Remarks

Your reading for this coming week is about points of finite order and how to work modulo $p$. The first section of chapter 2 characterizes the points of order $2$ or $3$ on an elliptic curve, and the second sectoin discusses the analytic way of viewing an elliptic curve as a complex torus. This analytic point of view makes it easy to see that the group of points of order dividing $m$ on an elliptic curve is isomorphic to ${\mathbb{Z}}/m \times {\mathbb{Z}}/m$. Section 3 contains some remarks about discriminants of cubics that are useful in the theorem that bounds torsion points, which you will read about next week. The reading from the appendix is concerned with how to define a reduction map from $\P ^2({\mathbb{Q}}) \to \P ^2({\mathbb{F}}_p)$. If $E$ is an elliptic curve with discriminant not divisible by $p$, this map induces a group homomorphism $E({\mathbb{Q}})\to E({\mathbb{F}}_p)$.

You should also read a proof that every finitely generated abelian group can be written as a product of cyclic groups.