Abstract
This note reformulates Mazur’s result on the possible orders of rational torsion points on elliptic curves over mathbb {Q} in a way that makes sense for arbitrary genus one curves, regardless whether or not the curve contains a rational point. The main result is that explicit examples are provided of ‘pointless’ genus one curves over mathbb {Q} corresponding to the torsion orders 7, 8, 9, 10, 12 (and hence, all possibilities) occurring in Mazur’s theorem. In fact three distinct methods are proposed for constructing such examples, each involving different in our opinion quite nice ideas from the arithmetic of elliptic curves or from algebraic geometry.
Highlights
A famous theorem [10, Thm. (7’)] by B
All positive divisors occur as order of rational points, for infinitely many pairwise non-isomorphic curves E
We reformulate Mazur’s result in terms of the group IsomQ(E) of invertible morphisms E → E defined over Q of an elliptic curve E/Q, as follows
Summary
A famous theorem [10, Thm. (7’)] by B. 110] and [13, Conjecture 1], asserts that if E is an elliptic curve defined over the rational numbers Q any point P ∈ E(Q) of finite order has the property that ord(P) divides one of {7, 8, 9, 10, 12}. We reformulate Mazur’s result in terms of the group IsomQ(E) of invertible morphisms E → E defined over Q of an elliptic curve E/Q, as follows. For every positive divisor d of one of {7, 8, 9, 10, 12}, examples of elliptic curves E/Q exist such that IsomQ(E) contains a dihedral group Dd of order 2d. Using Mazur’s result, the converse holds: let Dd ⊂ IsomQ(E) be a dihedral group of order 2d. In the remaining case one has τP = φ, d = ord(φ) = ord(τP) = ord(P) divides (at least) one of {7, 8, 9, 10, 12} by Mazur’s theorem
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