2013/08/07 by Joachim König, König, Joachim
Computer Science · Mathematics · #11 #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #math.NT #msc:11
paper · pdf · doi:10.48550/arxiv.1308.1566
6 pages
arxiv created 2013/08/07 · openalex publication_date 2013/08/07 · arxiv updated 2013/08/08 · openalex created_date 2016/09/16 · openalex updated_date 2026/07/28
We compute a family of coverings with four ramification points, defined over ℚ, with regular Galois group PSL5(2). On the one hand, this is (to my knowledge) the first explicit polynomial with group PSL5(2) over ℚ(t). On the other hand, it also positively answers the question whether PSL5(2) is the monodromy group of a rational function over ℚ. At least this does not follow from considering class triples in PSL5(2), as there are no rigid, rational genus-zero triples. Also, for 4-tuples, our family is the only one with a Hurwitz curve of genus zero (however it does not seem immediately clear without explicit computations whether this curve can be defined as a rational curve over ℚ). There are also genus zero families with five branch points, and maybe their Hurwitz spaces can be shown to have rational points; however, so far I have not seen such arguments.