Almost Rational Torsion Points on Semistable Elliptic Curves

If P is an algebraic point on a commutative group scheme A/K, then P is almost rational if no two non-trivial Galois conjugates σP , τP of P have sum equal to 2P . In this paper, we classify almost rational torsion points on semistable elliptic curves over Q. 1 Definitions and Results Let X be an algebraic curve of genus greater than one. Let J(X) be the Jacobian variety of X, and embed X in J(X). The Manin-Mumford conjecture states that the set of torsion points Xtors := X ∩ Jtors is finite. This conjecture was first proved in 1983 by Raynaud [9]. It has long been known that the geometry of X imposes strong conditions on the action of Galois on Xtors. An approach to the Manin-Mumford conjecture using Galois representations attached to Jacobians was first suggested by Lang [7]. Recently, by exploiting the relationship between the action of Galois on modular Jacobians and the Eisenstein ideal (developed by Mazur [8]) Baker [1] and (independently) Tamagawa [20] explicitly determined the set of torsion points of X0(N) for N prime. Developing these ideas further, Ribet defined the notion of an almost rational torsion point, and used this concept to derive the Manin-Mumford conjecture [12] using some unpublished results of Serre [13]. One idea suggested by these papers is that a possible approach to finding all torsion points on a curve X is to determine the set of almost rational torsion points on J(X). Moreover, the concept of an almost rational torsion point makes sense for any Abelian variety, or more generally any commutative group scheme, and the set of such points may be interesting to study in their own right. In this paper we consider almost rational points on semistable elliptic curves over Q, and prove (in particular) that they are all defined over Q( √ −3).