An~asymptotic formula for the number of representations by totally positive ternary quadratic forms

Yu. G. Teterin

Izvestiya. Mathematics , Tome 26 (1986) no. 2, pp. 371-403

Résumé

Suppose $\mathfrak o$ is a maximal order of a totally real algebraic number field $K$; $f(x_1,x_2,x_3)$ is a totally positive quadratic form over $K$; $\mathfrak a$ and $\mathfrak c$ are ideals of the ring $\mathfrak o$; $m\in K$; and $x_1,x_2,x_3\in\mathfrak o$. The author proves an asymptotic formula for the number of solutions of the system $$ f(x_1,x_2,x_3)=m,\quad\text{g.c.d.}(x_1,x_2,x_3)=\mathfrak c,\qquad x_1\equiv b_1,\ x_2\equiv b_2,\ x_3\equiv b_3\pmod{\mathfrak a} $$ in numbers $x_1,x_2,x_3\in\mathfrak o$. The proof is based on a discrete ergodic method. Bibliography: 19 titles.