\documentclass[a4paper]{article} \usepackage{amsmath,amssymb,amsthm,enumerate,eucal,tikz,enumitem} %\setlength\topmargin{-.1in} %\setlength\headheight{0in} %\setlength\headsep{0.2in} %\setlength\textheight{8.5in} %\setlength\textwidth{5.9in} %\setlength\oddsidemargin{0.4in} %\setlength\evensidemargin{0.4in} %% \setlength\parskip{0.3in} %% \setlength\parindent{0in} \def\le{\leqslant} \def\ge{\geqslant} \def\etq{{\cal E}_{\lower 1pt\hbox{\rm tors}}({\Bbb Q})} \def\etqp{{\cal E}_{\lower 1pt\hbox{\rm tors}}({\Bbb Q}_p)} \def\cotq{{\cal C}_{{\rm oddtors}}({\Bbb Q})} \def\dotq{{\cal D}_{{\rm oddtors}}({\Bbb Q})} \def\c{{\cal C}} \def\d{{\cal D}} \def\e{{\cal E}} \def\pk{\phi _\kappa} \def\im{{\hbox{\sl im}}} \def\hs{H_{\varsigma}} \def\hpk{\hat \phi _\kappa} \font\sc=cmssqi8 \def\scc#1{\hbox{\sc #1}} \def\sf{{\scc F}} \def\pnbq{{\Bbb P}^n(\overline {\Bbb Q} )} \def\hk{{\hat \kappa}} \def\bq{{\overline {\Bbb Q}}} \def\hq{{\hat q}} \def\pv{\prod\limits_v } \def\pnk{{\Bbb P}^n(K)} \def\mnkvw{{\Bbb M}^n(K[{\bf v}^2,{\bf w}^2])} \def\pnkv{{\Bbb P}^n(K[{\bf v}^2])} \def\kj{\kappa (J)} \def \qmods {{\Bbb Q}^*/({\Bbb Q}^*)^2} \def \qmodss { {\Bbb Q}^*/({\Bbb Q}^*)^2 \times {\Bbb Q}^*/({\Bbb Q}^*)^2 } \def \qs{{\Bbb Q}^*} \def \qss{({\Bbb Q}^*)^2} \def\bbQ{\Bbb Q} \def\bbF{\Bbb F} \def\bbZ{\Bbb Z} \def\bbR{\Bbb R} \def\bbC{\Bbb C} \def\notdiv{{\not\hskip-.5pt |\ }} \def\Q{{\Bbb Q}} \def\F{{\Bbb F}} \def\Z{{\Bbb Z}} \def\R{{\Bbb R}} \def\C{{\Bbb C}} % \def\g{{\cal G}} \def\h{{\cal H}} \begin{document} \noindent \centerline{\bf Elliptic Curves. MT 2025. Sheet 4.} \rm \bigskip %\par\noindent {\bf (d) [optional].} $Y^2 = X^3 - 219X + 1654$. %\par\noindent Note: (d) requires significant computation %(you should do an initial search for points with $x$-coordinate %in the range $|x| \leqslant 20$, with $x\in \Z$) %and is of course much more time consuming than anything which %would be asked in a timed exam. It is mainly intended to demonstrate %that interesting torsion groups can occur. %\par\noindent {\bf (a).} $Y^2 = X^3 + 1$. %\par\noindent {\bf (b).} $Y^2 = X^3 - 219X + 1654$. %%\par\noindent {\bf (c).} $Y^2 = X(X+1)(X+4)$. %\par\noindent {\bf (c).} $Y^2 = X(X+81)(X+256)$. %\par\noindent {\bf (d).} $Y^2 = X(X-1)(X-2)$. %\par\noindent {\bf (e).} $Y^2 = X^3 + 1/3^6$. %\par\noindent {\bf (f).} $Y^2 + Y = X^3 - X + 13$. %%\par\noindent {\bf (g).} $Y^2 = X^3 - X^2 + 1/4$. \noindent{\it Section A} \medskip \begin{enumerate}\item Let, as usual, $\c : Y^2 = X(X^2 + aX + b)$ and $\d : Y^2 = X(X^2 + a_1X + b_1)$, where $a,b\in\Z$, $a_1 = -2a, b_1 = a^2 - 4b$ and $b(a^2-4b)\not= 0$. Let $\cotq$ denote the set of torsion elements of $\c (\Q)$ which have odd order, and let $\dotq$ denote the set of torsion elements of $\d (\Q)$ which have odd order. Show that $\cotq$ and $\dotq$ are isomorphic. \item Let $\c$ and $\d$ be as in Question 1. Let the homomorphisms~$\phi, \hat\phi$ be defined as usual by \[\phi : \c \rightarrow \d : (x,y) \mapsto \Bigl( \bigl( {y\over x}\bigr)^2 , y - {by\over x^2} \Bigr),\] \[\hat\phi : \d \rightarrow \c : (u,v) \mapsto \Bigl( {1\over 4} \bigl( {v\over u} \bigr)^2, {1\over 8} \bigl( v - {b_1 v\over u^2}\bigr) \Bigr).\] What are the preimages of $(0,0)$ under $\hat\phi$? Show that $(0,0) \in 2\c (\Q)$ if and only if there exist~$m,n\in \Z$ such that $b = m^2$ and $a+2m = n^2$. \end{enumerate} \medskip \noindent{\it Section B} \begin{enumerate}[resume] \item As in lectures, write $\g = \c(\Q)$ and $\h = \d(\Q)$. Let $a$ and $b$ be the integers so that \[ \g / \hat{\phi}(\h) \cong C_2^a \mbox{ and } \h /\phi(\g) \cong C_2^b. \] \par\noindent {(a)} Explain why the map \[ \hat{\phi}: \h /\phi(\g) \rightarrow \hat{\phi}(\h) / 2 \g,\quad P + \phi(\g) \mapsto \hat{\phi}(P) + 2 \g \] is well-defined and surjective. Prove that the kernel of this map is generated by the class $[(0,0)] \in \h /\phi(\g)$ of $(0,0) \in \h$. %% Deduce that %% \[ \hat{\phi}(\h) / 2 \g \cong C_2^b \mbox{ or } C_2^{b-1} \] %% according to whether $(0,0) \in \phi(\g)$ or $(0,0) \not \in \phi(\g)$. \par\noindent {(b)} Hence show that \[ \g / 2 \g \cong C_2^{a + b} \mbox{ or } C_2^{a + b - 1}\] according to whether $(0,0) \in \phi(\g)$ or $(0,0) \not \in \phi(\g)$. \par\noindent {(c)} Show that $(0,0) \in \phi(\g)$ if and only if the $2$-torsion of $\g$ is isomorphic to $C_2^2$. Deduce that the rank of $\g$ is $a + b - 2$. \item Find the ranks of the following elliptic curves. \par\noindent {(a)} $Y^2 = X(X^2 + 2X + 3)$. \par\noindent {(b)} $Y^2 = X(X^2 + 14X + 1)$. %%\par\noindent {\bf (a).} $Y^2 = X(X^2 + 3X + 5)$. %\par\noindent {\bf (a).} $Y^2 = X(X^2 + 5X - 5)$. %\par\noindent {\bf (b).} $Y^2 = X(X^2 + 14X + 1)$. %\par\noindent {\bf (c).} $Y^2 = X(X^2 + 2X + 3)$. %%\par\noindent {\bf (d).} $Y^2 = X(X^2 + 2X + 9)$. %%\par\noindent {\bf (e).} $Y^2 = X(X^2 + 9X - 1)$. %%\par\noindent {\bf (f).} $Y^2 = X(X-12)(X-36)$. \item (\textit{2006 paper, Q 8(ii)}) For any prime $p \equiv 5 \bmod{8}$ show that the elliptic curve $Y^2 = X^3 + p^2 X$ has rank $0$. \item Let $A,+$ be an Abelian group. Let $h : A \rightarrow \R_{\ge 0}$ satisfy: \par\noindent \ \ \ \ (I) There exists a constant~$C$, independent of~$P,Q$, such that \par \ \ \ \ $ | h(P+Q) + h(P-Q) - 2 h(P) - 2 h(Q) | \le C $, for all $P,Q \in A$, \par\noindent \ \ \ \ (II) For any~$B\in\R$, the set $\{P\in A:h(P)\le B\}$ is finite. \par\noindent Show that~$h$ is a height function on~$A$. Show also that there exists a constant~$C_3$, independent of~$P$, such that $|h(3P) - 9 h(P)|\le C_3$, for all $P \in A$. \par (\textit{$\R_{\ge 0}$ denotes $\{ x\in\R : x \ge 0\}$}). \end{enumerate} \medskip \noindent {\it Part (b) of the next question is {\it Section B}, and the others parts {\it Section C}} \begin{enumerate}[resume] \item A four-letter word $L_1L_2L_3L_4$ has been divided into two pairs: $L_1L_2$ and $L_3L_4$. Each of these pairs has been converted into an integer (of at most 4 digits) via the standard map: $A \mapsto 01 , B \mapsto 02, \ldots , Z \mapsto 26$. These integers have been encoded by taking each to the power of $d=4085$, modulo $N=10481$. The encoded message reads: {\bf 6012},\ {\bf 3236}. \noindent You may assume that $N$ is the product of two primes. You should show, in your calculations, how you are only using numbers of length at most~$9$ digits. \par\noindent {(a)} Find a proper factor of~$N$ (that is, a factor~$d$ of~$N$ satisfying~$1 < d < N$) by applying Pollard's ``$p-1$'' method, using base~$2$ and exponent~$46$. \par\noindent {(b)} Factorise $N$ by applying the Elliptic Curve Method, using the curve $\e : Y^2 = X^3 - X + 1$ and~$3P$, where~$P=(5,11)$. \par\noindent {(c)} Use the factorisation of~$N$ to decode the message (which is the name of the town famous for being the country music capital of New Zealand). \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 7.} Show that any elliptic curve over $\Q$ with a rational point of order~$4$ is birationally equivalent to: $Y^2 + XY + v Y = X^3 + v X^2$, for some $v\in \Q$. \vfil \eject \end