On theta series vanishing at ∞ and related lattices
We answer three reducibility (or irreducibility) questions for -polynomials, those polynomials which have every coefficient either or . The first concerns whether a naturally occurring sequence of reducible polynomials is finite. The second is whether every nonempty finite subset of an infinite set of positive integers can be the set of positive exponents of a reducible -polynomial. The third is the analogous question for exponents of irreducible -polynomials.
We compute the numbers of locally principal ideals with given norm in a class of definite quaternion orders and the traces of the Brandt-Eichler matrices corresponding to these orders. As an application, we compute the numbers of representations of algebraic integers by the norm forms of definite quaternion orders with class number one as well as we obtain class number relations for some CM-fields.