# On the minima of positive definite binary hamiltonian forms

## TL;DR

This paper investigates the minima of positive definite binary Hamiltonian forms over maximal orders in quaternion algebras, establishing explicit minima related to the algebra's discriminant and providing algorithms for certain cases.

## Contribution

It determines the minimum of these forms as _A, offers explicit forms when the different is principal, and classifies all such forms when the order is principal.

## Key findings

- Minimum of forms is _A for discriminant .
- Explicit forms are provided when the different is principal.
- Algorithms are developed to check if the different is principal.

## Abstract

Let $A$ be a definite quaternion algebra over $\mathbb Q$, with discriminant $D_A$, and $O$ a maximal order of $A$. We show that the minimum of the positive definite hamiltonian binary forms over $O$ with discrimiminant $-1$ is $\sqrt{D_A}$. When the different of $O$ is principal, we provide an explicit form representing this minimum, and when $O$ is principal, we give the list of the equivalence classes of all such forms. We also give criteria and algorithms to determine when the different of $O$ is principal.