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.