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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
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References:
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