Torsion of Elliptic Curves Over Real Quadratic Fields of Smallest Discriminant

In [9] and [10], Filip Najman examined the torsion of elliptic curves over the number fields ℚ(−1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{Q}\left( {\sqrt { - 1} } \right)$$\end{document} and ℚ(−3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{Q}\left( {\sqrt { - 3} } \right)$$\end{document}. In this paper, we study the torsion structures of elliptic curves over the real quadratic number fields ℚ(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{Q}\left( {\sqrt 2 } \right)$$\end{document} and ℚ(5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{Q}\left( {\sqrt 5 } \right)$$\end{document}, which have the smallest discriminants among all real quadratic fields ℚ(d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{Q}\left( {\sqrt d } \right)$$\end{document} with d ≢ 1 mod 4 and d ≡ 1 mod 4 respectively.