turkmath.org

For elliptic curves over the rational numbers the possible torsion subgroups are known due to the famous result by Mazur. In an earlier result Ogg found parametrized infinite families for all possible torsion subgroups. For hyperelliptic curves nearly nothing about the rational torsion structure is known. There are some resluts by Cassels, Elkies, Flynn and Leprévost who give single curves with large torsion. We will show the construction of families of hyperelliptic curves with p-torsion for small primes p. Furthermore we will give an overview over the attempts that did not work. this situation.