A Kummer tower of function fields over $F_3$ and a question of Beelen, Garcia and Stichtenoth - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T00:47:28Z http://mathoverflow.net/feeds/question/103335 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103335/a-kummer-tower-of-function-fields-over-f-3-and-a-question-of-beelen-garcia-and A Kummer tower of function fields over $F_3$ and a question of Beelen, Garcia and Stichtenoth R. Toledano 2012-07-27T18:36:14Z 2012-07-27T18:36:14Z <p>The equation $y^m=x^df(x)$ where $f(x)$ is a polynomial with coefficients in a finite field $F_q$ defines a Kummer tower of function fields over $F_q$ with positive splitting rate if $f(0)\neq 0$ and $q\equiv 1\mod m$. Lesntra showed that a very well known sufficient condition (due to Garcia, Stichtenoth and Thomas) for proving the finiteness of the ramification locus can not be used in this case if $q$ is a prime number. Later Beelen, Garcia and Stichtenoth asked if the Kummer tower defined by $y^2=x(x+2)$ over $F_3$ can have finite genus. Does anyone know if this question was ever answered? Thanks! ricardo</p>