\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} %% \usepackage[T1]{fontenc} %% \usepackage[scaled]{beramono,berasans} %% \usepackage[charter]{mathdesign} \usepackage{microtype} \newfont{\Bb}{msbm10 scaled\magstep0} \newfont{\Bbl}{msbm10 scaled\magstep1} \newfont{\Bbs}{msbm10 scaled 800} \def\sqr{\ifmmode\square\else{$\square$}\fi} \def\square{\vcenter{ \hrule height.1mm \hbox{\vrule width.1mm height2.2mm\kern2.18mm\vrule width.1mm} \hrule height.1mm}} % This is a slimmer sqr. \null \def\le{\leqslant} \def\ge{\geqslant} \def\etq{{\cal E}_{\lower 1pt\hbox{\eightrm tors}}({\Bbb Q})} \def\etqp{{\cal E}_{\lower 1pt\hbox{\eightrm tors}}({\Bbb Q}_p)} \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}} % \begin{document} \centerline{\bf Elliptic Curves. MT 2025. Sheet 1.} \noindent {\it Section A} \medskip \begin{enumerate} \item { (a)} Find a birational transformation over~$\mathbb{Q}$ between the curves \[2X^2 - Y^2 = 1\] and \[X^2 + Y^2 - 6XY = 1.\] \emph{Hint: both curves are birational to a line.} \par\smallskip \noindent {(b)} Find a birational transformation over~$\mathbb{Q}$ between the curves \[Y^2=(X+2)^6(X^3+1)\] and \[Y^2 = X^3 + 1.\] \par\smallskip \noindent { (c)} Find a birational transformation over~$\mathbb{C}$ between the curves~\[Y^2=2X^2\] and \[Y^2=X^2.\] Is there a birational transformation over~$\mathbb{Q}$? \item For each of the following elliptic curves, find all the points (including, as always, the point at infinity) over ~$\bbF_5$. Draw a complete group table in each case and describe each group as a product of cyclic groups. \par\noindent {(a)} $Y^2 = X^3 + 2 X$. \ \ \ \ \ {(b)} $Y^2 = X^3 + 1$. \item Show that the point~$(2,4)$ is of order~4 on $Y^2 = X^3 + 4X$, defined over~$\bbQ$. \end{enumerate} \medskip \noindent {\it Section B} \begin{enumerate}[resume] \item (a) Let $m\in {\Bbb N}$ be odd or $f_m\in \qss$ (or both). Show that the curve \par \centerline{$Y^2 = f_mX^m + f_{m-1}X^{m-1} + \ldots + f_0$, where all $f_i\in \bbQ$ and $f_m\not= 0$,} \par\noindent can be birationally transformed over~$\bbQ$ to a curve of the form \par \centerline{$Y^2 = X^m + g_{m-1}X^{m-1} + \ldots+ g_0$, with all $g_i\in \bbZ$.} \smallskip \par\noindent {(b)} Birationally transform over $\bbQ$ the curve $Y^2 = {1\over 5}X^3 + 3 X^2 + 1$ to a curve of the form~$Y^2 = X^3 + AX + B$, where $A,B\in \bbZ$. \item Consider the elliptic curve with projective equation \[X^3 + Y^3 + aZ^3 = 0\] for some $a \in \Q^\times$ and identity point $(1,-1,0)$. \par\noindent Show that the group law satisfies \[-(x,y,z) = (y,x,z)\] and \[[2](x,y,z) = (y(x^3-az^3),-x(y^3-az^3),z(y^3-x^3).\] (You might find Lemma 1.43 in lecture notes helpful.) \newpage %\par\noindent {(b)} \emph{(Section C)} Find a formula for the sum of two (distinct) points $(x_1,y_1,z_1)+(x_2,y_2,z_2)$. \item (a) Let $p \equiv 2$~(mod~$3$) be prime and let $A \in \bbF_p^*$. Show that the number of points (including the point at infinity) on the curve $Y^2 = X^3 + A$ over $\bbF_p$ is exactly $p+1$. \smallskip \par\noindent {(b)} Let $p \equiv 3$~(mod~$4$) be prime and let $B \in \bbF_p^*$. Show that the number of points (including the point at infinity) on the curve $Y^2 = X(X^2 + B)$ over $\bbF_p$ is exactly $p+1$. \item (a) Show that the point $(2,0)$ is of order 2 on $Y^2 = (X-2)(X^2 + X + 1)$. \par\noindent {(b)} Find all $\bbQ$-rational points of order~2 and all $\bbC$-rational points of order~2 on each of the following elliptic curves: $Y^2 = X(X^2-3)$, $Y^2 = X^3 - 7$ and $Y^2 = X(X-1)(X-7)$. In each case, find the group structure (expressed as a product of cyclic groups) of the $\bbQ$-rational 2-torsion group (that is, the group of all $\bbQ$-rational points~$P$ such that $2P = {\bf o}$). \item Show that the point $(0,2)$ is of order 3 on $Y^2 = X^3 + 4$. \end{enumerate} \medskip %{\bf 8.}(a) %Let $Y^2 = (X-\alpha)(X^2 + aX + b)$ %be an elliptic curve with $a,b,\alpha\in K$ (characteristic $\not= 2$), %and ${\bf o} =$ point at infinity, as usual. Show that $(\alpha , 0)$ %is a point of order~2. Let $x',y'$ be defined by: $(x',y') %= (x,y) + (\alpha , 0)$, and define $T: K\rightarrow K : x\mapsto x'$. %Find $t_{11},t_{12},t_{21},t_{22}$ in terms of $a,b,\alpha$ such that: %$x' = \mu(x) = (t_{11} x + t_{12})/(t_{21} x + t_{22})$. %Check that $\mu^2 : x\mapsto x$. %\smallskip %\par\noindent {(b)} Consider~$Y^2 = (X-\alpha_1)(X-\alpha_2) %(X-\alpha_3)$, with $\alpha_1,\alpha_2,\alpha_3$ distinct, %and let $T_1,T_2,T_3$ be as in (a), but with $\alpha$ replaced %by $\alpha_1,\alpha_2,\alpha_3$, respectively. %Express each $T_i$ in terms of $x,\alpha_1,\alpha_2,\alpha_3$. %Show, directly from expressions, that $T_1,T_2,T_3$ %commute (i.e.\ $T_1 T_2 = T_2 T_1$, $T_1 T_2 = T_2 T_1$ and %$T_2 T_3 = T_3 T_2$), and that $T_1 T_2 T_3 : x\mapsto x$. %Find the fixed points of $T_1$ %and show that they are permuted by $T_2$. \noindent {\it Section C} \begin{enumerate}[resume] \item Let $K$ be any field with $\hbox{Char }K \not= 2,3$, and let \par\centerline{${\cal E}: F(X_0,X_1,X_2) = X_1^2 X_2 - (X_0^3 + A X_0 X_2^2 + B X_2^3), \hbox{ with }A,B \in K,$} \noindent be an elliptic curve (N.B. This is just the standard projective form, but with $X,Y,Z$ replaced by $X_0,X_1,X_2$). Let~$P$ be a point on~$\cal E$. \smallskip \par\noindent {(a)} Show that $3P = {\bf o}$ iff.\ the tangent line to $\cal E$ at~$P$ intersects $\cal E$ only at~$P$. \smallskip \par\noindent {(b)} Show that if $3P={\bf o}$ then the $3\times 3$ matrix $\bigl( \partial^2 F / \partial X_i \partial X_j (P) \bigr)$ has determinant~$0$. [This matrix is called the Hessian matrix]. \smallskip \par\noindent {(c)} Show that there are at most nine $3$-torsion points over~$K$. \end{enumerate} \end{document}