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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