William Stein's homepage

The torsion subgroup

I can compute upper and lower bounds on $\char93 A(\mathbf{Q})_{\tor}$, but I can not determine $\char93 A(\mathbf{Q})_{\tor}$ in all cases. Experimentally, the deviation between the upper and lower bound is reflected in congruences with forms of lower level; I hope to exploit this in a precise way. I also obtained the following intriguing corollary that suggests cancellation between torsion and c_p; it generalizes to higher weight forms, thus suggesting a geometric explanation for reducibility of Galois representations.

Corollary 2   Let n be the order of the image of $(0)-(\infty)$ in A(Q), and let m be the largest square dividing N. Then $$n\cdot L(A,1)/\Omega_A$$ is an integer, up to a unit in Z[1/(2m)].