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The truth about torsion in the CM case. (La vérité sur la torsion dans le cas CM.) (English. French summary) Zbl 1333.11051

The main result in the paper is an improvement on the upper limit for the torsion of complex multiplication elliptic curves. It is proven that for all number fields \(F\) of degree \(d\geq 3\) and all complex multiplication elliptic curves \(E\) defined over \(F\) \[ \sharp E(F)[\text{tors}]\leq C d\log\log d, \tag{1} \] where \(C\) is an absolute, effective constant. This improves a previous bound by M. Hindry and J. Silverman [C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 2, 97–100 (1999; Zbl 0993.11028)]. The above bound, combined with a result by F. Breuer [J. Number Theory 130, No. 5, 1241–1250 (2010; Zbl 1227.11072)], tell us that the maximum size \(T_{\text{CM}}(d)\) of the torsion subgroup of a CM elliptic curve over a degree \(d\) number field satisfies: \[ 0<\lim_{d\rightarrow\infty}\frac{T_{\text{CM}}(d)}{d\log\log d}<\infty. \] The key for the proof of the bound (1) is a uniform bound for Euler’s function in imaginary quadratic fields.

MSC:

11G05 Elliptic curves over global fields
11G15 Complex multiplication and moduli of abelian varieties
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References:

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