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


[\textit{For all elliptic curves in this paper, the identity point $O$ is the point at infinity.}]
\begin{questions}
\question 	Let $p$ be a prime number and let $F$ be a formal group defined over $\Z_p$.
\begin{parts}
	\part[10]\hfill
	\begin{subparts}
		\subpart Show that the multiplication by $p$ homomorphism on $F$ can be written as
		\[[p](T) = pf(T) + g(T^p) \] for power series $f(T), g(T) \in T\Z_p\lb T\rb$. 
		
		[\textit{You may assume existence and uniqueness of a normalized invariant differential for $F$ without proof.}]
		
		\subpart Prove that, for every $x \in p\Z_p$, \[\lim_{n \to \infty}[p^n](x) = 0.\]
	\end{subparts}
	
	\part[15]\hfill
	\begin{subparts}
		\subpart Suppose that a homomorphism $f : F \to F$ satisfies $f(T) = 0 \mod{T^2}$. Show that $f(T) = 0$. 
		
		\subpart Show that $[p^n](T) \in (p,T)^{n+1}$ for every $n \ge 1$. 
		
		[\textit{Here $(p,T)$ denotes the ideal in $\Z_p\lb T\rb$ generated by $p$ and $T$.}] 		
		\subpart Prove that, for every $a \in \Z_p$, there exists a unique homomorphism $[a] : F \to F$ satisfying $[a](T) = aT \mod{T^2}$.
		
		[\textit{Hint for uniqueness: to compare two homomorphisms $f(T)$ and $g(T)$, consider the power series $F(f(T),i\circ g(T))$, where $i(T)$ is the inverse for the formal group $F$.}]
		
	\end{subparts}
	
%	\part[6] Let $E: y^2 = x^3 + Ax + B$ be an elliptic curve over $\Q$, with identity the point at infinity and $A, B \in \Z$ and assume that $p\nmid 2(4A^3+27B^2)$. Let $P_1, P_2 \in E(\Q)$ be two rational points. 
%	\begin{subparts}
%\subpart Show that there is an integer $m$ such that $[m]P_1$ and $[m]P_2$ lie in $E_1(\Q_p)$ (the kernel of reduction). 
%
%\subpart If $P \in E_1(\Q_p)$, denote by $z(P) = -x(P)/y(P)$ the usual coordinate for the formal group law of $E$. We assume that $F = F_E$ is this formal group law. Show that there are $p$-adic integers $a,b \in \Z_p$ such that $[a](z([m]P_1)) = [b](z([m]P_1))$, with the power series $[a](T), [b](T)$ from part (b)(iii).
%\end{subparts}
\end{parts}

\question
\begin{parts}	
	
\part[7] Consider an elliptic curve over $\Q$ defined by the equation \[E: y^2 = x^3 + Ax + B,\] with $A, B \in \Z$. 

Show that if a point $P = (x_0,y_0) \neq O$ of $E(\Q)$ has finite order, then $x_0, y_0 \in \Z$. 

[{\it In this question, results on formal groups from lectures may be assumed, provided they are clearly stated.}]  
	
\part[10] Now suppose that $E$ has equation \[E: y^2 = x^3 + B,\] for a non-zero integer $B$. 
	\begin{subparts}
		\subpart For a point $P = (x_0,y_0) \in E(\Q)$ with $y_0 \neq 0$, find a formula for the $x$-coordinate of $2P$ in terms of $x_0$. You should write your formula in the form \[x(2P) = \frac{a(x_0)}{4b(x_0)},\] with $a(x)$ and $b(x)$ coprime polynomials.
		
		\subpart By considering the polynomial $3x^2 a(x) - (3x^3-27B)b(x)$, or otherwise, deduce that  \[y_0^2\left(12 x_0^2 x(2P) - (3x_0^3-27B)\right) = 27B^2.\]
		
	\subpart Show that if $P$ has finite order, then the integer $y_0$ divides $27B^2$.
	\end{subparts}
\part[8] Let $p$ be a prime number and consider the elliptic curve $E: y^2 = x^3 + p$. 
\begin{subparts}
\subpart	Show that the order of $E(\Q)_{\mathrm{tors}}$ is odd and less than or equal to $25$. 
\subpart Find a specific prime $p$ such that $E(\Q)_{\mathrm{tors}}$ is trivial, justifying your answer. 
\end{subparts}
\end{parts}

\newpage
\question
\begin{parts}	
	\part[5] Let $E$ be the elliptic curve $y^2 = x^3 + 10 x -2$, and consider the point \[P = (1,3) \in E(\Q).\] Explain how to use the computation of $3P$ to factorize the integer 4331. 
	
	[\textit{Note that $(13/6)^2 = 125 \mod{4331}$}.]
	
	\part[5] Find the torsion subgroup $C(\Q)_{\mathrm{tors}}$ for the elliptic curve over $\Q$ defined by \[ C: y^2 = x(x^2- 25).\]
	
	\part[15] Find the rank of $C(\Q)$. If the rank is greater than $0$, give (with justification) an example of a point of infinite order.
	
	[\textit{You might find it helpful to recall the formula $\hat{\phi}(u,v) = \left(\frac{1}{4}\left(\frac{v}{u}\right)^2,\frac{1}{8}\left(v - \frac{100v}{u^2}\right)\right)$ for the $2$-isogeny from the curve $D: v^2 = u(u^2 + 100)$ to $C$.}]
	
\end{parts}	

\end{questions}