Torsion of Elliptic Curves Over Real Quadratic Fields of Smallest Discriminant

Abstract

In [9] and [10], Filip Najman examined the torsion of elliptic curves over the number fields $\mathbb{Q}\left( {\sqrt { - 1} } \right)$ and $\mathbb{Q}\left( {\sqrt { - 3} } \right)$. In this paper, we study the torsion structures of elliptic curves over the real quadratic number fields $\mathbb{Q}\left( {\sqrt 2 } \right)$ and $\mathbb{Q}\left( {\sqrt 5 } \right)$, which have the smallest discriminants among all real quadratic fields $\mathbb{Q}\left( {\sqrt d } \right)$ with d ≢ 1 mod 4 and d ≡ 1 mod 4 respectively.