l-Torsion Bounds for the Class Group of Number Fields with an l -Group as Galois Group

Abstract

We describe the relations among the $\ell$-torsion conjecture for $\ell$-extensions, the discriminant multiplicity conjecture for nilpotent extensions and a conjecture of Malle giving an upper bound for the number of nilpotent extensions. We then prove all of these conjectures in these cases.