Sur le corps des modules de certaines vari\'et\'es

TL;DR

This paper explores the relationship between coverings of algebraic curves and associated varieties, providing examples of curves with rational moduli fields that lack models over the rational numbers but have models over their completions.

Contribution

It introduces a method to associate varieties to curve coverings with the same moduli and fields of definition, illustrating cases where rational moduli do not guarantee rational models.

Findings

Constructed examples of curves with Q as the field of moduli

Demonstrated curves with models over completions but not over Q

Established links between coverings and associated varieties

Abstract

To every covering of curves, we associate several varieties having the same field of moduli and same fields of definition. We deduce examples of curves having Q (the field of rationals) as field of moduli, that admit models over any completion of Q but no model over Q.