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\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}