The Fekete-Szego theorem with Local Rationality Conditions on Curves

TL;DR

This paper extends the Fekete-Szego theorem to curves over number and function fields, establishing conditions for infinitely many algebraic points with conjugates in specified adelic sets, including variants for Berkovich curves.

Contribution

It proves a strong Fekete-Szego type theorem for adelic sets on curves, with new variants for Berkovich curves and diverse examples.

Findings

Infinitely many points with conjugates in adelic sets are guaranteed under certain conditions.

Variants of the theorem are established for Berkovich curves.

Examples include the projective line, elliptic, Fermat, and modular curves.

Abstract

Let $K$ be a number field or a function field in one variable over a finite field, and let $K^{sep}$ be a separable closure of $K$. Let $C/K$ be a smooth, complete, connected curve. We prove a strong theorem of Fekete-Szego type for adelic sets $E = \prod_v E_v$ on $C$, showing that under appropriate conditions there are infinitely many points in $C(K^{sep})$ whose conjugates all belong to $E_v$ at each place $v$ of $K$. We give several variants of the theorem, including two for Berkovich curves, and provide examples illustrating the theorem on the projective line, and on elliptic curves, Fermat curves, and modular curves.