Modular Forms Whose Coefficients Possess Multiplicative Properties

Annals of Mathematics

Modular Forms Whose Coefficients Possess Multiplicative PropertiesAuthor(s): Morris NewmanSource: Annals of Mathematics, Second Series, Vol. 70, No. 3 (Nov., 1959), pp. 478-489Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1970326 .

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ANNALS OF MIATHEMATICS

Vol. 70, No. 3, November, 1959 Printed in Japan

MODULAR FORMS WHOSE COEFFICIENTS POSSESS MULTIPLICATIVE PROPERTIES

BY MORRIS NEWMAN*

(Received February 2, 1959) (Revised June 29, 1959)

In this paper we study the coefficients of the modular forms

X24(r sq)fJ (1 -xn)r(l - Xnq)s

where r, s are non-zero integers and q is a prime. From these coefficients we shall form modified Dirichlet series extended over 1 and integers divisible only by the members of a fixed set P of primes, and we shall be interested in determining when these modified Dirichlet series possess modified Euler products, extended only over P. Many of the identities so obtained are classical identities in the analytic theory of numbers. This happens for 147 sets of values of r and s, which are given in Theorem 3. In some of these instances the modified Dirichlet series and Euler products can be replaced by ordinary Dirichlet series and Euler products; in others this is not possible. When r = 0 or s = 0 these forms have been considered by Mordell in [5] and the author in [8]. Similar discussions for related modular forms have been given by Hecke, Rankin and others, which the reader may find in the articles referred to in this paper.

The method used is the 'subgroup' method, which concerns itself with functions invariant with respect to the substitutions of a suitable subgroup of the modular group F, regarded as linear fractional trans- formations. Here r is the group of matrices

A[~c d ] with rational integral elements and determinant 1. We shall be interested in the subgroups 1F(m) of F, where m is a positive integer and A e rF(m) if and only if

c 0 (mod in).

If z is a complex number then by Az we shall mean az- + b c: + d

All products will extend from 1 to o and all sums from 0 to oo, unless otherwise indicated.

* The preparation of this paper was supported (in part) by the Office of Naval Research. 47R



MODULAR FORMS 479

We set

B(z) = Br s(Z) = r (z)YrS(qz.)

where

Y(z-) exp 2 fJ (1X-n) x = exp 2wiTc 12

is the Dedekind modular form. Then B(z-) is an entire modular form of level q and dimension -e, where

? = (r + s) .

The (-function has been extensively studied and we shall make use of its transformation equation for substitutions of F to determine the transformation equations for B(z). A good discussion of this function may be found in [12] and the most important properties of the associated Dedekind sum s(a, c) are given by Rademacher and Whiteman in [10]. The author has shown in [6] that if

A =[a d]be a >0, c >0, (a, 6) = 1

then

( 1 ) s(a, c) - a + d la(c-b-3)- ' - (1 )- } (mod 2). 12C 12 2 ka)

Here (c/a) is the generalized Legendre-Jacobi symbol. We set

(Z.) (r jr) - II (1 - xn)r(1 _ Xnq)s c(n)X = E Cr,,(n)X

t r + sq 24

Thus

B(z) = xtp(z) .

If f = f(r, s) is a function of the variables r, s we shall write

f =f(s, r)

so that

t* s+ rq 24

We now prove a transformation formula for B(z).



480 MORRIS NEWMAN

LEMMA 1. Suppose that

A= aL d]eF0(q), c > 0, (a, 6)

Then

(2) B(Az-) {-i(qcz- q ) exp 27ria(bt - ct* + + ) B(z).

PROOF. Put

Ao Lc dbje ti.

Then by the transformation formula for Y(z-),

B(Az-) = yr(Az))s(AOqz) = {-i(qcz + d)}2 exp (-iwrX) - B(z-), where

X=r{s(a, qc)- a +d} ? s {(a, c) acd}

Suppose first that a > 0. Then (1) implies that

A r{ a(qc - b -3)- -2i1 _a))

+ s{ Il a(c - qb -3)- - 1 - c (mod 2)

and after simplification this becomes

X -2a(ct* - bt - +) + r q )-1 (mod 2)

which implies (2). If a < 0, we use instead the matrix

A-[-a bd L qc -dH

which also belongs to FO(q) and notice that for the corresponding X, X =- X since s(-a, c) =- s(a, c). The lemma is thus proved.

We note also that for the matrix

r0 11

we have B(Sz) = exp 2writ - B(zj;

and if we define the transformation Tq by

T, ftz) = ft -1 / q )



MODULAR FORMS 481

then r

(3) TqB(z)= (= _ .q2B*(z)

Formula (3) follows from the fact that if

T 1 0o then

Y(Tz) - (-iz))(z)

a special case of the transformation formula for r(z-). We now suppose that p is a prime different from q, and we shall also

assume that p > 3, although this is not essential. We put

Or~) g(-) B(pz-) g( = grs(zT) =B(z).

Then we may write

g(z) - { y((pz) r(q) } J r(pqz) As

Set

8 t(p - 1)

so that

8* t*(p-1)

Then Theorem 1 of [6] implies

LEMMA 2. g(z) is an entire modular function on Jo(pq), provided that ?, 8 and 8* are integers.

We shall assume from now on that this is indeed the case. Since p and q are relatively prime, integers po and qo may be determined

so that P -q0q = 1

Then the matrix

R - [qoq pop]

belongs to 1o(q). It is known (see [7]) that Fo(pq) is of index p + 1 in Fo(q), and that

Rk [q 1' < k _ p-1, RP = R



482 MORRIS NEWMAN

is a complete set of right coset representatives for F,(q) modulo F0(pq). Define

G(z) - Gr ,(Z) = Ek=O g(Rk)

Then Theorem 2.2 of [8] implies

LEMMA 3. G(z-) is an entire modular function on ro(q). We will study the function G(z-). Since G(z) is an entire modular func-

tion its singularities (if any) must occur at the two parabolic vertices j = i co, z = 0 of the fundamental region of Fo(q). At z = ico the singu-

larity is at most polar when measured by the uniformizing variable x = exp 2riz-. The function TqG(z) is also an entire modular function on Fo(q) (see [9]) and z = ioo for this function corresponds to z = 0 for G(z). The singularity of G(z-) at z = 0 may therefore be studied by considering the singularity of TqG(z) at z = ico, which is also at most polar when measured in terms of x.

We consider z = zoo first. Suppose that (k, p) = 1. We have

pt=[-kq 1

and since (p, kq) = 1 and p > 3 we may write

[ ] - [p 24ko]-[1 -24ko] L-kq 1 -kq -ki2 0 -P

where pk, - 24qkko = 1.

Thus when k runs over a reduced set of residues modulo p, so does ko. If we set

A-=[p 24ko] k Lkq - k,

then we have

B(Ak Z + 24ko) g(RJ) - B

Now the transformation formula (2) may be used to simplify this expression, and we find that

B Z + 24ko)

(4) g(Rj7) =P B()



MODULAR FORMS 483

where

=(12'e(p-1)( qr)

We shall write k: p in a summation to indicate that k runs over a reduced set of residues modulo p. Then (4) implies that

B(L

+ 24k)

Aok:p g(Rkj) = YP Tk:p B(z)

Also,

PRPZ tqoq popX [_qoq po p Mp

and Mp e F0(q). We find easily from (2) that

B(I) g(Rp'r) = p? P

Thus we can say

LEMMA 4.

BQr + 24k)

(5) G(r) - B(pz)+ P B(z) 5) G~~~zj B(z-) E= ~

In this form it is evident that G(z-) is derived from g(z) by the application of a Hecke operator.

We go on now to z = O. Suppose that O < k < p-1. Then for the representatives Rk, we have

RkT = TS Thus

g(Jh EP-) g(TSqkZ.) Ep-1 B( p- B(T(z + qk))

and so

B(T qz- + qk) Tq >Ij g(Rkj)

= p-0 ~?k Tq=Ek _o g(RtT) S B(T(qz + qk))

Making use of the transformation formula (3) and noting that 24k runs over a complete set of residues modulo p with k, we find that



484 MORRIS NEWMAN

B* ?+24k)

( 6 ) Tq Ekog(Rer) =P2 SkLB*r

Further, we have that

P [Lqoq pop] L[q0q po]Tpz= MPTpz, and Mp e U'o(q). Thus

Tqg(R~Z.) B (MP Tpqz-) B(Rp Tqz)

and this expression may be simplified by the transformation formulas (2) and (3) to give (7) Tqg(Rpz) = -g*(z)

Combining (6) and (7) we obtain

LEMMA 5. G(c) satisfies (8) TqG(z) =yG*(z) where

22(P-1) q )

Lemma 5 shows that it is only necessary to obtain the Fourier expansion of G(z) at z = ioo in order to have complete information about the singularities of G(z). If we define 80 as the least non-negative residue of 8 modulo p and 83 by the relationship 8% = 8-p81, then it is easy to find the expansion

(9) G(z) {x c (C(2 )X ?yp1-x-i E c(np, +

We note that if 8 > 0 then 8, = 8-p[8/p] and 81 = [8/p]. This expansion and Lemma 5 now give our first theorem.

THEOREM 1. At ioo, G(z-) has a pole of order [8/p] at most if 3 ? 0 and a pole of order -8 if 8 < 0. At 0, G(z-) has a pole of order [8*lp] at most if 8*>0and a pole of order -8*if 8*< 0.

Now we make the remark that if G(z) is pole-free at both ico and 0 then G(z-) must be constant, by a simple application of Liouville's theorem. This yields our second theorem.

THEOREM 2. Suppose that

(10) 0 < 8 < p, O < _* < P where 8 = t(p - 1), 8* = t*(p - 1). Then G(z) is constant and for the coefficients c(n) we have the identity



MODULAR FORMS 485

(11) c(np + 8) = /3c(n) - Ype-l( n-8

where

= C(8) + p1C 8 C(8) 8 > c

It is possible for G(z) to be constant without condition (1i0) holding, if the coefficients of the pole terms vanish. We do not consider these accidental' situations here. We must therefore determine all instances of (10), which are numerous.

We set r = +? n,

$ = ? - so

Then

4P 1(q + 1)? - (q - 1)?s} 24

8* P - 1 {(q + 1)s + (q - 1)e0}

24

The inequalities (10) then become

0 < (q + 1i)?- (q - 1i)eo < 24 P p-i

(12) 0 < (q + 1)? + (q - 1)e0o < 24 P

p-i Adding, we find that

(13) 0 < s 24 q?1 p-i

Further, we see that if (?, so) yields a solution (r, s) then (, -so) yields the solution (s, r). There is therefore no loss of generality in assuming so > 0, provided that we include with the solutions (r, s) so obtained the solutions (s, r). Now the pair (12) may be rewritten as

-(q + 1)e ? (q - 1)eo < 24 P (q + 1)s p-i

(q + 1)e - 24 P < (q - 1)e0o < (q + 1)? p-1 But we know that so > 0 and taking (13) into account these reduce to the conditions



486 MORRIS NEWMAN

? < (q- 1)s < 24 P I (q + 1)s (14)

o < (q - 1)so < (q + 1)s

Now (13) shows immediately that for q > 23 s must be 0 and (14) then implies that s0 is 0 as well, giving only the trivial solution r = s 0 O. Thus q < 23. This fact coupled with (13) and (14) makes possible a simple enumeration of cases, and we obtain

THEOREM 3. The values of r and s for which G(c) is constant are just those given in the following table, where with an entry (r, s) we must also include (s, r), and m is an integer such that p satisfies

(15) p -1 (mod m) .

q r S m r S m r S m r S m r S m

2 1 1 8 5 5 8 7 -1 24 9 5 24 11 -5 24 2 2 4 5 -1 8 7 -3 24 9 -1 24 12 -2 12 3 1 24 6 2 12 8 2 4 9 -3 8 12 -4 6 3 3 8 6 4 12 8 4 6 10 2 12 12 -6 4

3 -1 24 6 6 4 8 6 12 10 4 4 13 -3 24 4 2 12 6 -2 12 8 8 2 10 -2 4 13 -5 8 4 4 2 7 1 8 8 -2 12 10 -4 12 14 -4 4 4 -2 4 7 3 24 8 -4 2 11 1 24 14 -6 12 5 1 24 7 5 24 9 1 24 11 -1 8 15 -7 24 5 3 24 7 7 8 9 3 8 11 -3 24 16 -8 2

3 1 1 6 3 -1 6 5 3 12 6 4 12 7 3 6 2 2 6 4 2 12 5 5 6 6 6 2 7 -1 6 3 1 12 4 4 6 5 -1 12 6 -2 6 8 -2 12 3 3 2 5 1 6 6 2 6 7 1 12 9 -3 2

5 1 1 4 3 1 6 4 2 12 4 4 2 5 -1 2 2 2 2 3 3 4

7 1 1 6 2 2 6 3 1 12 3 3 2

11 1 1 2 2 2 2

13 1 1 12

17 1 1 4

19 1 1 6

23 1 1 2



MODULAR FORMS 487

If - = E c(n)xn is any of the functions of Theorem 3, then P,. will denote the class of primes p satisfying the congruence (15) of the theorem. We also denote by Q, the totality of integers divisible only by primes of P9, and 1. We set

d(n) = c(t(n -1)) , n e Q, Then, as is easy to verify, the numbers t(n - 1) are integers and (11) implies that for 8 > 0,

(16) d(np) = d(n)d(p) - ypei d( nf), neQ, peP,; P

while if 8 = 0

(17) c(np) - c(n) = ypl1-{c(n) - cQ ), p e P, .

Now (16) implies in the usual way that when 8 > 0, d(n) is multiplicative for n e Q, and the associated modified Dirichlet series possesses a modified Euler product. (See for example the discussion in Hardy's book [2].) The functions with 8 = 0 may be discussed as follows: Replace n by npk in (17), obtaining

c(npt+1) - c(npk) = yp'-`{c(np) -c(np1-')!

Forming the product of the two sides over k we obtain

c(npk) - c(npk1l) =(Yps-l)k{c(n) - c .

Summing over k and assuming that n and p are coprime we obtain

(18) c(np) p)+l1c(n) (n, p)= 1

If we now set

b(n) = c(n) c(1)

then (18) implies that b(n) is multiplicative for n e Q,. In this case the associated modified Dirichlet series also possesses a modified Euler product, and we have

THEOREM 4. If t > 0, then

(19) EnCQ, d(n)n-z = PE {1- d(p)p-z + yp-l2Z}l

If t = 0, then



488 MORRIS NEWMAN

(20) EnEQ b(n)n-z =f lP (1 - p-z)-l(l - pe-lZ-)-

and

(21) c(n) c(l) Ilk (ep)k+ - ] n e Q,

Of course c(1) is just -r. We do not go into questions of convergence here, and regard (19) and (20) as formal results.

It is perhaps worthwhile writing down an instance of (21). For the choice q = 5, r = 5, s =-1 we have that for all primes p > 5 and all integral n,

c(n p) - c(n) (pp{ (n) - Q)}

for the coefficients c(n) defined by

X (1 -cX8)5

Then (21) implies that for (n, 30) = 1,

c(n) - -5 JI k P ,

This result (without the restriction that (n, 30) = 1 which is superfluous) is implied by one of Ramanujan's formulas also.

The choices q = 2, r = 4, s = - 2; q = 2, r = 8, s =-4; q = 2, r = 12, s = - 6; and q = 2, r = 16, s = - 8 yield the usual theorems concerning the number of representations of an integer as the sum of 2, 4, 6, 8 squares, respectively.

The identity for q = 23 is due to van der Blij [1], whose proof uses ideal theory. Related discussions for certain other modular forms have been given by Hecke in [3] and van Lint in [4].

NATIONAL BUREAU OF STANDARDS, WASHINGTON

REFERENCES

1. F. van der BLIJ, Binary quadratic forms of discriminant -23, Nederl. Akad. Wetensch. Proc. Ser. A55= Indag. Math., 14 (1952), 498-503.

2. G. H. HARDY, Ramanujan, Cambridge University Press, 1940. 3. E. HECKE, Herleitung des Euler-Produktes der Zetafmnltion und einiger L-reihen

aus ihrer Funktionalgleichung, Math. Ann., 119 (1944), 266-287.



MODULAR FORMS 489

4. J. H. van LINT, Hecke operators and Euler products, Dissertation, University of Utrecht, 1957.

5. L. J. MORDELL, On Mr. Ramanujan's empirical expansions of modular functions, Proc. Cambridge Philos. Soc., 19 (1917), 117-124.

6. M. NEWMAN, Construction and application of a class of modular functions, II, Proc. London Math. Soc., (To appear).

7. , Structure theorems for modular subgroups, Duke Math. J., 22 (1955), 25-32. 8. , Remarks on some modular identities, Trans. Amer. Math. Soc., 73 (1952),

313-320. 9. , Further identities and congruences for the coefficients of modular forms,

Canad. J. Math., 10 (1958), 577-586. 10. H. RADEMACHER, and A. WHITEMAN, Theorems on Dedekind sums. Amer. J. Math.,

63 (1941), 377-407. 11. R. A. RANKIN, A certain class of multiplicative functions, Duke Math. J., 13 (1946),

281-306. 12 J. TANNERY et J. MOLK, El6ments de la Theorie des Fonctions Elliptiques. Tome 1,

Gauthier-Villars et fils, 1893.



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Modular Forms Whose Coefficients Possess Multiplicative Properties