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\begin{document}

\centerline{\bf Elliptic Curves. MT 2025. Sheet 3.}
\bigskip

\noindent{\it Section A}

\begin{enumerate}
	\item Let $K$ be a field, complete with 
respect to a
non-Archimedean valuation $|\ |$, and
let $ R = \{ x\in K : |x| \leqslant 1\}$.
Let $f(X) \in R[x]$ be a monic polynomial with discriminant~$D$, and let
$a_0 \in R$ satisfy $|f(a_0)| < |D|^2$. Show that $f(X)$
has a root $a\in R$.

(\emph{Hint: use Comment 1.17(a) in lecture notes}.)

\item Let $\e : Y^2 = X^3 + 17$, defined
over~$\Q$, and $\widetilde \e : Y^2 = X^3 + 2$, defined over
$\F_5$. What does $(-64/25 , 59/125) \in \e (\Q)$ map to
under the reduction map modulo~$5$?
\end{enumerate}
\medskip

\noindent{\it Section B}

\begin{enumerate}[resume] \item Show that the curve $2 Y^2 = X^4 - 17$
has points in $\R$ and every $\Q_p$, but not in~$\Q$. 
\par\noindent
(\emph{Hint: For showing that there are points in every $\Q_p$,
it is helpful to use Theorem~2.16. Note also that
the curve is birationally equivalent to $V^2 = 2 X^4 - 34$,
where $V = 2Y$. For showing there are
no points in~$\Q$,
first show that, if there were points in~$\Q$, then there would exist
$r,s,t\in \Z$ with $\hbox{gcd}(r,t) = 1$ such
that $2 s^2 = t^4 - 17 r^4$, and then show that any prime dividing
$s$ is a quadratic residue modulo~$17$}.)

\item Give examples of elliptic curves
defined over~$\Z_p$ ($p\not= 2$) such that $\widetilde \e$,
defined over~$\F_p$, has:
\par \noindent {(a)} A cusp which lifts to a point
in~$\e (\Q_p)$.
\par \noindent {(b)} A cusp which does not lift to a point
in~$\e (\Q_p)$.
\par \noindent {(c)} A node which lifts to a point
in~$\e (\Q_p)$.
\par \noindent {(d)} A node which does not lift to a point
in~$\e (\Q_p)$.

\item Let $m$ be a positive integer and $F(X,Y) \in R\lb X,Y\rb$ a formal group. Prove that
the multiplication by $m$ series $[m](T) \in R\lb T\rb$ is a homomorphism from $F$ to $F$.

\item Let $K$ be a field, complete with respect to a discrete
non-Archimedean valuation, with valuation ring $R$
and maximal ideal $\M = (\varpi)$ with uniformizer $\varpi$. Suppose the residue field $k = R/\M$ has characteristic $p$ and $K$ has characteristic $0$. Let~$F$ be a formal group defined over~$R$.

(\emph{You might find it helpful to use the result in Question 10 below.})
\par \noindent {(a)} Show that the power series $\log_F$ defines a homomorphism \[(F(\M),\oplus) \to (K,+)\]
\par\noindent {(b)} Suppose $r \ge 1$ is an integer such that $|\varpi|^r < |p|^{1/(p-1)}$. Show that for $x \in \M^r$, $|\log_F(x)| = |x|$, and deduce that the restriction of $\log_F$ to $(F(\M^r)$ is injective.
\par\noindent {(c)} Use part (b) to give an alternative proof of the $n=1$ case of Theorem 5.19 in the lecture notes.


\item Find the torsion group over~$\Q$
for each of:
\par\smallskip\par\noindent {(a)} $Y^2 = X^3 + 1$.\ \ \
{(b)} $Y^2 = X(X-1)(X-2)$.\ \ \
{(c)} $Y^2 = X^3 + 1/3^6$.
\end{enumerate}

\medskip
\noindent{\it Section C}

\begin{enumerate}[resume] 
	\item A {\it non-commutative formal group}
	over a ring~$R$ is a power series $F(X,Y) \in R\lb X,Y\rb$
	which satisfies:\begin{itemize}
	\item $F(X,Y) = X + Y +$ terms of degree~$\geqslant 2$,
	%\par and 
	\item $F(X, F(Y,Z)) = F(F(X,Y),Z)$ [associativity],
	\item\noindent but \textit{not} $F(X,Y) = F(Y,X)$ [commutativity].
\end{itemize}
	Let $R = \F_p[t]/I$, where $I = t^2 \F_p[t]$.
	Find a non-commutative formal group over~$R$.

(\emph{Hint: you may find it helpful to consider formal group laws of the form $F(X,Y) = X+Y+t G(X,Y)$, where $G(X,Y) \in R\lb  X,Y \rb$}.)
	
	\item Let $\e : Y^2 = X^3 + AX$, where
$A \in \Z$ and $A \not= 0$. Let $F(X,Y)$ be the formal group
associated to~$\e$ [as in lectures] and let $F(X,Y) = \sum F_n(X,Y)$,
where each $F_n(X,Y)$ is homogeneous of degree~$n$.
Show that $F_n(X,Y) = 0$ unless $n\equiv 1$~(mod~$4$).


What is the similar result when the elliptic curve
is of the form $\e : Y^2 = X^3 + B$?

\item The following result is useful for checking that composition of formal power series interacts well with composing the associated functions. 

Let $K$ be a field, complete with respect to a non-Archimedean valuation, with valuation ring $R$ and maximal ideal $\M$. Suppose we have a power series \[F(X_1,\ldots,X_k) = \sum_{n = (n_i)_{i=1}^k}a_n X_1^{n_1}\cdots X_l^{n_k}\in K\lb X_1,\ldots, X_k\rb\] and a $k$-tuple of power series
\[G_i(X_1,\ldots,X_l) \in R\lb X_1,\ldots, X_l\rb : 1 \le i \le k \] with no constant term.

Consider the composed formal power series
\[(F\circ G)(X_1,\ldots,X_l) := F(G_1(X_1,\ldots,X_l),\ldots,G_k(X_1,\ldots,X_l)) \in K\lb X_1,\ldots, X_l\rb.\]

Suppose that for each positive real number $\rho < 1$, the set \[\{|a_n|\rho^{\sum n_i} : n \in \Z_{\ge 0}^k \}\] is bounded. Prove that for all $(x_i) \in \M^l$ we have 
\[ (F\circ G)(x_1,\ldots,x_l) = F(G_1(x_1,\ldots,x_l),\ldots,G_k(x_1,\ldots,x_l)).\] (This includes the claim that the two sides of the equation converge!)

(\emph{You might think that the statement is automatic as long as everything converges, but this isn't the case. See Problem 148 and pg.~120 in Gouv\^{e}a's book `$p$-adic numbers' for examples illustrating this.})
\end{enumerate}


\end{document}

\medskip
\hrule
\medskip
{\it The following question is compulsory for students taking
the MSc in MFoCS (Mathematics and the Foundations of Computer
Science). For everyone else, it is optional.}
\medskip\noindent {\bf 9.} Let $\e : Y^2 = X^3 + AX$, where
$A \in \Z$ and $A \not= 0$. Let $F(X,Y)$ be the formal group
associated to~$\e$ [as in lectures] and let $F(X,Y) = \sum F_n(X,Y)$,
where each $F_n(X,Y)$ is homogeneous of degree~$n$.
Show that $F_n(X,Y) = 0$ unless $n\equiv 1$~(mod~$4$).
What is the similar result when the elliptic curve
is of the form $\e : Y^2 = X^3 + B$?
\vfil\eject\end

