the number of representions of n as the sum of k triangular numbers. Then (24) becomes

(26) tk(np2 + A) = r77tk(n) p2 tk( 2) k = 1, 3, 5, 7

where

A = k(p2 -1)/8, 7n = - -

(Ok)p(k-3)/2(f A),

c is a constant, and 0k = 2( 1)(1-k)2.

We are going to examine recurrence formula (24) somewhat more close- ly. Assume that (n, 6) = 1 so that t(n2 - 1) = An, is an integer. Define

d(n) = c(^,)

Let p be a prime > 3, p # q. Then (24) becomes

(27) d(np) = 7A&nd(n) - p28-2d( n)

where

yin = p28-2-C 0 () 3/2(t(n2 _ p2)-

{p28-2C pIn

p2e-2 c- ()P8-3/2(6)(24t) (P. n) = 1 .

Set

2e-2 a ( )pe-3/2 (6 24t)

It is not quite evident that (27) implies that d(n) is multiplicative for (n, 6q) = 1, but this is indeed true. The proof is straightforward and we omit it.

If we now define



250 MORRIS NEWMAN

(28) D(z) = E(, 6q,=ld(n)n-z

then the fact that d(n) is multiplicative for (n, 6q) = 1 implies

THEOREM 4. The Dirichlet series D(z) defined by (28) above satisfies

(29) D(z) = J(6q)=l (1-([) -(3/2)-z)(1 - fxpZ + p2e-2-2Z)-

where

, = ( _ 1)j12-eqs * 3(r + sq)

a= d(p) + (L)pe-3/2

This factorization is similar to, but somewhat more complicated than, the ones given in [6].

NATIONAL BUREAU OF STANDARDS, WASHINGTON

REFERENCES

1. J. VAN LINT, Hecke operators and Euler products, Dissertation, University of Utrecht, 1957.

2. M. NEWMAN, Construction and application of a class of modular functions, II, Proc. London Math. Soc., 9 (1959), 373-387.

3. , Structure theorems for modular subgroups, Duke Math. J., 22 (1955), 25-32. 4. , Remarks on some modular identities, Trans. Amer. Math. Soc., 73 (1952),

313-320. 5. , Further identities and congruences for the coefficients of modular forms,

Canad. J. Math., 10 (1958), 577-586. 6. , Modular forms whose coefficients possess multiplicative properties, Ann. of

Math., 70 (1959), 478-489. 7. , Subgroups of the modular group and sums of squares, Amer. J. Math., 82

(1960), 761-778. 8. G. PALL, On the arithmetic of quaternions, Trans. Amer. Math. Soc., 47 (1940), 487-500. 9. E. HECKE, Neuere Fortschritte in der Theorie der elliptischen Modulfunktionen, Compte

rendus du Congres international des Mathematiciens, Oslo (1936), 140-156.



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Modular Forms Whose Coefficients Possess Multiplicative Properties, II