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arXiv · 1905.04881

On the minima of positive definite binary hamiltonian forms

Gaëtan Chenevier·Frédéric Paulin
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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.

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Gaëtan Chenevier, Frédéric Paulin. 2019-05-13. On the minima of positive definite binary hamiltonian forms. https://arxiv.org/abs/1905.04881

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