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Torsion points on abelian varieties of CM-type. (English) Zbl 0683.14002
Compos. Math. 68, No. 3, 241-249 (1988); corrigendum ibid. 154, No. 3, 594 (2018).
Let $$A$$ be an abelian variety of dimension $$d$$ of CM-type over a number field $$k$$ and $$t$$ a torsion point on $$A$$ of order $$N$$.
Denote by $$D$$ the number of conjugates of $$t$$ over $$k$$. The author shows, as a consequence of the general theorems 1 and 2, that for every $$\varepsilon >0$$, there exists an explicitly given (contrary to Serre’s result) positive constant $$C_{k,d,\varepsilon}$$, depending only on $$k$$, $$d$$, $$\varepsilon$$ and not on $$A$$ (contrary to the results formerly obtained by Masser and Bertrand such that $D\geq C_ d(N)\cdot [k:\mathbb Q]^{-1}\geq C_{k,d,\varepsilon}\cdot N^{1-\varepsilon}$ where $C_ d(N)=\varphi (N)/((12)^ dd!2^{(d-1)\nu (N)+1}$ with Euler’s function $$\varphi$$ and the number $$\nu(N)$$ of prime divisors of $$N$$. This inequality shows that for given $$k$$ and $$d$$ there are only finitely many possibilities for $$k$$-torsion points on $$A$$.

##### MSC:
 11G10 Abelian varieties of dimension $$> 1$$ 11G15 Complex multiplication and moduli of abelian varieties 14G05 Rational points 14K22 Complex multiplication and abelian varieties
##### Keywords:
number of conjugates of torsion point; abelian variety
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##### References:
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