On units of a quaternion order of an indefinite anisotropic ternary quadratic form

U. M. Pachev ; T. A. Shakova

Čebyševskij sbornik, Tome 20 (2019) no. 4, pp. 270-280.

Résumé

The paper considers issues related to the group of units on the quaternion order $O_{f}$ corresponding to the indefinite anisotropic ternary quadratic form $$f = f {\left(x_{1},\ x_{2},\ x_{3}\right)} = x_{1}^{2} - b x_{2}^{2} - c x_{3}^{2},$$ where $b,\ c > 0$ are integers, an the number $c$ is not the norm from quadratic field $\mathbb{Q} {\left(\sqrt{b}\right)}$. With respect to the indicated group if units, we have proved that it contains an indefinite noncommutative $2$-generated subgroup described by means of the group of Pell units. The first studies relating to the group of units of division algebra were carried out in 1937 M. Eichler, who established their final generation. Of particular interest in connection with our work are the results obtained by Basilla J. M. and Bada H. in 2005 for equations of the form $x^{2} - d y^{2} = \pm m$, with can be applied in the further study of the group of units of the quaternion order under consideration. Another result to the question of the number of pairwise unassociated generalized quaternions of a given norm m from order $O_{f}$. This question is closely related to units of order $O_{f}$ and to groups of Pell units. It should be noted that, in the study of their arithmetic matrix algebras obtained simple exact formulas for the number of primitive unassociated right (left) whole matrices of a given determinant that have applications in the so-called discrete ergodic method of Yu. V. Linnik for solving the representability of integers by indefinite isotropic ternary quadratic forms. A series of results related to this issue were obtained by the first on the authors. As for the question of the number of unassociated generalized quaternions of a given norm and order $O_{f}$, as far as we know, the results of the presented species have not yet been met, and apparently this is due to the fact that the quaternion order under consideration, probably, it has a rather complicated structure. In the present paper, instead of exact formulas, it was possible to obtain upper and lower bounds for the number of pairwise unassociated quaternions of norm in from order $O_{f}$.

Détail

Keywords: ternary quadratic forms, algebra of generalized quaternions, quaternion order, quaternion norm, group of quaternion units, Pell equation, binary quadratic form, quaternion association.

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