# A Kummer tower of function fields over $F_3$ and a question of Beelen, Garcia and Stichtenoth

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

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