On $D_\ell$-extensions of odd prime degree $\ell$

TL;DR

This paper provides explicit formulas for counting $D_ ext{ell}$-extensions of odd prime degree with a fixed quadratic resolvent, linking number theory conjectures and improving bounds on their quantity.

Contribution

It generalizes previous work by deriving explicit Dirichlet series formulas and explores connections to key conjectures and relations in algebraic number theory.

Findings

Explicit formulas for Dirichlet series of $D_ ext{ell}$-extensions

Connections to the Ankeny-Artin-Chowla conjecture and Ohno-Nakagawa relation

Improved upper bounds for the number of such extensions

Abstract

Generalizing the work of A. Morra and the authors, we give explicit formulas for the Dirichlet series generating function of $D_{\ell}$-extensions of odd prime degree $\ell$ 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.