Torsion of Elliptic Curves Over Real Quadratic Fields of Smallest Discriminant

Abstract

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