Distribution of lattice points on surfaces of second order

Let $f_1(x_1,\dots,x_{l_1})$ and $f_2(y_1,\dots,y_{l_2})$ be positive definite primitive quadratic forms in $l_1$ and $l_2$ variables, respectively. We obtain new results in the well-known problem on the number of lattice points on the cone $f_1(x_1,\dots,x_{l_1})=f_2(y_1,\dots,y_{l_2})$, in the domain $f_1(x_1,\dots,x_{l_1})\le N$ for $N\to\infty$. Our technical tool is the Rankin–Selberg convolution. In several special cases the results can be sharpened by other methods. In addition, new facts concerning the uniform distribution of lattice points on ellipsoids in $l$ variables, $l$ odd, $l\ge5$ are obtained. Bibliography: 40 titles.