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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 \(E\).
MSC:
11G05 Elliptic curves over global fields
11F80 Galois representations
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[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
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