p-torsion in elliptic curves over subfields of Q(? p ) (original) (raw)

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.

Criteria for p-ordinarity of families of elliptic curves over infinitely many number fields

International Journal of Number Theory, 2015

Let Ki be a number field for all i ∈ ℤ>0 and let ℰ be a family of elliptic curves containing infinitely many members defined over Ki for all i. Fix a rational prime p. We give sufficient conditions for the existence of an integer i0 such that, for all i > i0 and all elliptic curve E ∈ ℰ having good reduction at all 𝔭 | p in Ki, we have that E has good ordinary reduction at all primes 𝔭 | p. We illustrate our criteria by applying it to certain Frey curves in [Recipes to Fermat-type equations of the form xr + yr = Czp, to appear in Math. Z.; http://arXiv.org/abs/1203.3371 ] attached to Fermat-type equations of signature (r, r, p).

Elliptic curves and class field theory

Arxiv preprint math/0304235, 2003

Suppose E is an elliptic curve defined over Q. At the 1983 ICM the first author formulated some conjectures that propose a close relationship between the explicit class field theory construction of certain abelian extensions of imaginary quadratic fields and an explicit construction that (conjecturally) produces almost all of the rational points on E over those fields. Those conjectures are to a large extent settled by recent work of Vatsal and of Cornut, building on work of Kolyvagin and others. In this paper we describe a collection of interrelated conjectures still open regarding the variation of Mordell-Weil groups of E over abelian extensions of imaginary quadratic fields, and suggest a possible algebraic framework to organize them.