## Exercises

Exercise 1. For each of the following conics, either find a rational point or prove that there are no rational points.

1. $x^2+y^2=6$
2. $3x^2+5y^2=4$
3. $3x^2+6y^2=4$
 Exercise 2. Define a composition law on the points of a cubic $C$ by the following rules: Given $P, Q\in C$, then $P*Q$ is the point on $C$ so that $P, Q$ and $P*Q$ are colinear. Explain why this law is commutative. Prove that there is no identity element for this composition law. Prove that this composition law is not associative. Explain why $P*(P*Q)=Q.$ Exercise 3. Let $E$ be the cubic curve $y^2=x^3-43x+166$, and take $P=(3,8)\in E$. Compute $2P,4P$ and $8P.$ Exercise 4. Show that on the cubic curve $y^2=x^3+17$, the points $(-1, 4),(4, 9)$ and $(8, 23)$ are in the subgroup generated by $(-2, 3)$ and $(2, 5).$ Exercise 5. Let $f:\mathbb{C}/L_1\to\mathbb{C}/L_2$ be a holomorphic map between complex tori. Prove that there exists $\alpha, z_0\in\mathbb{C}$ such that $f(z+L_1)=\alpha z+z_0+L_2.$ Exercise 6. Prove that for most (in some sense that you should make precise) complex tori $\mathbb{C}/L$, the abelian group ${\rm End}(\mathbb{C}/L):=\{f:\mathbb{C}/L\to \mathbb{C}/L: f \text{ is holomorphic, } f(0+L)=0+L\}$ is equal to $\mathbb{Z}.$