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Sur les points de torsion des courbes elliptiques. (On torsion points of elliptic curves). (French) Zbl 0737.14004
Les pinceaux de courbes elliptiques, Sémin., Paris/Fr. 1988, Astérisque 183, 25-36 (1990).
[For the entire collection see Zbl 0702.00011.]
Let \(E\) be an elliptic curve defined over a number field \(K\). The number of torsion points of \(E\) which are rational over \(K\) is finite, and a classical conjecture states that it is bounded by a constant depending only on \(K\). Bounds for \(\text{Card}(E(K)_{tors})\) in terms of the number \(\beta(E)=\log(N_{K/\mathbb{Q}}(\Delta(E))/\log(N_{K/\mathbb{Q}}(N(E))\) (\(\Delta(E)\)=discriminant, \(N(E)\)=conductor of \(E\)) were determined analytically by Hindry and Silverman. The authors, using algebraic methods, establish an upper bound, depending only on \(K\), for \(\text{Card}(E(K)_{tors})\) when either (i) \(E\) has good reduction everywhere, (ii) \(E\) has at least one place with bad additive reduction, (iii) \(E\) has at least two places with bad additive reduction; the last bound is proven to be sharp. This result, together with a theorem of Frey, also implies that the number of torsion points of \(E\) over \(K\) is bounded by \(\beta(E)\).

MSC:
14G05 Rational points
14H52 Elliptic curves