Numbers Represented by a Finite Set of Binary Quadratic Forms

TL;DR

This paper investigates the integers represented by collections of positive-definite binary quadratic forms, using class group structure and class field theory to determine when nontrivial common representations occur.

Contribution

It extends the understanding of simultaneous representations by binary quadratic forms with different discriminants through class field theory methods.

Findings

Characterization of integers represented by multiple forms

Application of class field theory to non-discriminant-matching forms

Conditions for nontrivial common representations

Abstract

Every quadratic form represents 0; therefore, if we take any number of quadratic forms and ask which integers are simultaneously represented by all members of the collection, we are guaranteed a nonempty set. But when is that set more than just the "trivial" 0? We address this question in the case of integral, positive- definite, reduced, binary quadratic forms. For forms of the same discriminant, we can use the structure of the underlying class group. If, however, the forms have different discriminants, we must apply class field theory.