Bounding the torsion in CM elliptic curves (Q2741206)

In [Contemp. Math. 133, 175-193 (1992; Zbl 0787.14028)] \textit{A. Silverberg}, using the main theorem of complex multiplication of Shimura and Taniyama, obtained a bound for the order of a point of finite order on a complex multiplication abelian variety over a number field only in terms of the degree of the number field and the dimension of the abelian variety. As a corollary, if \(E/K\) is an elliptic curve defined over a number field \(K\) of degree \(d\) with complex multiplication by an imaginary quadratic order \(\mathcal{O}\) of a quadratic field \(k\), then \(\phi(N)\leq\delta\mu d\), where \(\mu\) denotes the number of roots of unity in \(\mathcal{O}\), \(\delta=1\) or \(1/2\), depending on whether \(k\subset K\) or not. The first result also implies a bound for the torsion subgroup of complex multiplication elliptic curves.NEWLINENEWLINENEWLINEIn this paper the authors' purpose is to give an estimate for the order of the torsion subgroup of a complex multiplication elliptic curve just using the result of Deuring on supersingular primes and elementary algebraic number theory. Let \(M=l_1^{j_1}\ldots l_r^{j_r}\), \(M'=l_1^{(j_1+\delta_1)/2}\ldots l_r^{(j_r+\delta_r)/2}\), where \(\delta_i=0\), if \(j_i\equiv 0\pmod{2}\). Under the same hypotheses, if \(M=|E(K)_{\text{tor}}|\), then \(\phi(M)\leq 2d\), if \(K\cap k=\mathbb{Q}\) and \(\phi(M')\leq 2d\), if \(K\subset K\), but \(k\neq\mathbb{Q}(i),\mathbb{Q}(w)\), where \(w\) denotes a primitive third root of unity. When \(k\) is equal to one of the latter fields then the author's method gives an extra factor of \(2^{d(M)+1}\), where \(d(M)\) is equal to the number of distinct prime divisors of \(M\).