The growth of class numbers of quadratic Diophantine equations

Abstract

Let F be a totally real algebraic number field, with OF the ring of algebraic integers in F, let L be an OF-lattice on a d-dimensional (d≥2) positive definite quadratic space V over F and let u0 be a primitive vector in L. The main objective of this paper is to study when a is a fixed (non-unit) algebraic integer in OF and n is a positive rational integer, how the class numbers of lattices translation L+u0an grow as n tends to infinity.