  Elementary abelian groups are finite groups in the form of
$A=(\mathbb{Z}/p\mathbb{Z})^r$ for a prime number $p$. For every integer
$\ell>1$ and $r>1$, we prove a non-trivial upper bound on the $\ell$-torsion in
class groups of every $A$-extension. Our results are pointwise and
unconditional. When $r$ is large enough, the pointwise bound we obtain also
breaks the previously best known bound shown by Ellenberg-Venkatesh under GRH.
