zbMATH — the first resource for mathematics

Torsion on elliptic curves in isogeny classes. (English) Zbl 1116.11037
Let \({\mathcal C}\) be the \(K\)-isogeny class of an elliptic curve over a number field \(K\). For a prime number \(l\) and \(E\in{\mathcal C}\) let \[ E(K)_{(l)}\cong{\mathbb Z}/l^s{\mathbb Z}\oplus{\mathbb Z}/l^t{\mathbb Z} \] with \(s\geq t\geq 0\) be the \(l\)-primary part of the torsion subgroup of \(E(K)\). The authors investigate which types of such groups can occur as \(E\) varies over \({\mathcal C}\), describing the subset of corresponding tuples \((s,t)\) in the \(s,t\)-plane. Some results of N. Katz [Invent. Math. 62, 481–502 (1981; Zbl 0471.14023)] are also refined. The proofs are based on the same approach as in that paper, namely reinterpretation of the problem in terms of Gal\((\overline{K}/K)\)-stable lattices in the \(l\)-adic Tate-module of \(E\).
11G05 Elliptic curves over global fields
11F80 Galois representations
Full Text: DOI
[1] Nicholas M. Katz, Galois properties of torsion points on abelian varieties, Invent. Math. 62 (1981), no. 3, 481 – 502. · Zbl 0471.14023 · doi:10.1007/BF01394256 · doi.org
[2] Tetsuo Nakamura, Cyclic torsion of elliptic curves, Proc. Amer. Math. Soc. 127 (1999), no. 6, 1589 – 1595. · Zbl 0922.11051
[3] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. · Zbl 0585.14026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.