Problems for next time.

1. (Jenna) Let $P$, $P_1$, $P_2$, $P_3$ be points in $\P ^2$ and let $L$ be a line in $\P ^2$.
   1. If $P_1$, $P_2$, and $P_3$ do not lie on a line, prove that there is a projective transformation of $\P ^2$ so that
      $$P_1 \mapsto (0:0:1), \qquad P_2 \mapsto (0:1:0), \qquad P_3 \mapsto (1:0:0).$$

   2. If no three of $P_1$, $P_2$, $P_3$ and $P$ lie on a line, prove that there is a unique projective transformation as in (a) which also sends $P$ to $(1:1:1)$.
   3. Prove that if $P$ does not lie on $L$, then there is a projective transformation of $\P ^2$ so that $L$ is sent to the line $Z=0$ and $P$ is sent to the point $(0:0:1)$.

2. (Jennifer) Let $C$ be the cubic curve $y^2=x^3+1$.
   1. For each prime $5\leq p<30$, describe the group $C({\mathbb{F}}_p)$ of points on this curve having coordinates in the finite field of order $p$. (Use a computer.)
   2. For each prime in (a), let $N_p$ be the number of points in $C({\mathbb{F}}_p)$. (Don't forget the point at infinity.) 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 about the value of $N_p$ when $p\equiv 2\pmod{3}$ and prove that your conjecture is correct.
   3. Find a conjectural pattern for the values of $N_p$ for the set of primes $p\equiv 1\pmod{3}$, and give evidence for your conjecture. Feel free to try to find the answer to this question by looking in other books or asking around the department, since this problem is double starred in Silverman-Tate.

3. (Mauro) Let $C$ be a nonsingular cubic curve given by a Weierstrass equation
   $$y^2 = f(x) = x^3+ax^2 + bx + c.$$
   1. Prove that
      $$\frac{d^2 y}{dx^2} = \frac{2f''(x) f(x) - f'(x)^2}{4yf(x)} = \frac{\psi_3(x)}{4yf(x)}.$$
      Deduce that a point $P=(x,y)\in C$ is a point of order three if and only if $P\neq \O$ and $P$ is a point of inflection on the curve $C$.
   2. Suppose that $a,b,c\in{\mathbb{R}}$. Prove that $\psi_3(x)$ has exactly two real roots, say $\alpha_1, \alpha_2$ with $\alpha_1 < \alpha_2$. Prove that $f(\alpha_1)<0$ and $f(\alpha_2) > 0$. Deduce that the points in $C({\mathbb{R}})$ of order dividing $3$ form a cyclic group of order $3$.

4. (Alex) Let $A$ be an abelian group, and for every integer $m\geq 1$, let $A[m]$ be the set of elements $P\in A$ satisfying $m P = \O$. (Note that $A[m]$ is denoted $A_m$ in [Silverman-Tate].)
   1. Prove that $A[m]$ is a subgroup of $A$.
   2. Suppose that $A$ has order $M^2$ and that for every integer $m$ dividing $M$, the subgroup $A[m]$ has order $m^2$. Prove that $A$ is the direct product of two cyclic groups of order $M$.
   3. Find an example of a non-abelian group $G$ and an integer $m\geq 1$ so that the set $G[m] = \{g \in G : g^m = 1\}$ is not a subgroup of $G$.

5. (Jeff)
   1. Let $f(x) = x^2+ax+b=(x-\alpha_1)(x-\alpha_2)$ be a quadratic polynomial with the indicated factorization. Prove that
      $$(\alpha_1 - \alpha_2)^2 = a^2 - 4b.$$

   2. Let $f(x) = x^3 + ax^2 + bx + c = (x-\alpha_1)(x-\alpha_2)(x-\alpha_3)$ be a cubic polynomial with the indicated factorization. Prove that
      $$(\alpha_1 - \alpha_2)^2 (\alpha_1 - \alpha_3)^2 (\alpha_2-\alpha_3)^2 = -4a^3 c + a^2 b^2 + 18abc - 4b^3 - 27c^2.$$