Problems

1. (Jenna)
   1. Prove that the additive group of rational number $({\mathbb{Q}},+)$ is not finitely generated.
   2. Prove that the multiplicative group of nonzero ratoinal numbers $({\mathbb{Q}}^*,*)$ is not finitely generated.
   3. Prove that the group of real point $E({\mathbb{R}})$ on an elliptic curve is not finitely generated.

2. (Jeff) Let $p$ be a prime and let $C_p$ be the curve $y^2=x^3+px$.
   1. Prove that the rank of $C_p({\mathbb{Q}})$ is either 0, $1$, or $2$.
   2. If $p\equiv 7 \pmod{16}$, prove that $C_p$ has rank 0.
   3. If $p\equiv 3\pmod{16}$, prove that $C_p$ has rank either 0 or $1$. (Can the rank ever be 0?)

3. (Mauro/Alex) Using the method developed in Section III.6 of [Silverman-Tate], find the rank of each of the following curves. Check your answers with the output from MAGMA and/or mwrank.
   1. (Mauro) $y^2 = x^3 + 3x$
   2. (Alex) $y^2 = x^3 + 5x$
   3. (Mauro) $y^2 = x^3 + 73x$
   4. (Alex) $y^2 = x^3 + 7x$

4. (Jennifer) Let $p\geq 3$ be a prime, and let $m\geq 1$ be an integer which is relatively prime to $p-1$.
   1. Prove that the map $x\mapsto x^m$ is an isomorphism form ${\mathbb{F}}_p^*$ to itself.
   2. Prove that the equation
      $$x^m + y^m + z^m = 0$$
      has exactly $p+1$ projective solutions with $x,y,z\in {\mathbb{F}}_p$.