Reduction and torsion points of abelian varieties

This thesis consists of two parts. First, reduction of elliptic curves (and more generallyabelian varieties) with torsion points is studied. In the second part, we prove a conjecture of Agashe, which is a consequence of the Birch–Swinnerton-Dyer (BSD) conjecture for elliptic curves over the rationals. The connection between the two parts is that we can put restrictions on the torsion subgroups of certain twists of elliptic curves using reduction.