I aim to study the binary forms $ax^2 + bxy + cy^2 = (a,b,c)$ where $a,b,c \in {F_q}[T]$ (charasteristic of $F_q$ not 2) in particular those such that the discriminant $D = b^2 - 4ac \in F_q[T]$ has even degree and sign ${D} \in {F_q^*}^2$ – in other words its principal coefficient is a square.

This case is interesting because $\sqrt{D}$ exists as a Laurent series, and you can even consider its expression as a continued fraction (with polinomial coefficients), which has may interesting properties such as its periodicity. In our definition of reduced forms, we say $(a,b,c)$ with discriminant $D$ is reduced if $$|\sqrt{D} - b| < |a| < |\sqrt{D}|$$ and the sign of $a$ is either $1$ or a fixed square, although this property is just for unicity and has no importance. The inequality is the important part.

I would like to follow the plot normally used with the usual binary forms with integers coefficients: For the indefinite case with discriminant $D >0$ not a square, the reduced forms of one equivalence class can be arranged into cycles, these cycles have an even number forms, and the most important part, two reduced forms are equivalent iff they belong to the same cycle. The last point is the problem I would like to share with you:

Do you know how I could achieve this result?

I can give more details: in the usual case in $\mathbb{Z}$, what you normally do (for example is Buell's book) is to provide yourself with the following results:

  • Every cycle has an even number of reduced forms.

  • Let $x,y \in \mathbb{R}$. If there exist integers $a,b,c,d$ so that $ad - bc = 1$ and $$y = \frac{a x + b }{c x + d},$$ then we can express $y$ as $$y = [u; a_1, \cdots a_{2r}, v, x],$$ where $a_1, \cdots a_{2r}$ are positive integers and $u,v$ are integers.

  • If a continued fraction contains only a limited number of negative or zero partial quotients, then it is possible, by a finite number of steps, to convert it into a scf. During this process, almost all coefficients are shitfed an even numer of positions.

What you normally do in $\mathbb{Z}$ is making the most os the parellelism between the continued fraction of the principal root of one reduced form in the cycle and moving in the cycle, and then use these propositions. In $F_q[T]$ everything behaves almost identically, except these three points.

However I haven't found their "equivalent propositions" for the present case. For example, regarding the parity of the cycle, in the case in $\mathbb{Z}$ you just consider the alternance of sign positive-negative when you move to the adjacent form in the cycle, hence if you return to the initial one it means you did an even number of steps. In ${F_q}[T]$ the degree is not enough to prove it and I don't find any other invariant.

If someone could give me any advide my nightmares will be over :D I don't find this matter anywhere in the bibliography (I have been recommended a book written by Gernstein in Stackexchange (Click here), but I think his idea is different than mine and I got no more answers. I guess this field of binary forms is not the most popular). All suggestions are welcome. Thanks in advance!

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    $\begingroup$ You should definitely read Artin's thesis; you can find it in his collected works. $\endgroup$ – Franz Lemmermeyer Jun 4 '15 at 9:46
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    $\begingroup$ Do you want to say right at the start that $q$ is odd? You use the notation $F_q$ with no conditions imposed on $q$ at all. $\endgroup$ – KConrad Jun 4 '15 at 15:17

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