Selmer groups of twists of elliptic curves over K with K-rational torsion points

Jackson Salvatore Morrow

doi:10.1007/s40993-016-0059-1

Abstract

We generalize a result of Frey on Selmer groups of twists of elliptic curves over \(\mathbf Q \) with \(\mathbf Q \)-rational torsion points to elliptic curves defined over number fields of small degree K with a K-rational torsion point. We also provide examples of elliptic curves coming from Zywina that satisfy the conditions of our Corollary D.

Highlights

Let be an odd, rational prime and let E/K be an elliptic curve defined over a number field K
If K = Q, Frey [2] provides explicit examples of quadratic twist of elliptic curves over Q with Q-rational points of odd, prime order whose -Selmer groups are non-trivial; a theorem of Mazur [3] implies that ∈ {3, 5, 7}
Theorem 1.1 ([2], Corollary) Suppose that E/Q is an elliptic curve with a Q-rational torsion point P of odd prime order, and suppose that P is not contained in the kernel of reduction modulo ; in particular, this means that E is not supersingular modulo if ord ≥ 0

Summary

Introduction

Let be an odd, rational prime and let E/K be an elliptic curve defined over a number field K. If K = Q, Frey [2] provides explicit examples of quadratic twist of elliptic curves over Q with Q-rational points of odd, prime order whose -Selmer groups are non-trivial; a theorem of Mazur [3] implies that ∈ {3, 5, 7}. We show th√at for specific quadratic twists Ed, the order of the -torsion of some ray class group of K ( d) divides the order of Sel (Ed, K ), and th√e order of Sel (Ed, K ) divides the order of the -torsion of a different ray class group of K ( d) times the degree of some maximal abelian extension of exponent with prescribed ramification and Galois conditions (cf Theorems A, B for precise statements and Remark 3.1 for a colloquial statement) These results allow us to give explicit applications to elliptic curves defined over Q (cf Corollaries C, D), and we provide explicit examples of elliptic curves over Q satisfying Corollary D in Sect. In order to state our results, we first need to recall some facts concerning prime torsion of elliptic curves defined over number fields of small degree. The exact value of the set S(n) is currently known for n ≤ 5, but reasonable good bounds on S(6) and S(7) are given in [7]

Some remarks about the proofs

Background and notation

Notation We set the following notation:

Galois structure of splitting fields of -covers of E

Proof of Corollary E