In papers in 1977 and 1978, Barry Mazur proved the torsion conjecture giving a complete list of the possible torsion subgroups of elliptic curves over the rational numbers. Mazur's first proof of this theorem depended upon a complete analysis of the rational points on certain modular curves.[22][23] In 1996, the proof of the torsion conjecture was extended to all number fields by Loïc Merel.[24]