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Author: William A. Stein
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William Stein's Publications: The field generated by the points of small prime order on an elliptic curve

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.