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
by the following rules: Given
\[P, Q\in C\]
, then
\[ P*Q\]
is the point on
so that
\[P, Q\]
are colinear.
  1. Explain why this law is commutative.
  2. Prove that there is no identity element for this composition law.
  3. Prove that this composition law is not associative.
  4. Explain why
Exercise 3. Let
be the cubic curve 
, and take 
\[P=(3,8)\in E\]
. Compute 
Exercise 4. Show that on the cubic curve 
, the points 
\[(-1, 4),(4, 9)\]
\[(8, 23)\]
are in the subgroup generated by 
\[(-2, 3)\]
\[(2, 5).\]
Exercise 5. Let 
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
, 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