Bounds for the torsion of elliptic curves over number fields
Let \(p\) be a prime number and \(X_0(p)\) be the modular curve associated to the group \(\Gamma_0(p)= \left\{\left( \begin{smallmatrix} a &b\\ c &d\end{smallmatrix} \right)\in \text{SL}_2(\mathbb{Z}):c\equiv 0\pmod p \right\}\). Let \(J_0(p)\) be the Jacobian variety of \(X_0(p)\). We put \(H= H_1(X_0(p), \mathbb{Z})\) and we denote by \(T\) the subring of \(\text{End} (J_0(p))\) generated by the Hecke operators \(T_r\) and the Atkin-Lehner involution. Let \({\mathcal T}_e\) be the kernel of the morphism of \(T\)-modules \(T\to H\otimes \mathbb{Q}\) defined by \(t\to te\), where \(e\) is the winding element. The image of the map \({\mathcal T}_e\times J_0(p)\to J_0(p)\) defined by \((t,x)\to tx\) generates an abelian subvariety of \(J_0(p)\) which we denote by \({\mathcal T}_e J_0(p)\). We define the winding quotient \(J_e\) as the quotient \(J_0(p)/{\mathcal T}_e J_0(p)\) which is an abelian variety defined over \(\mathbb{Q}\). The author proves that the group \(J_e(\mathbb{Q})\) is finite and the elements \(T_1e,\dots, T_d e\) are linearly independent in \(H\otimes \mathbb{Q}\) provided \(p> d^{3d^2}\). Finally, combining these two results he proves the following interesting theorem: Let \(E\) be an elliptic curve defined over a number field \(K\) of degree \(d>1\) over \(\mathbb{Q}\). If \(E(K)\) has a point of prime order \(p\), then \(p<d^{3d^2}\).
scientific article; zbMATH DE number 866098