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

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.