Almost rational torsion points on semistable elliptic curves.(English)Zbl 1002.14004

An almost rational torsion point on an abelian variety $$A$$ over a perfect field $$K$$ is a geometric torsion point $$P$$ of $$A$$ with the property that whenever $$\sigma$$ and $$\tau$$ are elements of the absolute Galois group of $$K$$ such that $$P^\sigma + P^\tau = 2P$$, we must have $$P = P^\sigma = P^\tau$$. The definition is due to Ribet, who used the concept to show that the Manin-Mumford conjecture follows from results of Serre. Ribet’s argument requires the analysis of almost rational torsion points on abelian varieties of dimension greater than $$1$$.
The present paper considers the case of almost rational torsion points on semistable elliptic curves over $$\mathbb{Q}$$. The main theorem is that a non-rational torsion point $$P$$ on such an elliptic curve $$E$$ is almost rational if and only if it can be written as a sum $$Q + R + S$$, where $$Q$$ generates a $$\mu_3$$ subgroup of $$E[3]$$, where $$R$$ is an element of $$E(\mathbb{Q})[9]$$, and where $$S$$ is an element of $$E(\mathbb{Q}(\zeta_3))[16]$$. Using this theorem the author easily produces an elliptic curve whose set of almost rational torsion points does not form a group.
The proof of the main theorem depends on the study of the Galois module $$M$$ generated by the Galois conjugates of a non-trivial almost rational torsion point $$P$$. The author uses the modularity of $$E$$ to show that if $$\ell$$ is the largest prime dividing the order of $$P$$, then $$E[\ell]$$ is a reducible Galois module, so that $$\ell$$ is at most $$7$$ (by Mazur’s theorem). The proof is completed by a case-by-case analysis of the possible ramification of the module $$M$$ at the primes $$p \leq 7$$.

MSC:

 14G05 Rational points 14K05 Algebraic theory of abelian varieties 14H52 Elliptic curves 11G05 Elliptic curves over global fields
Full Text: