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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.

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I deleted an incorrect answer. Thanks to Torsten Ekedahl for pointing out the mistake. –  Tom Church Apr 29 '10 at 16:38

3 Answers 3

up vote 8 down vote accepted

If the group has order 84(g-1), then the curve is a Galois cover of the (2,3,7) orbifold. By Belyi's theorem, it is defined over a number field.

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Thanks! I'm still curious about other questions connected with my original one. (For example, does the presence of any extra automorphisms whatsoever imply that the curve is defined over a number field?) –  Jonah Sinick Apr 29 '10 at 4:45
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The point of Belyi's theorem is that a curve defined over a number field is a cover of P^1 ramified over only three points. The converse is essentially trivial (assuming elementary algebraic geometry). –  ulrich Apr 29 '10 at 4:52
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For those not familiar with Hurwitz's theorem, it says that the 84(g-1) here is the maximal number of automorphisms of a curve of genus g. This bound can be pushed down to 12(g-1) using the fact that the (2,2,2,3) orbifold, with area pi/3, has the smallest area of any non-triangular hyperbolic orbifold. Thus any curve with more than 15% of the possible number of automorphisms is defined over a number field. –  Tom Church Apr 29 '10 at 4:54

Consider the more general case (just to lighten the notation) of a curve $X$ and a group of automorphism $G$ (assumed to act faithfully). Consider the map $X \rightarrow X/G$. This covering has a topological component, namely the genus of $X/G$, the number of branch points and some supplementary (essentially) combinatorial which can be most succinctly formulated as a surjection of the fundamental of $X/G$ minus the branch points. The geometric situation is then specified by specifying an actual curve $Y$ (which is to be $X/G$) and the right number of points on it. For any such choice there is a $G$-covering $X \rightarrow Y$branched at the points and with the right combinatorial data which in particular gives the right genus to $X$.

Now, it is very easy to see when the geometric data is rigid or forms a positive dimensional family: It is rigid precisely when the genus of $X/G$ is $0$ and there are at most three branch points. In the rigid case $X$ is defined over a number field (a theorem due to Weil I believe) and in the non-rigid case $X$ may or may not be defined over a number field (both cases will occur).

The rigidity condition is easy to check: Using the (useful) result that the rational cohomology of a group quotient is given by the invariants on cohomology we get a formula for the genus of the quotient. Hence $2-2g$ is given by the number of times the trivial representation appears in the $G$-Euler characteristic which can be computed using the Lefschetz fixed point formula. The branch points correspond to $G$-orbits of points with non-trivial stabilisers so they are (in practice at least) easy to count (and that kind of information is needed anyway to use the Lefschetz formula).

To get examples one can also go backwards, i.e., start with one of the rigid situations and construct an epimorphism form the fundamental group to some finite group $G$. The only interesting case is $\mathbb P^1$ with three branch points (which may, and usually are, specified to be $0$, $1$ and $\infty$) and the fundamental group is free on $2$ generators; the generators represent the monodromy around two of the three points and their product represents the monodromy around the third. Scott's example refers to the case when $G$ achieves the the Hurwitz bound in which case the ramification around the branch points is of order $2$, $3$ and $7$.

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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.

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Another equivalent condition, which is sometimes useful, is that there are no non-zero quadratic differential that are invariant for the action of the automorphism group. –  Angelo Apr 29 '10 at 6:17

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