On Three-Term Linear Relations for Theta Series of Positive-Definite Binary Quadratic Forms

TL;DR

This paper characterizes all three-term linear relations among theta series of positive-definite binary quadratic forms, revealing a unique non-trivial relation involving forms with specific discriminants.

Contribution

It extends Schiemann's methods and develops a novel algorithm to fully classify three-term linear relations among these theta series.

Findings

Exactly one non-trivial three-term relation found

Relation involves forms with discriminants -3, -12, -48

Relation occurs within the same rational squareclass

Abstract

In this paper, we investigate three-term linear relations among theta series of positive-definite integral binary quadratic forms. We extend Schiemann's methods to characterize all possible three-term linear relations among theta series of such forms, providing necessary and sufficient conditions for such relations to exist. To accomplish this, we develop, implement, and execute a novel extended refinement algorithm on polyhedral cones. We show that there is exactly one non-trivial three-term linear relation: it involves quadratic forms with discriminants $-3, -12, -48$, all in the same rational squareclass $-3(\mathbb{Q}^\times)^2$.