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

Henri Cohen; Frank Thorne

doi:10.1112/plms.12339

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arXiv · 1609.09153

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

Henri Cohen·Frank Thorne

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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.

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Henri Cohen, Frank Thorne. 2016-09-28. On $D_\ell$-extensions of odd prime degree $\ell$. https://doi.org/10.1112/plms.12339

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