CONGRUENCES AND CONJECTURES FOR THE PARTITION FUNCTION 5
Using classical facts (see, for example
[GH, Nl, N2],
we see that
fp(z)
is a
modular form in
Mu_
1);
2
(ro(i:'),
(7))
with integral coefficients. We have (see, for
example,
[An, Th. 1.1]),
the generating function
Using this with
(3.1)
and
(3.2),
we find that
(3.3)
~
ac(n)qn
=
(~p(n)qnHt)
·
g
(1 lnt
Recall that r5c
=
(1:'2

1)/24.
We have
fe(z)
=
~ 8 t(z)
(mod£), where
~(z)
=
7]24(z)
is the unique normalized cusp form of weight
12
on 81
2
(Z). By
[SwD, §3, Lemma
6],
we see that
wc(~ 8 t)
=
(1:'2

1)/2;
therefore Proposition
3.3
implies that
W£ (
8~£1)/2 (~Ot(z)))
=
£2
1.
Further, notice that
e~£ 1 )/ 2 (~
8
£(z))
=
~
('l)ac(n)qn
(mod£).
Therefore, there is a cusp form
Po,c(z)
in
Scz_
1
n
Z[[q]]
for which
Po,c(z)
=
~
('l)ac(n)qn
(mod£).
Let
P1,e(z)
E Sc2_
1
n
Z[[q]]
be the cusp form defined by
Pu(z)
:=
~ 8 £(z) · E~~"i 1 )/
2
(z).
Define the cusp form
P2,e(z)
in Sc2_
1
n
Z[[q]]
by
Using
(3.3),
we obtain
(3.4)
P2,c(z)
:=
P1,c(z) EePo,e(z).
P2,c
= {
L
p(n r5e)qn
+
2
L
p(n bc)qn}
IT
(1 ln)£
(mod
1:').
n:=O (mod
C)
(7)=E£ n=1
By construction, the first exponent in
P2
,c(z)
E
Scz_
1 whose coefficient could be
nonzero is
be+
1. Since Sc2_
1
has a basis of the form
{
~(z)l
E4(z)
£2;1
3j :
1 :::; j :::;
£2121}'
we see that there exists
C(z)
E M(£2_
25
);
2
n
Z[[q]]
such that
P2,e(z)
=
~o£+ 1 (z)
· C(z).