Mordell-Weil Groups and Torsion subgroups

The Mordell-Weil group of an abelian variety (and its torsion subgroup) is again an isomorphism invariant, which varies up to finite index in the isogeny class. In the case of the above elliptic curves $E_1$ and $E_2$ we have

$$E_1(\mathbb{Q}) \cong \mathbb{Z}/5\mathbb{Z}$$ while $$E_2(\mathbb{Q}) = \{O\}.$$

Computing Mordell-Weil groups is very well studied, though still there is no provably-correct algorithm for computing it. Computing torsion subgroups for modular abelian varieties is less well studied. The paper [AS05] describes some algorithms that give upper and lower bounds.

Problem 8.4.1   Given a modular abelian variety $A$ attached to a newform $f\in S_2(\Gamma_0(N))$ find an algorithm to compute $\char93 A(\mathbb{Q})_{{\mathrm{tor}}}$ .

Problem 8.4.2   Given a modular abelian variety $A$ attached to a newform $f\in S_2(\Gamma_0(N))$ find an algorithm to compute the group structure of $A(\mathbb{Q})_{{\mathrm{tor}}}$ .