finite quotients of fundamental groups in positive characteristic - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T15:43:41Z http://mathoverflow.net/feeds/question/58673 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58673/finite-quotients-of-fundamental-groups-in-positive-characteristic finite quotients of fundamental groups in positive characteristic shenghao 2011-03-16T18:36:06Z 2011-03-16T22:10:10Z <p>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. </p> <p>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.)</p> <p>In particular, let $G$ be the automorphism group of the supersingular elliptic curve in char. $p=2$ or $3$ (see <a href="http://mathoverflow.net/questions/58455/supersingular-elliptic-curve-in-char-2-or-3" rel="nofollow">http://mathoverflow.net/questions/58455/supersingular-elliptic-curve-in-char-2-or-3</a> 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!).</p> http://mathoverflow.net/questions/58673/finite-quotients-of-fundamental-groups-in-positive-characteristic/58691#58691 Answer by Holger Partsch for finite quotients of fundamental groups in positive characteristic Holger Partsch 2011-03-16T22:10:10Z 2011-03-16T22:10:10Z <p>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 $\pi_1(C)$.</p> <p>This is explained in section 3 of <a href="http://arxiv.org/abs/1005.2142v3" rel="nofollow">http://arxiv.org/abs/1005.2142v3</a></p> <p>This is an easy application of a general theory of finite quotients of fundamental groups of smooth curves as explained in the paper</p> <p>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</p> <p>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.</p>