CONGRUENCES AND CONJECTURES FOR THE PARTITION FUNCTION 3
THEOREM
1. If£
~
5 is prime, m is a positive integer, and (3
E
Se, then a
positive proportion of the primes
Q
=
-1 (mod 24£) have the property that
p(Q
3
;
4
+
1)
=
0 (mod
£ffi)
for all n
=
1- 24(3 (mod 24£) with gcd(Q, n)
=
1.
We remark that the case when (3
=
-De
(mod £) contains the main results in
[02J
and
[A2J.
Further, we note that given (3
E
Se and a prime Q as in Theorem 1,
fixing
n
in an appropriate residue class modulo 24£Q gives a Ramanujan-type con-
gruence within the progression
Cn
+
(3. This yields the following
THEOREM
2.
If£~
5 is prime, m is a positive integer, and (3
E
Se, then there
are infinitely many non-nested arithmetic progressions
{An+ B}
;;; {
Cn
+
(3} such
that
p(An +B)= 0 (mod
£ffi)
for every integer n.
Finally, we note that if M is an integer coprime to 6, then Theorem 2 and the
Chinese Remainder Theorem guarantee the existence of infinitely many congruences
modulo M. The results in
[A-OJ
provide a theoretical framework which (to our
knowledge) explains every known partition function congruence.
3. A new construction
All of the results in
[A-OJ
rely on the construction of half-integral weight
modular forms whose coefficients capture values of
p( n)
modulo gm. In this section
we present an alternate construction in the case
m
=
1 using the theory of modular
forms modulo£ as developed by Serre and Swinnerton-Dyer
[SwDJ.
This approach
yields an elegant proof of Theorem 1 in the case when m
=
1, and is particularly
convenient for constructing examples. The main advantage of this approach is that
it allows us to work with modular forms on SL
2
(Z); although the construction in
[A-OJ
is more general, it requires the use of modular forms of much higher level.
Throughout we use the notation q
:=
e21riz, and we adopt standard notation from
the theory of modular forms. Define the character
x12
by
x12
(d)
:=
(11).
Our aim
in this section is to prove the following:
THEOREM
3.1. Suppose that
£
~
5 is prime. Then there exists a cusp form
Pe(z) in
Su2_ 2J;2 (fo(576),
xd
n Z[[q]]
such that
Pe(z)
=
n=O
(mod£)
('l)=-,
We recall (see, for example
[S-St])
that if
f(z)
=
I:~=l
a(n)qn
E
S.+~
(f
1
(N) ),
and
r
and
t
are positive integers, then
n=r
(mod
t)
Let Pe(z) be as in Theorem 3.1, and suppose that (3
E
Se, where Se is defined in
(2.7). Extracting those terms from Pe(z) whose exponents are congruent to 24(3 -1
(mod£), we obtain the following corollary, which implies Theorem 2.1 of
[A-0]
in
the case
m
=
1.
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