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.
  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
    \[P*(P*Q)=Q.\]
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
\[\mathbb{Z}.\]