\newcommand{\lb}{[\![}
\newcommand{\rb}{]\!]}


\begin{questions}
\question
\begin{parts}
\part[10] Let $E$ be the elliptic curve over $\mathbb{Q}$ with affine equation \[E: Y^2 - Y = X^3 - X^2\] and identity element the point at infinity.
	\begin{subparts}
		\subpart Find a birational transformation from $E$ to a curve with equation of the form \[Y^2 = X^3 + aX + b,\] with $a, b \in \mathbb{Z}$.
		\subpart Find the group of rational torsion points, $E(\mathbb{Q})_{\mathrm{tors}}$. If you find it helpful, you may use without proof the fact that the rank of $E(\Q)$ is $0$.
		
		[\textit{Your answer should give the structure of $E(\mathbb{Q})_{\mathrm{tors}}$, together with explicit rational points of $E$ which are generators for this group.}]
	\end{subparts}
\part[15] Let $\alpha \in \mathbb{Z}_{> 0}$ be a positive integer, and consider the elliptic curve $E_{\alpha}$ over $\mathbb{Q}$ with affine equation \[E_\alpha: Y^2 = X(X^2+\alpha)\] and identity the point at infinity. 

\begin{subparts}
	\subpart For a point $P$ of $E_\alpha$, explain the definition of the point $2P$, and show that $2P \in E_\alpha(\mathbb{Q})$ if $P \in E_{\alpha}(\mathbb{Q})$. 
	\subpart Find a necessary and sufficient condition on $\alpha$ for $E_\alpha$ to have a rational point of order $4$. 
\end{subparts}
\end{parts}	
	
	
\question 
\begin{parts}
\part[10]
Let $F$ be a formal group defined over a characteristic $0$ field $K$. 
\begin{subparts}
	\subpart Explain what a \textit{normalized invariant differential} for $F$ is, and prove that there is at most one normalized invariant differential.
	
	\subpart Assuming the existence of a normalized invariant differential for $F$, define the \textit{logarithm} $\log_F \in K\lb X \rb$ and show that it defines an isomorphism from $F$ to the additive formal group $\widehat{\mathbb{G}}_a$ with group law $X+Y$. 
\end{subparts}

\part[15] 

Let $R$ be the ring of integers in a characteristic $0$ field $K$, complete with respect to a non-Archimedean valuation $|\cdot|$. Suppose that the maximal ideal $\{z \in R: |z| < 1 \}$ is the principal ideal $pR$ for a prime number $p$, and that the residue field of $R$ has order $q = p^k$ for a positive integer $k$.
\begin{subparts}
	
	\subpart Consider the polynomial $h(X) = pX + X^q \in R[X]$. Show that for each integer $a \in \Z$ there is a unique power series $g_a(X) \in R\lb X \rb$ satisfying: \[h(g_a(X)) = g_a(h(X))\text{ and }g_a(X) = aX + (\text{terms of degree}\ge 2).\] 
	
[\emph{Hint: For each $r \ge 1$, show that there is a unique polynomial $g_{a,r}(X) = aX + \cdots \in R[X]$ of degree $\le r$ such that $h(g_{a,r}(X)) = g_{a,r}(h(X)) + (\text{terms of degree}\ge r+1)$. You may use without proof the fact that $z^q = z$ mod $p$ for all $z \in R$.}]

\subpart Suppose that $F(X,Y) \in R\lb X, Y\rb$ is a formal group such that the polynomial $h(X)$ from part (i) defines a homomorphism from $F$ to $F$. Deduce from part (i) that the power series $[p](X)$ giving multiplication by $p$ on the formal group $F$ satisfies $[p](X) = h(X)$. 

%\subpart For which elements $\alpha \in R$ does there exist $z \in R$ with $|z| < 1$ and $pz + z^q = \alpha$? Justify your answer. 	
	
	
\end{subparts}
\end{parts}
\newpage 
\question
\begin{parts}
	\part[10] Consider an elliptic curve $C: Y^2 = X(X^2 + aX + b)$, with $a,b\in\mathbb{Z}$ satisfying $b(a^2-4b) \ne 0$, together with the curve $D: V^2 = U(U^2 - 2aU + (a^2-4b))$. Recall that there is a \emph{$2$-isogeny} $\phi: C \to D$ defined by \[{\phi}(x,y) = \left(\left(\frac{y}{x}\right)^2,y - \frac{by}{x^2}\right).\] 
	
	Define the non-zero map \[q: D(\mathbb{Q}) \to \mathbb{Q}^\times/\left(\mathbb{Q}^\times\right)^2, \] and prove that the preimage of the identity coset $1\cdot \left(\mathbb{Q}^\times\right)^2$ is $\phi(C(\mathbb{Q}))$. 
	
	\part[15] Consider the elliptic curve $C: Y^2 = X(X^2 + 5X + 3)$ defined over $\mathbb{Q}$. 
	\begin{subparts}
		\subpart Find the group of rational torsion points, $C(\mathbb{Q})_{\mathrm{tors}}$.
		
		[\textit{Your answer should give the structure of this group, together with explicit points of $C(\mathbb{Q})$ which are generators.}]
		\subpart Find the rank of the group $C(\Q)$. If the rank is positive, give an example of a point of infinite order. 
	\end{subparts} 
\end{parts}

\end{questions}