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For affine smooth curves over $k=\bar{k}$ of char. $p,$ Abhyankar's conjecture (proved by Raynaud and Harbater) tells us exactly which finite groups can be realized as quotients of their fundamental groups.

What about complete smooth curves, or more generally higher dimensional varieties? Are there results or conjectural criteria (or necessary conditions) for finite quotients of their $\pi_1?$ (Definitely, not too much was known around 1990; see Serre's Bourbaki article on this.)

In particular, let $G$ be the automorphism group of the supersingular elliptic curve in char. $p=2$ or $3$ (see supersingular elliptic curve in char. 2 or 3 for various descriptions of its structure). Is there (and if yes, how to construct) a projective smooth variety in char. $p$ having $G$ as a quotient of its $\pi_1?$ Certainly there are lots of affine smooth curves with this property (e.g. $\mathbb G_m$), and I wonder if for some of them, the covering is unramified at infinity (so that we win!).

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I think you can get $G$ as a quotient of the fundamental group of any curve of genus $g>1$, since such groups have only one (topological) relation. – S. Carnahan Mar 16 '11 at 19:59
That's my hope too. But is there any reference for an Abhyankar-type conjectural statement for complete curves? – shenghao Mar 16 '11 at 20:19
In fact, I don't know if we know now that $\pi_1$ of projective smooth varieties (or just curves) in char. $p$ are of finite presentation in general, although in char. 0 it is the case. At least at the time when SGA1 was written this was not known; cf. SGA1, Exp.X, 2.8. – shenghao Mar 16 '11 at 20:51
A naive Abhyankar-type statement would claim that a finite group $G$ is a quotient of $\pi_1(X)$ is $G/p(G)$ is such a quotient in characteristic zero, where $p(G)$ is the characteristic subgroup of $G$ generated by its $p$-Sylow subgroups. Unfortunately, it fails miserably already for $X$ the projective line. – ACL Mar 17 '11 at 9:36
up vote 2 down vote accepted

For a supersingular elliptic $E$ over an algebraically closed field of characteristic two or three there exists a smooth curve $C$ of higher genus such that $Aut_0(E)$ is a finite quotient of $\pi_1(C)$.

This is explained in section 3 of

This is an easy application of a general theory of finite quotients of fundamental groups of smooth curves as explained in the paper

Amilcar Pacheco and Katherine F. Stevenson. Finite quotients of the algebraic fundamental group of projective curves in positive characteristic. Pacific J. Math., 192(1):143–158, 2000

In this paper, it is explained how to realize groups which have the property that their maximal $p$-Sylow subgroup ($p$ being the characteristic) is normal. The automorphism groups of supersingular elliptic curves satisfy this property.

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Thanks for the two papers, Holger! – shenghao Mar 16 '11 at 22:32

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