field of moduli



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

Automorphisms of F¯\overline{F} which fix FF are fundamental in Galois theory, forming the absolute Galois group of FF, 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.


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).


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


The field of moduli of XX with respect to FF is a finite extension of FF.


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


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)

Last revised on March 30, 2021 at 02:00:15. See the history of this page for a list of all contributions to it.