  In this article, we consider indecomposable Specht modules with abelian
vertices. We show that the corresponding partitions are necessarily $p^2$-cores
where $p$ is the characteristic of the underlying field. Furthermore, in the
case of $p\geq 3$, or $p=2$ and $\mu$ is 2-regular, we show that the complexity
of the Specht module $S^\mu$ is precisely the $p$-weight of the partition
$\mu$. In the latter case, we classify Specht modules with abelian vertices.
For some applications of the above results, we extend a result of M. Wildon and
compute the vertices of the Specht module $S^{(p^p)}$ for $p\geq 3$.
