On binary quadratic forms with the semigroup property
Труды Математического института им. В.А. Стеклова РАН. 2007. Т. 258. № 1. С. 23-43.
A quadratic form f is said to have semigroup property if its values at points of the integer lattice form a semigroup under multiplication. A problem of V. Arnold is to describe all binary integer quadratic forms with semigroup property. If there is an integer bilinear map s such that f(s(x,y))=f(x)f(y) for all vectors x and y from the integer 2-dimensional lattice, then the form f has semigroup property. We give an explicit description of all pairs (f,s) with the property stated above. We do not know any other examples of forms with semigroup property. It turns out that certain pairs (f,s are closely related with order 3 elements in class groups.