Sur le nombre d'idéaux dont la norme est la valeur d'une forme binaire de degré 3

Abstract

Let $\mathbb{K}$ be a cyclic extension of degree $3$ of $\mathbb{Q}$. Take $G=\mathrm{Gal}(\mathbb{K}/\mathbb{Q})$ 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 $$r_3(n):=(1\ast \chi \ast {\chi }^2)(n),$$ 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 $$Q(\xi ,\mathcal{R},F):=\sum _{\mathbf{x}\in \mathcal{R}(\xi )}{r}_3(F(\mathbf{x})),$$ for a binary form $F$ of degree $3$ irreducible over $\mathbb{K}$ and $\mathcal{R}$ a good domain of ${\mathbb{R}}^2$, with $$\mathcal{R}(\xi ):=\{\mathbf{x}\in {\mathbb{R}}^2:\frac{\mathbf{x}}{\xi }\in \mathcal{R}\}.$$ We also give a geometric interpretation of the main constant of the asymptotic estimate when the ring $O_{\mathbb{K}}$ is principal.

In French

References in corpus (2)

• Nair-Tenenbaum bounds uniform with respect to the discriminant
• Sur la fonction Delta de Hooley associée à des caractères