摘要：Our work is motivated by a question posed by Conway and Sloane in their book Sphere Packings, Lattices and Groups: “If two rational quadratic forms are in the same genus, find an explicit rational equivalence whose denominator is prime to any given number.” I will present a work in progress, joint with Wai Kiu Chan and Jacob Tolman, in which we establish a search bound over a general number field. Our approach combines the arithmetic theory of quadratic forms with ergodic theory on Lie groups over non-Archimedean local fields. In particular, we develop a quantitative version of strong approximation for the group O'(f)—the kernel of the spinor norm—which may be of independent interest.
