TITLE: Extra automorphisms of curves and definability over \bar Q
QUESTION [12 upvotes]: A generic elliptic curves over C has automorphism group of order 2. The elliptic curves with extra automorphisms are C/Z[i] (automorphism group of order 4) and C/Z[w] where w is a primitive 3rd root of unity (automorphism group of order 6). One can use their extra automorphisms to prove that they can be defined over Q
Similarly, the Klein quartic (a genus 3 curve with 168 automorphisms, the maximum possible number for genus 3 curves) can be defined over Q.

Suppose X is a genus g curve over C.
  Is there a condition on the group
  Aut(X) that guarantees that X is
  defined over a number field?

I feel like maybe one could try to approach this question by passing to the Jacobian of X and arguing that extra automorphisms of X give extra automorphisms of J(X) which means that J(X) has "large" endomorphism ring and hence must be defined over a number field (just as CM elliptic curves are defined over number fields), but I don't know whether "large" is large enough.

REPLY [11 votes]: One has the notion of a compact Riemann surface with "many automorphisms", a concept for which there exist many equivalent definitions.  You have correctly identified the "genus one Riemann sufaces with many automorphisms" -- a situation which is slightly complicated by the fact that, strictly speaking, the automorphism group of any complex elliptic curve as a Riemann surface is infinite -- so let's concentrate on the case of Riemann surfaces $X$ of genus $g \geq 2$, in which case the automorphism group is always finite.
The following three conditions on $X$ are equivalent:
(1) On the moduli space $\mathcal{M}_g$ of genus $g$ complex algebraic algebraic curves, the function $C \mapsto \# \operatorname{Aut}(C)$ has a strict local maximum at $X$ -- explicitly, there exists a neighborhood of $U$ of $X$ such that every curve in $U \setminus \{X\}$ has smaller automorphism group than $X$.  
(2) The natural map $X \rightarrow X/\operatorname{Aut}(X)$ is a Belyi map -- i.e., the quotient has genus zero and the map is ramified only over three points.  By the "easy" direction of Belyi's theorem, it follows that $X$ can be defined over some number field, and it is interesting to study the field of moduli of such curves $X$ (which in this case is known to be the minimal field of definition).
(3) $X$ is uniformized by a finite index torsion free subgroup $\Gamma$ of a hyperbolic triangle group $\Delta(a,b,c)$.  This makes clear that these curves are a significant generalization of the Hurwitz curves, which correspond to the "first" hyperbolic triangle group $\Delta(2,3,7)$.  At times I have expressed the opinion that it is strange that there has been so much research done on the $\Delta(2,3,7)$ case -- i.e., when the automorphism group is numerically as large as it could possibly be -- rather than on the more general case of $\Delta(a,b,c)$, i.e., the case where the automorphism group is sufficiently large to have the above very interesting analytic / topological / arithmetic implications.  
For more information on these Riemann surfaces, see the bibliography of 
http://math.uga.edu/~pete/triangle-091309.pdf
Especially highly recommended is [50], a paper by J. Wolfart.  In general Wolfart is one of the pioneers of this subject and has written many interesting papers, mostly from the perspective of complex function theory.