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\title{567 Number Theory I, Fall Term 2008, Problems 13}\endtitle
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\centerline{\it Return by Tuesday 2nd December}
\medskip
\noindent $\Gamma$ denotes the modular group and $S$, $T$ are its generators,
$S(z)=-1/z$, $T(z)=z+1$. Given a quadratic form $Q(x,y)=ax^2+bxy+cy^2$ with
real coefficients, $d=d_Q=b^2-4ac$ is called the discriminant of $Q$.
\medskip
\noindent 1. (i) Find all elements $A$ of $\Gamma$ which commute with $S$.\par
\noindent (ii) Find all elements $A$ of $\Gamma$ which commute with $T$.\par
\noindent (iii) Find the smallest $n>0$ such that $(ST)^n=I$.\par
\noindent (iv) Determine all $A$ in $\Gamma$ which leave $i$ fixed.\par
\noindent (v) Determine all $A$ in $\Gamma$ which leave $\rho=e(1/3)$ fixed.\par
\medskip
\noindent 2. Prove that if $A\in\Gamma$, and
$(x,y)^T=A(x',y')^T$, then the quadratic form $Q'$ defined by $Q'(x',y') =
Q(x,y)$ satisfies $d_{Q'}=d_Q$. Two forms related in this way are called
equivalent. This relation separates all forms into equivalence
classes. The forms in the same class have the same discriminant and the ranges
$Q(\Bbb Z^2)$ coincide.\par
\medskip
\noindent In the remaining exercises it will be supposed that the quadratic forms
have positive coefficients of $x^2$ and $y^2$ and negative discriminant. The associated
polynomial $Q(z,1)$ has two complex roots. The one in $\Bbb H$ is called the
representative of $Q$.
\medskip
\noindent 3. (i) If $d$ is fixed, prove that there is a bijection between the set of
forms with discriminant $d$ and the members of $\Bbb H$.\par
\noindent (ii) Prove that two quadratic forms with discriminant $d$ are equivalent
iff their representatives are equivalent under $\Gamma$.
\medskip
\noindent A reduced form is one whose representative lies in the fundamental domain
$\Bbb D$, the set of $z$ such that either $|z|>1$ and $-1/2\le\Re z<1/2$ or $|z|=1$
and $-1/2\le\Re z\le 0$. Thus two reduced forms are equivalent iff they are identical,
and moreover each equivalence class contains exactly one reduced form.
\medskip
\noindent 4. Prove that $Q(x,y)=ax^2+bxy+cy^2$ is reduced iff either $-a