Proper cycles of indefinite quadratic forms and their right neighbors. (Q954631)

Let \(F=(a,b,c)\) be an indefinite binary quadratic form with integer coefficients and discriminant \(\Delta\): \(F(x,y)=ax^2+bxy+cy^2\), \(\Delta=b^2-4ac\). Let \[ \rho(F)=(c,-b+2cs,cs^2-bs+a), \] with \[ s=\begin{cases} \text{sign}(c) \lfloor\frac{b}{2|c|}\rfloor,&|c|\geq \sqrt{\Delta},\\ \text{sign}(c) \lfloor\frac{b+\sqrt{\Delta}}{2|c|}\rfloor,&|c|<\sqrt{\Delta}, \end{cases} \] \[ \chi(F)=(-c,b,-a), \] \[ \tau(F)=(-a,b,-c). \] The cycle of \(F\) is the sequence \(((\tau\rho)^i(G))_{i\in {\mathbb Z}}\), where \(G=(k,l,m)\) is a reduced form with \(k>0\) which is equivalent to \(F\). The proper cycle of \(F\) is the sequence \((\rho^i(G))_ {i\in {\mathbb Z}}\), where \(G\) is a reduced form with \(k>0\) which is properly equivalent to \(F\). The right neighbor of \(F\) is the form \(R(F)=(A,B,C)\) with \(A=c\), \(b+B\equiv 0 (\bmod 2A)\) and \(\sqrt{\Delta}-2|A|<B<\sqrt{\Delta}\), \(B^2-4AC=\Delta\). Let \(F=F_0\sim F_1\sim\dots F_{l-1}\) be the cycle of \(F\) of length \(l\), and let \(R^i(F)\) be the consecutive right neighbors of \(F\) for \(i\geq 0\). The author proves that if \(l\) is odd, then the proper cycle of \(F\) is \[ F\sim R^1(F)\sim R^2(F) \sim\dots\sim R^{2l-2}(F) \sim R^{2l-1}(F) \] of length \(2l\). If \(l\) is odd, then \[ \chi(F_i)=F_{l-1-i} \] for \(0\leq i\leq l-1\), and the cycle of \(\chi(F)\) is \[ \chi(F_l)\sim \chi(F_{l-1}) \sim \chi(F_{l-2})\sim\dots\chi(F_1). \] If \(l\) is even, then the proper cycle of \(F\) is \[ F\sim R^1(F)\sim R^2(F) \sim\dots\sim R^{l-2}(F) \sim R^{l-1}(F). \]