Let
p
(
n
)
p(n)
be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form
p
(
Q
3
ℓ
n
+
β
)
≡
0
(
mod
ℓ
)
p( Q^3 \ell n+\beta )\equiv 0\pmod \ell
where
ℓ
\ell
and
Q
Q
are prime and
5
≤
ℓ
≤
31
5\leq \ell \leq 31
; these lie in two natural families distinguished by the square class of
1
−
24
β
(
mod
ℓ
)
1-24\beta \pmod \ell
. In recent decades much work has been done to understand congruences of the form
p
(
Q
m
ℓ
n
+
β
)
≡
0
(
mod
ℓ
)
p(Q^m\ell n+\beta )\equiv 0\pmod \ell
. It is now known that there are many such congruences when
m
≥
4
m\geq 4
, that such congruences are scarce (if they exist at all) when
m
=
1
,
2
m=1, 2
, and that for
m
=
0
m=0
such congruences exist only when
ℓ
=
5
,
7
,
11
\ell =5, 7, 11
. For congruences like Atkin’s (when
m
=
3
m=3
), more examples have been found for
5
≤
ℓ
≤
31
5\leq \ell \leq 31
but little else seems to be known.
Here we use the theory of modular Galois representations to prove that for every prime
ℓ
≥
5
\ell \geq 5
, there are infinitely many congruences like Atkin’s in the first natural family which he discovered and that for at least
17
/
24
17/24
of the primes
ℓ
\ell
there are infinitely many congruences in the second family.
We study the principal series representations of central extensions of a split reductive algebraic group by a cyclic group of order n. We compute the Plancherel measure of the representation using Eisenstein series and a comparison method. In addition, we construct genuine central characters of the metaplectic torus in the simply-laced case.
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