The field generated by the points of small prime order on an elliptic curve

Loic Merel and William A. Stein


Abstract
Let Q be an algebraic closure of Q, and for any prime number p, denote by Q(u_p) the cyclotomic subfield of Q generated by the pth roots of unity.

Theorem.  Let p be a prime. If there exists an elliptic curve E over Q(up) such that the points of order p of E(Q) are all Q(u_p)-rational, then p=2,3,5,13 or p>1000.



The paper: pdf, dvi, ps. There is also a page with some software and tables related to the paper.