3
$\begingroup$

Let $d\geq 1$ be an integer. I'm not assuming anything here about $d$ (it is not necessarily squarefree, for example)

For all $a,b,c\in\mathbb{Z}$ such that $ac-b^2=d,$ set $[a,b,c]_d=\begin{pmatrix}b & -c \cr a & -b\end{pmatrix}\in M_2(\mathbb{Z})$.

(Note that the relation $ac-b^2=d$ forces $a\neq 0$ and $c\neq 0$.)

Let's say that $[a,b,c]_d$ is reduced if $0<a\leq c, \ -\dfrac{a}{2}<b\leq \dfrac{a}{2}$, and $b\geq 0$ whenever $a=c$.

Note the similarity with the notion of reduced positive binary form à la Gauss (up to a factor $1/2$ which might be easily explained); this is not a coincidence.

Question. Without using the theory of binary forms, is it possible to prove that, if $[a,b,c]_d$ and $[a',b,'c']_d$ are reduced and similar over $\mathbb{Z}$ (that is, conjugate by an element of $GL_2(\mathbb{Z})$), then $a=a',b=b',c=c'$.

[Let me explain why it should be true. The set of conjugacy classes of integral matrices with characteristic polynomial $X^2+d$ and the set of equivalences of ideals of $\mathbb{Z}[\sqrt{-d}]$, where two ideals $I$ and $J$ are equivalent if there exist $z,z'\in \mathbb{Q}(\sqrt{-d})$ such that $zI=z'J$.

But if I am not mistaken, it is known that the set of classes of ideals is in bijection with the set of equivalent positive definite binary forms of discriminant $-4d$. And such a form is properly equivalent to a unique reduced binary form. Now, my matrix $[a,b,c]_d$ is reduced if and only if $ax^2+2bxy+cy^2$ is reduced.]

Thanks in advance for your help!

G.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.