\begin{questions}
	
\question 
\begin{parts}

\part[6]
\begin{subparts}\subpart State Hensel's lemma.
	\subpart Let $a \in \Q_7^\times$. Show that $a$ has a cube root in $\Q_7$ if and only if $a = 7^nu$ with $n \in \Z$ divisible by $3$, and $u \in \Z_7$ with $u\equiv \pm1 \mod{7}$.
	
	
\end{subparts}
\part[19] Consider the elliptic curve with (projective) equation \[\mathcal{C}: X^3 + Y^3 = 2Z^3\] and distinguished identity point $(1:-1:0) \in \mathcal{C}(\Q)$. 
\begin{subparts}
	\subpart For a point $P = (x_1:y_1:1)$ of $\mathcal{C}$, find a formula for the doubled point $2P$ in terms of rational functions in $x_1$ and $y_1$.

	[\textit{You may find it useful to note that the tangent line to $\mathcal{C}$ at $(1:-1:0)$ has equation $X+Y = 0$.}]  
	\subpart What are the $2$-torsion points of $\mathcal{C}(\Q_7)$?
	
		\subpart By considering your formula for $2P$, or otherwise, show that $\mathcal{C}(\Q_7)$ contains no point $P$ of order $4$. 

[\textit{You may assume without proof that the points in $\mathcal{C}(\Q_7) \cap \{ Z = 0\}$ do not have order $4$.}]
\end{subparts}
\end{parts}


\question Let $p$ be a prime with $p > 5$, and consider the elliptic curve over $\Q$ with equation ${\mathcal E} : y^2 = x^3 + p$.

[{\it Throughout this question, results on formal groups from lectures may be assumed provided they are clearly stated.}]
\begin{parts}
\part[7] Let $q$ be a prime with $q \nmid 6p$. Show that the order $\#{\mathcal E}(\Q)_{{\rm tors}}$ of the torsion subgroup of
${\mathcal E}$ divides the order $\#\widetilde{\mathcal E}_{q}(\mathbb{F}_q)$ of the group of
points on the reduction $\widetilde{\mathcal E}_{q}$ of the curve modulo $q$.

[{\it You may assume without proof that the reduction map modulo a prime of good reduction is a homomorphism.}]

\part[8]

\begin{subparts}
\subpart Compute the order $\#\widetilde{\mathcal E}_{5}(\mathbb{F}_5)$ of the group of
points on the reduction $\widetilde{\mathcal E}_{5}$ of the curve modulo $5$.
\subpart Compute the number of points $\#\widetilde{\mathcal E}_{p}(\mathbb{F}_p)$ of the reduction $\widetilde{\mathcal E}_{p}$ of the curve modulo $p$.
\end{subparts}

\part[10] 
\begin{subparts}
	
\subpart Show that there is no point of ${\mathcal E}(\Q_p)$ reducing to the singular point \[(0,0) \in \widetilde{{\mathcal E}}_{p}(\mathbb{F}_p).\]
\subpart Deduce that $\#{\mathcal E}(\Q)_{{\rm tors}} = 1$.

\end{subparts}

\end{parts}

\newpage
\question  
\begin{parts}
\part[10] Consider an elliptic curve over $\Q$ defined by the equation \[\mathcal{C}: y^2 = x(x^2 + B),\] with $B$ a non-zero integer, together with the 2-isogeny $\hat{\phi}: \mathcal{D}\to \mathcal{C}$ from the curve
\[\mathcal{D}: v^2 = u(u^2 - 4B)\] given by \[\hat{\phi}(u,v) = \left(\frac{1}{4}\left(\frac{v}{u}\right)^2,\frac{1}{8}\left(v + \frac{4Bv}{u^2}\right)\right).\] 

Show that if there is a point $(u,v) \in \mathcal{D}(\Q)$ with $u = rt^2$, for $r$ a squarefree integer and $t \in \Q^\times$, then $r$ divides $4B$ and there are integers $l, m, n$, not all $0$, with $\gcd(l,m) = 1$, satisfying the equation
\[rl^4 - \left(\frac{4B}{r}\right)m^4 = n^2.\]
\part[10] Suppose $p$ is a prime with $p \equiv 3 \mod 8$. Show that the elliptic curve over $\Q$ \[{\mathcal E} : y^2 = x(x^2-p^2)\] has rank zero.

[\textit{You may use any results from lectures provided they are clearly stated. It may also help to	recall that for an odd prime $p$ we have that $2$ is a quadratic residue modulo $p$ if and only if $p \equiv \pm1 \mod{8}$ and $-1$ is a quadratic residue modulo $p$ if and only if $p \equiv 1 \mod{4}$.}]
\part[5] Find a point of infinite order on the curve with equation $y^2 = x(x^2 - 7^2)$, justifying why it has infinite order.
\end{parts}
\end{questions}