nLab field of moduli

Idea

Given a curve $X$ in algebraic geometry over some field $F$, especially some kind of number field, the field of moduli of $X$ with respect to $F$ is, roughly speaking, the largest field invariant under those automorphisms of $\overline{F}$, the algebraic closure of $F$, which do not change $X_{\overline{F}}$, the base change of $X$ to a scheme over $\overline{F}$.

Automorphisms of $\overline{F}$ which fix $F$ are fundamental in Galois theory, forming the absolute Galois group of $F$, and indeed in mathematics generally: the absolute Galois group of the rational numbers is a very deep object, for example. Considering the field of moduli of a curve is a kind of geometric analogue of this.

Definition

There are slightly different ways to give a precise definition. We choose the following formulation (see Mochizuki2015 for example, although the notion goes back at least to the era of André Weil).

Definition

Let $X$ be a curve in the sense of algebraic geometry, defined over a field $F / \mathbb{Q}$. Given an automorphism $\sigma$ of $\overline{F}$, one can ask that the scheme $X_{\overline{F}} = X \times_{F} \overline{F}$ (where the projection morphisms are the obvious ones) is isomorphic to the base change of $X_{\overline{F}}$ along $\sigma$. The field of moduli of $X$ with respect to $F$ is the sub-field of $\overline{F}$ fixed by all such automorphisms $\sigma$ which belong to the Galois group $Gal\left(\overline{F} / \mathbb{Q} \right)$, namely which fix $\mathbb{Q}$.

Remark

The field of moduli of $X$ with respect to $F$ is a finite extension of $F$.

Remark

A curve $X$ over $F$ need not be definable over $F_{mod}$, that is, there may not exist any scheme over $F_{mod}$ which is isomorphic to $X_{\overline{F}}$ after base change along the canonical morphism $\overline{F} \rightarrow F_{mod}$.

References

Definition is 5.1 (ii) in the following.

• Shinichi Mochizuki, Topics in absolute anabelian geometry. III: Global reconstruction algorithms, J. Math. Sci., Tokyo 22, No. 4, 939-1156 (2015)