![[Abstract]](http://www.math.ufl.edu/~frank/icons/scroll.gif)
F.G. Garvan;
Congruences for Andrews' spt-function modulo 32760
and extension of Atkin's Hecke-type partition congruences,
preprint.
Dedicated to the memory of A.J. (Alf) van der Poorten, my former teacher
Abstract:
New congruences are found for Andrews' smallest parts partition function
$\spt(n)$. The generating function for $\spt(n)$ is related to
the holomorphic part $\alpha(24z)$
of a certain weak Maass form $\mathcal{M}(z)$ of weight $\tfrac{3}{2}$.
We show that a normalized form of the generating function for $\spt(n)$ is
an eigenform modulo $72$ for the Hecke operators $T(\ell^2)$ for
primes $\ell > 3$, and an eigenform modulo $p$ for $p=5$, $7$ or $13$
provided that $(\ell,6p)=1$. The result for the modulus $3$
was observed earlier by the author and
considered by Ono and Folsom. Similar congruences for higher
powers of $p$ (namely $5^6$, $7^4$ and $13^2$) occur for the coefficients
of the function $\alpha(z)$. Analogous results for the partition
function were found by Atkin in 1966. Our results depend
on the recent result of Ono that $\mathcal{M}_{\ell}(z/24)$ is
a weakly holomorphic modular form of weight $\tfrac{3}{2}$ for the full
modular group where
$$
\mathcal{M}_{\ell}(z) = \mathcal{M}(z) \vert T(\ell^2) -
\leg{3}{\ell} (1 + \ell) \mathcal{M}(z).
$$
This abstract uses MathJax.
The url of this page is http://www.math.ufl.edu/~frank/abstracts/spt3.html.
Created by
F.G. Garvan
(fgarvan@ufl.edu) on
Saturday, November 6, 2010.
Last update made Wed Nov 24 19:14:33 EST 2010.
fgarvan@ufl.edu
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