Torsion of elliptic curves over cubic fields

Although it is not known which groups can appear as torsion groups of elliptic curves over cubic number fields, it is known which groups can appear for infinitely many non-isomorphic curves. We denote the set of these groups as $S$. In this paper we deal with three problems concerning the torsion of elliptic curves over cubic fields. First, we study the possible torsion groups of elliptic curves that appear over the field with smallest absolute value of its discriminant and having Galois group $S_3$ and over the field with smallest absolute value of its discriminant and having Galois group $\Z/3\Z$. Secondly, for all except two groups $G\in S$, we find the field $K$ with smallest absolute value of its discriminant such that there exists an elliptic curve over $K$ having $G$ as torsion. Finally, for every $G\in S$ and every cubic field $K$ we determine whether there exists infinitely many non-isomorphic elliptic curves with torsion $G$.