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

Elementary abelian groups are finite groups in the form of A = (ℤ/pℤ) r for a prime number p. For every integer ℓ > 1 and r > 1, we prove a non-trivial upper bound on the ℓ-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-torsion in class groups are bounded non-trivially for every G-extension and every integer ℓ > 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. [ABSTRACT FROM AUTHOR]