The classification of the torsion subgroup structures E(K)_tors that arise when E is an elliptic curve over a quartic number field K and its j-invariant j(E) is rational. This is Hamada's determination of the set Phi_{j in Q}(4): the complete, finite list of groups Z/NZ (N in 1-10,12,13,15,16,17,20,21,24) and the products Z/2Z+Z/2NZ, Z/3Z+Z/3NZ, Z/4Z+Z/4NZ, Z/5Z+Z/5Z, Z/6Z+Z/6Z, each realized by an explicit curve. The proof restricts the Derickx-Najman degree-4 torsion list by ruling out the orders 11, 14, 18, 22 (and Z/2+Z/14, Z/2+Z/18, Z/3+Z/9) using quadratic twists over Q, complex-multiplication input, and constraints on the field of definition of prime-order points.
An elliptic curve E over a number field K has a finitely generated Mordell-Weil group E(K), whose finite part is the torsion subgroup E(K)_tors. B. Mazur classified the possible torsion subgroups ove…