  We study convergence of ergodic averages along squares with polynomial
weights. For a given polynomial $P\in \mathbb{Z}[\cdot]$, consider the set of
all $\theta\in[0,1)$ such that for every aperiodic system $(X,\mu, T)$ there is
a function $f\in L^1(X,\mu)$ such that the weighted averages along squares $$
{\frac{1}{N}\sum_{n=1}^N} e(P(n)\theta)T^{n^2}f $$ diverge on a set with
positive measure. We show that this set is residual and includes the rational
numbers as well as a dense set of Liouville numbers. This on one hand extends
the divergence result for squares in $L^1$ of the first author and Mauldin and
on the other hand shows that the convergence result for linear weights for
squares due to the second author and Krause in $L^p$, $p>1$ does not hold for
$p=1$.
