Groups acting on Riemann Surfaces - MathOverflow

Groups acting on Riemann Surfaces

2011-02-24T10:00:57Z

By Hurwitz theorem, order of a group $G$ of automorphisms (conformal homeomorphisms) of a compact Riemann surface of genus $g\geq 2$ is bounded above by $84(g-1)$.

1. Is there any example of a compact Riemann surface whose automorphism group is trivial?

2. Does $C_2$ act on every compact Riemann surface of genus $g\geq 2$ ? ($C_2$ acts on any compact surface of genus $g$).

3. If all Sylow-subgroups of a finite group act on the a compact Riemann surface, does it imply that the whole group acts on Riemann surface?

4. Can one suggest a survey article about groups acting on Riemann surfaces/ automorphisms of Riemann surfaces?

Answer by Charles Matthews for Groups acting on Riemann Surfaces

2011-02-24T10:29:37Z

Q1: A typical Riemann surface has no holomorphic automorphisms, and this implies a negative answer to Q2. I don't see that Q3 can work: the p-subgroups surely don't uniquely determine the group in general.

The literature on these questions is quite large. http://www.jstor.org/pss/2160738 is a paper on the issue of surfaces with no non-trivial automorphisms. It is a little hard to tell what you want, but some of the material on the inverse Galois theory problem (which does use curves) might help you.

Answer by JSE for Groups acting on Riemann Surfaces

2011-02-24T16:07:15Z

I agree with the answers above. Allen Broughton at Rose-Hulman is a guy who has written a lot about automorphisms of Riemann surfaces: his paper Classifying finite group actions on surfaces of low genus, J. of Pure & Appl. Algebra 69 (1990), 233-270 will probably be of use.

Answer by Yuri Zarhin for Groups acting on Riemann Surfaces

2011-02-24T18:36:26Z

A counterexample to Q3 is provided by the genus 2 compact Riemann surface $X$ of $y^2=x^5-1$. Indeed, the order 10 cyclic group $C_{10}$ acts on $X$ (by changing sign of $y$ and multiplying $x$ by $5$th roots of unity). It is known that the jacobian of $X$ has endomorphism ring $Z[\zeta_5]$ - the $5$th cyclotomic ring of integers and any finite multiplicative subgroup of $Z[\zeta_5]$ is a subgroup of $\mu_{10}\cong C_{10}$. This implies that $Aut(X)=C_{10}$. On the other hand, the dihedral group $D_{10}$ of order $10$ has the same Sylow subgroups as $C_{10}$ but is not isomorphic to it. In other words, there is no faithful action of $D_{10}$ on $X$ while its Sylow subgroups $C_5$ and $C_2$ act faithfully on $X$.

If $Y$ is a compact Riemann surface of genus $g$ and its jacobian $J$ has no nontrivial automorphisms (i.e., $End(J)$ is the ring of integers $Z$) then either $Y$ is non-hyperelliptic and $Aut(Y)={1}$ or $Y$ is hyperelliptic and $Aut(Y)=C_2$. For example, if $g>1$ and $Y_g$ is the hyperelliptic Riemann surface $y^2=x^{2g+1}-x-1$ then its jacobian $J_g$ has no nontrivial endomorphisms (Math. Research Letters 7 (2000), 123--132) and therefore $Aut(Y_g)=C_2$. If $p$ is an odd prime then for each integer $n \ge 5$ the automorphism group of the compact Riemann surface $y^p=x^n-x-1$ is the cyclic group $C_{p}$. Indeed, the endomorphism ring of the jacobian is the $p$th cyclotomic ring $Z[\zeta_p]$ (Math. Proc. Cambridge Philos. Soc. 136 (2004), 257--267) and one may easily check, using the differentials of the first kind that the curve is non-hyperelliptic.

Using Del Pezzo surfaces of degree 2, one may construct non-hyperelliptic genus 3 curves $Y$, whose jacobian has no nontrivial endomorphisms (AMS Translations Series 2, vol. 218 (2006), 67--75; MR2279305, 2007k:14060) and therefore $Aut(Y)={1}$. For the genus 4 case see a paper of Anthony Várilly-Alvaradoa and David Zywina (LMS Journal of Computation and Mathematics (2009), 12: 144-165); their approach makes use of Del Pezzo surfaces of degree 1 (see also Math. Ann. 340 (2008), 407--435).