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)
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