Pointwise bound for ℓ-torsion in class groups: Elementary abelian extensions

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