Given positive integers $a,b,c$, and knowledge that there exist two matrices $U,V$ in $SL(2,\mathbb{Z})$ (2x2 integer matrices with determinant 1) that satisfy:
$$ b = \text{Tr}\left(\begin{bmatrix}-1 & 0 \\0 & a\end{bmatrix}U \begin{bmatrix} -1 & 0 \\ 0 & c \end{bmatrix} V \right), $$ how can I characterize/parameterize the possible $U,V$ solving this given a particular $(a,b,c)$?
As far as I can tell, there is no closed form solution, and am not expecting one here. What I would like to learn is how the solutions relate and hopefully how to parameterize this relationship as fully as possible. For instance:
- Are all the solutions for $U,V$ essentially unrelated, depending heavily on the number theory details of the equations given $a,b,c$?
- Or will there be a single pair of $U,V$ (the values of which depend non-trivially on $a, b, c$) from which all other solutions can then be generated from in some simple way?
- Or is it something inbetween: many pairs of $U,V$ depending non-trivially on $a,b,c$; with each pair specifying a "class" of solutions generated in a simple way from the pair?
Forgive me if there is an obvious symmetry in the trace operation which means we can immediately pull out some class of solutions. If it is there, I have over-looked it, so even a partial answer would be much appreciated.
The following may be miscellaneous info, so can be ignored, but is supplied in case knowing where this problem comes from helps give ideas of other tools to apply.
The equation came up in the context of exploring some of the freedoms in constructing a Bhargava cube with two fixed faces referring to the binary quadratic form $$f(x,y) = -\text{det}(Mx + Ny) = a x^2 + b xy + c y^2,$$ where $M$ and $N$ are 2x2 matrices of integers. The form parameters are then related by: $$ a = -\det(M)$$ $$ c = -\det(N)$$ $$ b = \text{Tr} \left( M^T \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} N \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \right). $$ Using Smith normal form for $M$ and $N$ we can see that there is always $U,V$ in $SL(2,\mathbb{Z})$ which satisfy the equation in question.
In case some extra conditions help, I'm particularly interested in the case where we can also assume $b^2-4ac<0$, and $\gcd(a,b)=\gcd(b,c)=1$.