## Proper cycles of indefinite quadratic forms and their right neighbors.(English)Zbl 1216.11042

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 $F\sim R^1(F)\sim R^2(F) \sim\dots\sim R^{l-2}(F) \sim R^{l-1}(F).$

### MSC:

 110000 Quadratic forms over general fields 1.1e+13 Quadratic forms over global rings and fields 1.1e+17 General binary quadratic forms
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### References:

 [1] J. Buchmann: Algorithms for Binary Quadratic Forms. Springer-Verlag, accepted. · Zbl 0948.11051 [2] D. E. Flath: Introduction to Number Theory. Wiley, New York, 1989. · Zbl 0651.10001
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