4

SCOTT AHLGREN AND KEN ONO

COROLLARY 3.2. Suppose that

I!

~

5 is prime and that (3 E Se. Then there

exists a cusp form Fe,(J(z)

E S(£2_

2);2(fi(576£2))

n Z[[q]]

for which

(X)

Fe,(J(z)

=

LP(f!n

+

f3)lMn+

24

/3-l

(mod£).

n=O

Applying the arguments in Section 3 of

[A-0]

(which rely on certain facts

arising from the theory of Galois representations associated to modular forms and

Shimura's theory of half integral weight modular forms), the forms given in Corol-

lary 3.2 yield a proof of Theorem 1 in the case when

m

=

1.

Before beginning the proof of Theorem 3.1, we briefly recall certain facts about

the theory of modular forms modulo

I!

(see

[Sw-D]

for details). If k is an even

integer, then let Mk (resp. Sk) denote the ((vector space of weight k modular

(resp. cusp) forms with respect to SL2 (Z). Let Mk,£ and Sk,e denote the lFe-vector

spaces given by

Mk,P

:= {

f(z)

= '[;

a(n)qn (mod£) : f(z) E Mk

n Z[[q]]},

Sk,P

:= {

f(z)

=

~

a(n)qn (mod£) : f(z) E Sk

n Z[[q]]}.

As usual, let Ek(z) denote the normalized weight k Eisenstein series on SL2 (Z).

Using the fact that

Ee-I(z)

=

1 (mod£),

one sees that the set of modular forms modulo

I!

forms a graded algebra.

We recall that if f(z)

=

I:~=O

a(n)qn is a modular form with integral coeffi-

cients, then

we(!),

the filtration off modulo £, is defined by

we(!):=

min{k :

f

(mod£) E

Mk,d·

Also, if f(z)

=

I:~=O

a(n)qn has integral coefficients, then the Ramanujan theta

operator is defined by

(X)

8e(J)

:=

L na(n)qn (mod£).

n=O

Finally, we record the following

PROPOSITION 3.3.

[Sw-D, §3, Lemma 5]

Suppose that f(z)

=

I:~=O

a(n)qn

is a modular form with integral coefficients and that

we (f)

"¥-

0 (mod £). Then

we(8e(J))

=we(!)+ I!+

1.

PROOF OF THEOREM 3.1. We begin by recalling Dedekind's eta function

(X)

(3.1)

rJ(z)

:=

ql/24

II (

1

_ qn).

n=l

If

I!~

5 is prime, then define fc(z)

=

I:~=l

ae(n)qn by

~

n

rJR(f!z)

fc(z)

=

~ ae(n)q

:=

-(z) .

n=l

fJ

(3.2)