Some remarks on almost rational torsion points par

Let G be a commutative algebraic group defined over a perfect field k. Let k be an algebraic closure of k and Γk be the Galois group of k over k. Following Ribet ([1], [19], see also [7]), we say a point P ∈ G(k) is almost rational over k if whenever σ, τ ∈ Γk are such that σ(P )+ τ(P ) = 2P , then σ(P ) = τ(P ) = P . We denote the almost rational points of G over k by Gar k . Let Gtors denote the torsion subgroup of G(k) and G ′ tors the subgroup of points of order prime to the characteristic of k. Let Gar tors,k = G ar k ∩Gtors and Gar,′ tors,k = G ar k ∩Gtors. For any N ≥ 1, we let G[N ] denote the subgroup of Gtors consisting of points of order dividing N , and O denote the origin of G. Using unpublished results of Serre [22], Ribet showed that if K is a number field and G is an abelian variety over K, then Gar tors,K is a finite set [1], [19]. Let C be a nonsingular projective curve of genus at least 2 over K, and φQ : C → J an Albanese embedding of C into its Jacobian J with a K-rational point Q as base point. Then for any P ∈ C(K) which is not a hyperelliptic Weierstrass point, φQ(P ) ∈ Jar K . Hence Ribet’s result gives a new proof of the Manin-Mumford conjecture, originally proved by Raynaud [18], that the torsion packet φQ(C)∩Jtors is finite. In [7], Calegari determined all the possibilities for the Q-almost rational torsion points on a semi-stable elliptic curve over Q. In [4] and [5] the authors defined and studied the notion of the set of singular torsion points Esing on an elliptic curve E over a field of characteristic different from 2, which is an analogue of torsion packets for elliptic curves. Since singular torsion points of order at least 3 are almost rational, Ribet’s result also shows that Esing is a finite set when E is defined over a field of characteristic 0. The purpose of this paper is to prove a number of properties of almost rational torsion points on various classes of commutative algebraic groups over fields of arithmetic interest. Our first topic concerns uniform bounds