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

Elementary abelian groups are finite groups in the form of A = ( ℤ / p ⁢ ℤ ) r {A=(Z/pZ)ʳ} for a prime number p . For every integer ℓ > 1 {>1} and r > 1 {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 {>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.