Idélic Approach in Enumerating Heisenberg Extensions

For odd primes $$\ell $$ and number fields k, we study the asymptotic distribution of number fields L/k given as a tower of relative cyclic $$C_\ell $$ -extensions L/F/k using the idélic approach of class field theory. This involves a classification for the Galois group of L/k based on local conditions on L/F and F/k, and an extension of the method of Wright in enumerating abelian extensions. We call the possible Galois groups for these extensions generalized and twisted Heisenberg groups. We then prove the strong Malle–conjecture for all these groups in their representation on $$\ell ^2$$ points.