On the torsion of rational elliptic curves over quartic fields

Let E be an elliptic curve defined over Q and let G = E(Q)tors be the associated torsion subgroup. We study, for a given G, which possible groups G ⊆ H could appear such that H = E(K)tors, for [K : Q] = 4 and H is one of the possible torsion structures that occur infinitely often as torsion structures of elliptic curves defined over quartic number fields. Let K be a number field, and let E be an elliptic curve over K. The Mordell-Weil theorem states that the set E(K) of K-rational points on E is a finitely generated abelian group. It is well known that E(K)tors, the torsion subgroup of E(K), is isomorphic to Z/nZ × Z/mZ for some positive integers n,m with n|m. In the rest of the paper we shall write Cn = Z/nZ for brevity, and we call Cn × Cm the torsion structure of E over K. The characterization of the possible torsion structures of elliptic curves has been of considerable interest over the last few decades. Since Mazur’s proof [36] of Ogg’s conjecture,1 and Merel’s proof [37] of the uniform boundedness conjecture, there have been several interesting developments in the case of a number field K of fixed degree d over Q. The case of quadratic fields (d = 2) was completed by Kamienny [29], and Kenku and Momose [31] after a long series of papers. However, there is no complete characterization of the torsion structures that may occur for any fixed degree d > 2 at this time.2 Nevertheless, there has been significant progress to characterize the cubic case [27, 24, 39, 23, 3, 50] and the quartic case [28, 25, 26, 40]. Let us define some useful notations to describe more precisely what is known for d ≥ 2: • Let S(d) be the set of primes that can appear as the order of a torsion point of an elliptic curve defined over a number field of degree ≤ d. • Let Φ(d) be the set of possible isomorphism torsion structures E(K)tors, where K runs through all number fields K of degree d and E runs through all elliptic curves over K. • Let Φ∞(d) be the subset of isomorphic torsion structures in Φ(d) that occur infinitely often. More precisely, a torsion structure G belongs to Φ∞(d) if there are infinitely many elliptic curves E, non-isomorphic over Q, such that E(K)tors ' G. Mazur established that S(1) = {2, 3, 5, 7} and Φ(1) = {Cn | n = 1, . . . , 10, 12} ∪ {C2 × C2m | m = 1, . . . , 4} . Kamienny, Kenku and Momose established that S(2) = {2, 3, 5, 7, 11, 13} and Φ(2) = {Cn | n = 1, . . . , 16, 18} ∪ {C2 × C2m | m = 1, . . . , 6} ∪ {C3 × C3r | r = 1, 2} ∪ {C4 × C4} . Date: November 1, 2016. 2010 Mathematics Subject Classification. Primary: 11G05; Secondary: 14H52,14G05,11R16.