Torsion of elliptic curves over quadratic cyclotomic fields

In this paper we study the possible torsions of elliptic curves over $\mathbb Q(i)$ and $\mathbb Q(\sqrt {-3})$.Comment: 9 pages, to appear in Math. J. Okayama Univ

Complete classification of torsion of elliptic curves over quadratic cyclotomic fields

In a previous paper, the author examined the possible torsions of an elliptic curve over the quadratic fields $\mathbb Q(i)$ and $\mathbb Q(\sqrt{-3})$. Although all the possible torsions were found if the elliptic curve has rational coefficients, we were unable to eliminate some possibilities for the torsion if the elliptic curve has coefficients that are not rational. In this note, by finding all the points of two hyperelliptic curves over $\mathbb Q(i)$ and $\mathbb Q(\sqrt{-3})$, we solve this problem completely and thus obtain a classification of all possible torsions of elliptic curves over $\mathbb Q(i)$ and $\mathbb Q(\sqrt{-3})$.Comment: 6 pages, to appear in J. Number Theor

Smooth values of some quadratic polynomials

In this paper, using a method of Luca and the author, we find all values $x$ such that the quadratic polynomials $x^2+1,$ $x^2+4,$ $x^2+2$ and $x^2-2$ are 200-smooth and all values $x$ such that the quadratic polynomial $x^2-4$ is 100-smooth.Comment: 12 pages, to appear in Glas. Mat. Ser. II