Sur le nombre d'id\'eaux dont la norme est la valeur d'une forme binaire de degr\'e 3

TL;DR

This paper provides an asymptotic estimate for counting ideals of a cyclic cubic extension of rationals with norms represented by a binary cubic form, using new results on Hooley's Delta function.

Contribution

It introduces a novel asymptotic estimate for a sum involving the number of ideals with norms given by a binary cubic form over a cyclic cubic field, utilizing a new result on Hooley's Delta function.

Findings

Established an asymptotic formula for the sum Q(ξ, R, F)

Connected the main constant to geometric properties when the ring is principal

Extended understanding of ideal counting in cyclic cubic fields

Abstract

Let $\mathbb{K}$ be a cyclic extension of degree $3$ of $\mathbb{Q}$. Take $G={\rm Gal}(\mathbb{K}/ \mathbb{Q})$ and $\chi$ the character of a non trivial representation of $G$. In this case, $\chi$ is a non principal Dirichlet character of degree $3$ and the quantity $r_3(n)$ defined by $$r_3(n):=\big(1*\chi*\chi^2\big)(n){\rm ,}$$ counts the number of ideals of $O_{\mathbb{K}}$ of norm $n$. In this paper, using a new result on Hooley's Delta function, we prove an asymptotic estimate, in $\xi$, of the quantity $$Q(\xi,\mathcal{R},F):=\sum\limits_{\boldsymbol{x} \in \mathcal{R}(\xi)}{r_3\big(F(\boldsymbol{x})\big)}{\rm ,}$$ for a binary form $F$ of degree $3$ irreducible over $\mathbb{K}$ and $\mathcal{R}$ a good domain of $\mathbb{R}^2$, with $$\mathcal{R}(\xi):=\Big\{\boldsymbol{x} \in \mathbb{R}^2\;:\: \frac{\boldsymbol{x}}{\xi} \in \mathcal{R}\Big\}{\rm .}$$ We also give a geometric interpretation of the main constant of the asymptotic estimate when the ring $O_{\mathbb{K}}$ is principal.