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 \(p \leq 7\).


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