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In "Supersingular K3 surfaces" by TetsuJi Shioda, when proving Theorem 3.5 (Deligne) he considers a supersingular elliptic curve $C$ over an algebraic closed field of $\text{char}\ p>0$ and let $R = End(C)$, which is a maximal order in a quaternion algebra over $\Bbb Q$. Then he uses the fact that any projective module of rank $g \geq 2$ over $R$ is free by Eichler's theorem $[5]$.

The reference $[5]$ is an old paper written in German, so why is it free? More generally, what do we know about finitely generated projective modules (non only rank $1$ ones) over a maximal order inside some central simple algebras over a number field?

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  • $\begingroup$ Possibly this is discussed in Maximal Orders by Curtis and Reiner? I don't have a copy to hand so this is only a tentative suggestion based on a vague memory. $\endgroup$
    – inkspot
    Commented Nov 8, 2018 at 19:30
  • $\begingroup$ @inkspot，thank you. I checked that book but it defines $Cl$ as stable isomorphism classes of ideals and always input the Eichler condition. However, here $R$ is the maximal order in the quaternion algebra over $\mathbb Q$ ramified at $p$ at $\infty$, and it's class number is not $1$ when $p$ grows. The theorem only states for higher rank, which is quite wild. $\endgroup$
    – Zhiyu
    Commented Nov 10, 2018 at 5:19
  • $\begingroup$ But we know maximal orders are hereditary rings, so I am very skeptical about the theorem used in that paper. $\endgroup$
    – Zhiyu
    Commented Nov 10, 2018 at 5:21

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