Croissance asymptotique de nombres de Weil appartenant \`a un corps de nombres fix\'e

TL;DR

This paper establishes an asymptotic count for algebraic integers in a fixed CM number field with bounded norm, linking it to the properties of a height zeta function and providing a new proof of Manin's conjecture for a specific toric variety.

Contribution

It proves an asymptotic formula for algebraic integers in CM fields and analyzes the meromorphic continuation of a related height zeta function, offering a new proof of Manin's conjecture.

Findings

Asymptotic formula for algebraic integers in CM fields

Meromorphic continuation of the height zeta function to Re(s)>1/2

New proof of Manin's conjecture for the toric variety

Abstract

We prove an asymptotic formula as $x\to +\infty$ for the number of algebraic integers $\alpha$ belonging to a fixed CM number field and satisfying $\alpha\overline{\alpha}\leq x$. This problem is related to the height zeta function $Z_h(X^K,s)$ associated to the anticanonical class of a certain toric variety $X^K$ over $\mathbb{Q}$ and we show that $Z_h(X^K,s)$ has a meromorphic continuation to the half-plane $\{\Re(s)>\frac{1}{2}\}$ where it is holomorphic except at $s=1$. Along the way we obtain a new proof of Manin's conjecture on the asymptotic growth of points on $X^K(\mathbb{Q})$ of bounded height.