Problems

1. (Jenna) Prove that the set of rational numbers $x$ with height $H(x)$ less than $\kappa$ contains at most $2\kappa^2 + \kappa$ elements.

2. (Jeff) Let $\alpha: \Gamma\to {\mathbb{Q}}^*/{{\mathbb{Q}}^*}^2$ be the map defined in Section 5 of Chapter III of [Silverman-Tate] by the rule

   $$\alpha(\mathcal{O}) = 1 \pmod{{{\mathbb{Q}}^*}^2},$$

   $$\alpha(T) = b \pmod{{{\mathbb{Q}}^*}^2}$$

   $$\alpha(x,y) =x \pmod{{{\mathbb{Q}}^*}^2} x\neq 0$$

   
   
   Prove that if $P_1 + P_2 + T = \mathcal{O}$, then
   $$\alpha(P_1)\alpha(P_2)\alpha(T)\equiv 1\pmod{{{\mathbb{Q}}^*}^2}.$$

3. (Jeff/Mauro) Let $A$ and $B$ be abelian groups and let $\phi:A \to B$ and $\psi:B\to A$ be homomorphisms. Suppose there is an integer $m\geq 2$ such that

   $$\psi\circ \phi(a) = ma a \in A,$$

   $$\phi\circ \psi(b) = mb b \in B$$

   
   
   Suppose further that $\phi(A)$ has finite index in $B$, and $\psi(B)$ has finite index in $A$.
   1. (Jeff) Prove that $mA$ has finite index in $A$, and that the index satisfies the inequality
      $$[A:mA] \leq [A:\psi(B)]\cdot [B:\phi(A)].$$

   2. (Mauro) Give an example to show that it is possible for the inequality in (a) to be a strict inequality.

4. (Jennifer) Let $P\in E({\mathbb{Q}})$ be a point on an elliptic curve. The canonical height of $P$ is
   $$\hat{h}(P) = \lim_{n\to \infty} \frac{\log_e(H(2^nP))}{4^n},$$
   where $H$ is as in Chapter III of [Silverman-Tate]. Define a function $d : {\mathbb{Q}}\to {\mathbb{Z}}$ by letting $d(a/b)$ be the maximum of the number of digits of $a$ and $b$ (where we assume $\gcd(a,b)=1$), and extend $d$ to points $P=(x,y)$ by letting $d((x,y)) = d(x)$. Prove that
   $$\hat{h}(P) = \log_e(10)\cdot \lim_{n\to \infty} \frac{d(2^nP)}{4^n}.$$