Exercises

6..

1. Write down an equation $y^2=x^3+ax+b$ over a field $K$ such that $-16(4a^3+27b^2)= 0$ . Precisely what goes wrong when trying to endow the set $E(K) = \{ (x,y)\in K\times K : y^2 = x^3 + ax +b \} \cup \{\O\}$ with a group structure?

2. One rational solution to the equation $y^2=x^3-2$ is $(3,5)$ . Find a rational solution with $x\neq 3$ by drawing the tangent line to $(3,5)$ and computing the second point of intersection.

3. Let $E$ be the elliptic curve over the finite field $K=\mathbb{Z}/5\mathbb{Z}$ defined by the equation
   $$y^2 = x^3 + x +1.$$
   1. List all $9$ elements of $E(K)$ .
   2. What is the structure of $E(K)$ , as a product of cyclic groups?

4. Let $E$ be the elliptic curve defined by the equation $y^2 = x^3 +1$ . For each prime $p\geq 5$ , let $N_p$ be the cardinality of the group $E(\mathbb{Z}/p\mathbb{Z}{})$ of points on this curve having coordinates in $\mathbb{Z}/p\mathbb{Z}{}$ . For example, we have that $N_{5} = 6, N_{7} = 12, N_{11} = 12, N_{13} = 12, N_{17} = 18, N_{19} = 12, , N_{23} = 24,$    and $N_{29} = 30$ (you do not have to prove this).
   1. For the set of primes satisfying $p\equiv 2\pmod{3}$ , can you see a pattern for the values of $N_p$ ? Make a general conjecture for the value of $N_p$ when $p\equiv 2\pmod{3}$ .
   2. (*) Prove your conjecture.

5. Let $E$ be an elliptic curve over the real numbers  $\mathbb {R}$ . Prove that $E(\mathbb{R})$ is not a finitely generated abelian group.

6. (*) Suppose $G$ is a finitely generated abelian group. Prove that the subgroup $G_{\tor }$ of elements of finite order in $G$ is finite.

7. Suppose $y^2=x^3+ax+b$ with $a,b\in\mathbb{Q}$ defines an elliptic curve. Show that there is another equation $Y^2=X^3+AX+B$ with $A,B\in\mathbb{Z}$ whose solutions are in bijection with the solutions to $y^2=x^3+ax+b$ .

8. Suppose $a$ , $b$ , $c$ are relatively prime integers with $a^2+b^2=c^2$ . Then there exist integers $x$ and $y$ with $x>y$ such that $c=x^2+y^2$ and either $a=x^2-y^2$ , $b=2xy$ or $a=2xy$ , $b=x^2-y^2$ .

9. (*) Fermat's Last Theorem for exponent $4$ asserts that any solution to the equation $x^4+y^4=z^4$ with $x,y,z\in\mathbb{Z}$ satisfies $xyz=0$ . Prove Fermat's Last Theorem for exponent $4$ , as follows.
   1. Show that if the equation $x^2+y^4=z^4$ has no integer solutions with $xyz\neq 0$ , then Fermat's Last Theorem for exponent $4$ is true.
   2. Prove that $x^2+y^4=z^4$ has no integer solutions with $xyz\neq 0$ as follows. Suppose $n^2+k^4=m^4$ is a solution with $m>0$ minimal amongst all solutions. Show that there exists a solution with $m$ smaller using Exercise 6.8 (consider two cases).

10. This problem requires a computer.
    1. Show that the set of numbers $59+1\pm s$ for $s\leq 15$ contains $14$ numbers that are $B$ -power smooth for $B=20$ .
    2. Find the proportion of primes $p$ in the interval from $10^{12}$ and $10^{12}+1000$ such that $p-1$ is $B=10^5$ power-smooth.

11. (*) Prove that $1$ is not a congruent number by showing that the elliptic curve $y^2=x^3-x$ has no rational solutions except $(0,\pm 1)$ and $(0,0)$ , as follows:
    1. Write $y=\frac{p}{q}$ and $x=\frac{r}{s}$ , where $p,q,r,s$ are all positive integers and $\gcd(p,q)=\gcd(r,s)=1$ . Prove that $s\mid q$ , so $q=sk$ for some $k\in\mathbb{Z}$ .
    2. Prove that $s=k^2$ , and substitute to see that $p^2=r^3-rk^4$ .
    3. Prove that $r$ is a perfect square by supposing that there is a prime $\ell$ such that $\ord _{\ell}(r)$ is odd and analyzing $\ord _{\ell}$ of both sides of $p^2=r^3-rk^4$ .
    4. Write $r=m^2$ , and substitute to see that $p^2=m^6-m^2k^4$ . Prove that $m\mid p$ .
    5. Divide through by $m^2$ and deduce a contradiction to Exercise 6.9.