  The arithmetic properties of the ordinary partition function $p(n)$ have been
the topic of intensive study for the past century. Ramanujan proved that there
are linear congruences of the form $p(\ell n+\beta)\equiv 0\pmod\ell$ for the
primes $\ell=5, 7, 11$, and it is known that there are no others of this form.
On the other hand, for every prime $\ell\geq 5$ there are infinitely many
examples of congruences of the form $p(\ell Q^m n+\beta)\equiv 0\pmod\ell$
where $Q\geq 5$ is prime and $m\geq 3$. This leaves open the question of the
existence of such congruences when $m=1$ or $m=2$ (no examples in these cases
are known). We prove in a precise sense that such congruences, if they exist,
are exceedingly scarce. Our methods involve a careful study of modular forms of
half integral weight on the full modular group which are related to the
partition function. Among many other tools, we use work of Radu which describes
expansions of such modular forms along square classes at cusps of the modular
curve $X(\ell Q)$, Galois representations and the arithmetic large sieve.
