A remark on torsion points of elliptic curves on small extensions of Q Tr^ a n Qu^ o c D^ an a and Nguy^ e~ n Qu^ o c Th a ng b y (original) (raw)

Torsion of Rational Elliptic Curves over Abelian Extensions of Q

2017

Let E be an elliptic curve de ned over Q. We investigate E(K)tors for various abelian extensions K of Q. For number elds, a theorem of Merel implies a uniform bound on the size of the torsion subgroup based on the degree of the number eld. We discuss a number of results that classify torsion subgroups of elliptic curves over number elds of a xed degree. We prove a classi cation of torsion subgroups for elliptic curves E/Q base extended to quartic Galois number elds. For in nite extensions of Q, the Mordell-Weil theorem no longer applies, and so the torsion subgroup of E/Q is a priori not even nite. We prove that when base extended to Q, the size of the torsion subgroup of an elliptic curve E/Q is uniformly bounded. Moreover, we classify all groups that arise as E(Q)tors for elliptic curve E/Q. Torsion of Rational Elliptic Curves over Abelian Extensions of Q

On the torsion of rational elliptic curves over quartic fields

Mathematics of Computation, 2017

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 C n = Z/nZ for brevity, and we call C n × C m 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.