## Bounds for the torsion of elliptic curves over number fields. (Bornes pour la torsion des courbes elliptiques sur les corps de nombres.)(French)Zbl 0936.11037

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 $$p<d^{3d^2}$$.

### MSC:

 11G05 Elliptic curves over global fields 14H25 Arithmetic ground fields for curves 14H52 Elliptic curves
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