2-universal -lattices over real quadratic fields

Abstract

Let Open image in new window be a real quadratic field with m a square-free positive rational integer, and Open image in new window be the ring of integers in F. An Open image in new window-lattice L on a totally positive definite quadratic space V over F is called r-universal if L represents all totally positive definite Open image in new window-lattices l with rank r over Open image in new window. We prove that there exists no 2-universal Open image in new window-lattice over F with rank less than 6, and there exists a 2-universal Open image in new window-lattice over F with rank 6 if and only if m=2, 5. Moreover there exists only one 2-universal Open image in new window-lattice with rank 6, up to isometry, over Open image in new window.