Problem 1 (yau-contest 2014)
Section: Algebra and Number Theory

Let the special linear group (of order 2)
$$SL_2(\mathbb{R}) = \left\{ g = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in M_2(\mathbb{R}) : \det g = 1 \right\}$$
act on the upper half plane $$\mathbb{H} = \{ z = x + i y \in \mathbb{C} : y > 0 \}$$ linearly fractionally:
$$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot z = \frac{a z + b}{c z + d}.$$

(a) (5 points) Prove that the action is transitive, i.e., for any two $z_1, z_2 \in \mathbb{H}$, there is $g \in SL_2(\mathbb{R})$ such that $g z_1 = z_2$.

(b) (5 points) For a fixed $z \in \mathbb{H}$, prove that its stabilizer $$G_z = \{ g \in SL_2(\mathbb{R}) : g z = z \}$$ is isomorphic to $$SO_2(\mathbb{R}) = \{ g \in M_2(\mathbb{R}) : g g^t = 1 \},$$ where $g^t$ is the transpose of $g$.

(c) (10 points) Let $\mathbb{Z}$ be the set of integers and let
$$\Gamma(2) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathbb{Z}) : a,b,c,d \in \mathbb{Z},\; a - 1 \equiv d - 1 \equiv b \equiv c \equiv 0 \pmod{2} \right\}$$
be a discrete subgroup of $SL_2(\mathbb{Z})$ (no need to prove this), and let it act on $\mathbb{Q} \cup \{\infty\}$ linearly fractionally as above. How many orbits does this action have? Give a representative for each orbit.

Problem 2 (yau-contest 2014)
Section: Algebra and Number Theory

Let $p \ge 7$ be an odd prime number.

(a) (5 points) (to warm up) Evaluate the rational number $\cos(\pi/7)\cdot \cos(2\pi/7)\cdot \cos(3\pi/7)$.

(b) (15 points) Show that $$\prod_{n=1}^{(p-1)/2} \cos\big(\tfrac{n\pi}{p}\big)$$ is a rational number and determine its value.

Problem 3 (yau-contest 2014)
Section: Algebra and Number Theory

(20 points, 10 points each) For any $3 \times 3$ matrix $A \in M_3(\mathbb{Q})$, let $A^{\mathrm{db}}$ be the $6 \times 6$ matrix
$$A^{\mathrm{db}} := \begin{pmatrix} 0 & I_3 \\ A & 0 \end{pmatrix}.$$

(a) Express the characteristic and minimal polynomials of $A^{\mathrm{db}}$ over $\mathbb{Q}$ in terms of the characteristic and minimal polynomial of $A$.

(b) Suppose that $A, B \in M_3(\mathbb{Q})$ are such that $A^{\mathrm{db}}$ and $B^{\mathrm{db}}$ are conjugate in the sense that there exists an element $C \in GL_6(\mathbb{Q})$ such that $$C \cdot A^{\mathrm{db}} \cdot C^{-1} = B^{\mathrm{db}}.$$ Are $A$ and $B$ conjugate? (Either prove this statement or give a counterexample.)

Problem 4 (yau-contest 2014)
Section: Algebra and Number Theory

(20 points) Classify all groups of order $8$.

Problem 5 (yau-contest 2014)
Section: Algebra and Number Theory

Let $V$ be a finite dimensional vector space over complex field $\mathbb{C}$ with a non-degenerate symmetric bilinear form $(\, , \,)$. Let
$$O(V) = \{ g \in GL(V) \mid (g u, g v) = (u, v),\; u, v \in V \}$$
be the orthogonal group.

(a) (10 points) Prove that
$$(V \otimes_{\mathbb{C}} V)^{O(V)} \cong \operatorname{End}_{O(V)}(V),$$
and construct one such isomorphism. Here $O(V)$ acts on $V \otimes_{\mathbb{C}} V$ via $g(a \otimes b) = g a \otimes g b$, and $(V \otimes_{\mathbb{C}} V)^{O(V)}$ is the fixed point subspace of $V \otimes V$.

(b) (10 points) Prove that the fixed point subspace $(V \otimes_{\mathbb{C}} V)^{O(V)}$ is $1$-dimensional.

Problem 6 (yau-contest 2014)
Section: Algebra and Number Theory

(20 points) Let $c$ be a non-zero rational integer.

(a) (6 points) Factorize the three variable polynomial
$$f(x,y,z) = x^3 + c y^3 + c^2 z^3 - 3 c x y z$$
over $\mathbb{C}$ (you may assume $c = \theta^3$ for some $\theta \in \mathbb{C}$).

(b) (7 points) When $c = \theta^3$ is a cube for some rational integer $\theta$, prove that there are only finitely many integer solutions $(x,y,z) \in \mathbb{Z}^3$ to the equation $f(x,y,z) = 1$.

(c) (7 points) When $c$ is not a cube of any rational integers, prove that there are infinitely many integer solutions $(x,y,z) \in \mathbb{Z}^3$ to the equation $f(x,y,z) = 1$.