Over Q, the definite quaternion algebras with a unique conjugacy class of maximal orders, i.e. "with class number one", are those with discriminant 2,3,5,7, and 13.
Three questions:
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What is a reference for this result?
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What are some examples of definite quaternions of class number one, over other totally real fields?
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For a fixed real quadratic field, is there a known classification of definite quaternion algebras class number one?
Best Answer
For the first question, the statement goes back to Brzezinski (http://archive.numdam.org/ARCHIVE/JTNB/JTNB_1995__7_1/JTNB_1995__7_1_93_0/JTNB_1995__7_1_93_0.pdf), who treats all definite quaternion orders over $\mathbb{Z}$, not just maximal ones. For maximal orders, this is an almost-immediate consequence of Eichler's mass formula. The simplest proof I know of the mass formula follows in the same way as the proof of the analytic class number formula and can be found in his "Lectures on Modular Correspondences"; for prime discriminant this has many nice proofs (Is there a nice proof of the fact that there are (p-1)/24 supersingular elliptic curves in characteristic p?).
For the second and third questions, you can see my paper with Markus Kirschmer (http://www.math.dartmouth.edu/~jvoight/articles/quatideal-fixed-errata-052613.pdf) where we find all Eichler orders of class number one: you're asking for Table 8.2. Kirschmer and David Lorch are working on generalizing this in several directions. The simplest (and arguably most beautiful) example is to take a maximal order in $B=(-1, -1 \mid \mathbb{Q}(\sqrt{5}))$, the base change of the Hamilton quaternions to $\mathbb{Q}(\sqrt{5})$. Such a maximal order is unique up to conjugation in $B^\times$ and has unit group (modulo scalars) given by the rigid motions of the icosahedron (or dodecahedron); explicitly, it is generated as a $\mathbb{Z}[(\sqrt{5}+1)/2]$-algebra by $i$ and the element $$ \frac{1}{2}\left(\frac{\sqrt{5}+1}{2}+\frac{\sqrt{5}-1}{2}i + j\right).$$