Persiflage

Can you help settle a disagreement? This is a debate about notation I am having with a colleague; I will try to present it without prejudice (and probably fail).

Let \(G\) be a group, and let \(V\) be a finite dimensional complex irreducible representation of \(G\). Suppose that the traces of \(G\) actually lie inside a number field \(K\). The field \(K\) may well be much smaller than \(\mathbf{C}\). For example, if \(G\) is a finite group, then \(K\) will be a number field. But it is not always the case that the representation itself is conjugate to a representation over the field \(K\) itself. The smallest example is the quaternion group \(Q_8\) of order \(8\) and its \(2\)-dimensional irreducible representation \(V\). This has traces in \(\mathbf{Q}\), but \(Q_8\) is not a subgroup of \(\mathrm{GL}_2(\mathbf{Q})\).

Now consider:

Definition: A field of definition of \(V\) is an extension \(L/K\) inside \(\mathbf{C}\) so that the representation can be realized over \(L\).

Returning to the example of \(Q_8\), basic representation theory tells us that the fields of definition are controlled by the corresponding quaternion algebra \(B/\mathbf{Q}\); in this case, \(B\) is Hamilton’s quaternion algebra over \(\mathbf{Q}\) which is ramified only at \(2\) and \(\infty\) and has Hilbert symbol \((-1,-1)\). So we very much understand what the fields of definition are in this case by global class field theory. More generally, for any irreducible representation of a finite group, we obtain a corresponding division algebra over the trace field \(K\) which controls the fields of definition.

One context in which to view this definition is with respect to the pair of expressions “fields of definition” and “fields of moduli”. Let’s consider some class of objects \(X\), for example (for concreteness) an algebraic variety defined over the algebraic closure of \(K\). Given \(X\), the term “field of moduli” usually refers to the smallest field \(L\) such that there exists an automorphism \(\psi_{\sigma}: X^{\sigma} \simeq X\) for every \(\sigma \in \mathrm{Gal}(\overline{K}/L)\), whereas the “field of definition” is a field \(L\) so that \(X\) has a model over \(L\). The reason that a field of moduli is not always a field of definition is that the \(\psi_{\sigma}\) do not always give compatible descent data because \(X\) may have extra automorphisms. This typically arises when one considers points on moduli spaces which are not representable as schemes but only exist as stacks, the canonical example being the moduli space of elliptic curves and more generally the moduli space of principally polarized abelian varieties. In the first case, because of the \(j\) invariant, the field of moduli is always field of definition, but in the second case this does not always happen.

Returning to the case of representations of \(G\), one can of consider the the moduli space \(\mathcal{M}_{G,n}\) of \(n\)-dimensional representations up to isomorphism. (One usually thinks about as some sort of GIT quotient which makes it a variety but perhaps it’s better to think about the moduli problem more directly, which will not in general be representable). The field of moduli in this case is just the trace field, and the fields of definition are the fields where one can realize the representation.

An objection:

Let us suppose that \(\Gamma \subset \mathrm{PSL}_2(\mathbf{R})\) is a Fuchsian group. More precisely, suppose concretely that \(\Gamma\) is a cocompact triangle group

\(\Gamma = \langle x,y,z | x^p, y^q, (xy)^r \rangle\)

where \(1/p+1/q+1/r<1\). For rigidity reasons, the representation \(\Gamma\) has traces in a number field \(K \hookrightarrow \mathbf{R}\) which one can write down explicitly. So now one might ask for what extensions \(L/K\) the group \(\Gamma\) is conjugate to a subgroup of \(\mathrm{PSL}_2(L)\). What is wrong with calling these fields of definition?

One objection is as follows. For these triangle groups, the quotient spaces \(\mathbf{H}/\Gamma\) and their covers are projective curves (sometimes with orbifold points). By rigidity, they can now as curves be defined over number fields (See, for example, this paper.) But now one might worry there is a possible conflict of notation when saying “fields of definition” in this setting between the “fields of definition of the curves” and the “fields of definition of the representation.” Adding to the confusion is that these fields will not be the same thing.

So now the basic question is whether one should avoid using “fields of definition” for representations (in this precise setting of triangle groups) precisely because of the possible confusion with the (standard use of) “fields of definition” of the corresponding curves?

Out of three people so far (which includes me) there is one person who thinks there is no problem, one person who has an objection, and a third who seems a little ambivalent. So I thought I might try to resolve it in the public square!