On Dℓ‐extensions of odd prime degree ℓ

Abstract

Generalizing the work of Morra and the authors, we give explicit formulas for the Dirichlet series generating function of D ℓ -extensions of odd prime degree ℓ with given quadratic resolvent. Over the course of our proof, we explain connections between our formulas and the Ankeny–Artin–Chowla conjecture, the Ohno–Nakagawa relation for binary cubic forms, and other topics. We also obtain improved upper bounds for the number of such extensions (over Q) of bounded discriminant.