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.