\documentclass[a4paper]{article} \usepackage{amsmath,amssymb,amsthm,enumerate,eucal,tikz} %\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 \qmodsquares {{\Q}^*/({\Q}^*)^2} \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 0.} \rm \bigskip {\it This sheet is not intended to be handed in or discussed in classes. It is for you to use to reinforce the background material discussed in the preliminary reading file.} \bigskip \noindent {\bf 1.} Determine whether the following are groups. \par\noindent {\bf (a).} The set of all $2\times 2$ matrices under matrix multiplication. \par\smallskip \noindent {\bf (b).} The set of all $2\times 2$ matrices under matrix addition. \par \medskip \noindent {\bf 2.} For each of the following, decide whether $\phi$ is a homomorphism. When $\phi$ is a homomorphism, decide whether~$\phi$ is injective, surjective, bijective, and find the kernel of~$\phi$. \par\noindent {\bf (a).} $\phi : \mathbb{Z} , + \rightarrow \mathbb{Q}^* , \times : x \mapsto x^2+1$. \par\smallskip \noindent {\bf (b).} $\phi : \mathbb{Q} , + \rightarrow \mathbb{R} , + : w \mapsto \sqrt{2}\, w$. \par\smallskip \noindent {\bf (c).} $\phi : \mathbb{Z} , + \rightarrow \mathbb{Z} / 3\mathbb{Z} , + : x \mapsto 2x$. \par \medskip \noindent {\bf 3.} \par\noindent {\bf (a).} In $\qmodsquares$, decide whether the following are true or false: $3=1/27$, $-4=4$, $3=5/6$. \par\smallskip \noindent {\bf (b).} In $\qmodsquares$, write each of the following as a square free integer: $-2/27$, $16$, $12$, $1/3$. \par\smallskip \noindent {\bf (c).} Perform each of the following in $\qmodsquares$, writing your answer as a square free integer: $6\times 10$, $10 / 21$, $15^{101}$, $3^{-1}$. \par\smallskip \noindent {\bf (d).} How many elements are in each of the groups: $\qmodsquares$, ${\mathbb{R}}^*/({\mathbb{R}}^*)^2$, ${\mathbb{C}}^*/({\mathbb{C}}^*)^2$? \par\medskip \noindent {\bf 4.} \par\noindent {\bf (a).} Find all singular points on the curve $$ \mathcal{C} : f(X,Y) = X^4 + Y^3 - 3 X^2 Y = 0. $$ \noindent Find all tangents to $\mathcal{C}$ at the point $(0,0)$. \par\smallskip \noindent {\bf (b).} Find all singular points on the curve $$ \mathcal{C} : f(X,Y) = Y^2 - X(X^2-1)^2 = 0.$$ \noindent Find all tangents to $\mathcal{C}$ at the points $(0,0)$ and $(1,0)$. \par \medskip \noindent {\bf 5.} Show that $\mathcal{C} : Y^2 = X^3 + AX + B$ is smooth if $4A^3 + 27B^2 \not= 0$ and we work over a field with characteristic $\neq 2$. What happens in characteristic $2$? \par \medskip\noindent {\bf 6.} For each of the following curves, find the irreducible components over~$\mathbb{Q}$ and the irreducible components over~$\mathbb{C}$. \par\noindent {\bf (a).} $\mathcal{C} : Y^2 = X^5$. \par\smallskip \noindent {\bf (b).} $\mathcal{C} : Y^3 = X^3$. \par\smallskip \noindent {\bf (c).} $\mathcal{C} : Y^2 = X^3 + 1$. %\par\medskip\noindent{\bf 7.} %\par\noindent {\bf (a).} Find a birational transformation over~$\mathbb{Q}$ %between the curves $2X^2 - Y^2 = 1$ and $X^2 + Y^2 - 6XY = 1$. %\par\smallskip %\noindent {\bf (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 {\bf (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~$\mathbb{Q}$? \par\medskip \noindent {\bf 7.} \par\noindent {\bf (a).} Find the discriminant of~$X^4-2$. \par\smallskip \noindent {\bf (b).} Find the resultant of $X^3 - a$ and $X^2 - b$, where $a,b$ are constants. \par\medskip\noindent {\bf 8.} Find all intersection points (with multiplicities) over~$\mathbb{C}$ of the curves: $X^3 + Y^3 = Z^3$ and $X^2 + Y^2 = Z^2$. \par \medskip \newpage \noindent {\bf 9.} \par\noindent {\bf (a).} Decide whether each of $2,3,5,10,15$ are quadratic residues modulo~1009 (if you use quadratic reciprocity, this should not involve any lengthy computations). \par\smallskip \noindent {\bf (b).} Describe all primes~$p$ such that $3$ is a quadratic residue modulo~$p$. Describe all primes~$p$ such that $5$ is a quadratic residue modulo~$p$. Describe all primes~$p$ such that $10$ is a quadratic residue modulo~$p$. \par\medskip \noindent {\bf 10.} Are there integers $a,b,c$, not all~$0$, such that $2a^2 + 5b^2 = c^2$? \par\medskip \noindent {\bf 11.} For any $n\in{\mathbb{N}}$ define, as usual, Euler's $\phi$-function by: $$ \phi(n) = \# \{ x : 1 \leqslant x \leqslant n \hbox{ and gcd} (x,n) = 1 \}. $$ For any prime~$p$, what is $\phi(p^r)$? For any distinct primes $p_1,p_2$, what is $\phi(p_1 p_2)$? \par For each of the following examples of the type $a^b \ (\hbox{mod }n)$, reduce $a^b \ (\hbox{mod }n)$ to a member of $\{ 0, \ldots , n-1 \}$. \par\noindent $2^{12} \ (\hbox{mod }13)$, $3^{12} \ (\hbox{mod }13)$, $3^{24} \ (\hbox{mod }13)$, $3^{12000} \ (\hbox{mod }13)$, $3^{12002} \ (\hbox{mod }13)$, \hfill\par\noindent $4^{24} \ (\hbox{mod }35)$, $4^{48} \ (\hbox{mod }35)$, %$4^{48000} \ (\hbox{mod }35)$, $4^{48000001} \ (\hbox{mod }35)$, \hfill\par\noindent $7^{24} \ (\hbox{mod }35)$, $7^{48} \ (\hbox{mod }35)$, %$7^{48000} \ (\hbox{mod }35)$, $7^{48000001} \ (\hbox{mod }35)$. \end{document}