Proceedings of the London Mathematical Society Advance Access published online on November 27, 2006
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdl008
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© 2006 London Mathematical Society
Diophantine analysis and torsion on elliptic curves
Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, BC, Canada V6T 1Z2 pingram{at}math.utoronto.ca
Received 14 October 2005. Revision received 8 February 2006.
In a recent paper of Bennett and the author, it was shown that the elliptic curve defined by y2=x3+Ax+B, where A and B are integers, has no rational points of finite order if A is sufficiently large relative to B (at least if one assumes the abc Conjecture of Masser and Oesterlé). In the present article we show, perhaps surprisingly, that the rational torsion on the above curve is also quite restricted if B is sufficiently large relative to A. In particular, we demonstrate that for any 
>0 there is a constant c such that if A and B are integers satisfying |B|>c |A|6+, then the elliptic curve defined above has no rational torsion points, other than a possible point of order 2 (again making use of the abc Conjecture in some cases). We then extend this by proving similar results for elliptic curves admitting non-trivial 
-isogenies, elliptic curves written in other forms, and elliptic curves over certain number fields. Curiously, the results on isogenies lead to two unexpected irrationality measures for certain algebraic numbers.