the analogous problem in the category of supersingular elliptic curves, on which the security of the Charles, Goren and Lauter hash function [@Charles2009] is based, has to date resisted attack. This dichotomy poses several questions on the extent to which the information from the algebraic category can be transported to the geometric one. In particular, one expects an algorithm for computing the endomorphism ring of a given elliptic curve to provide an effective reduction to the algebraic setting, making the hardness of this problem critical to the underlying security.

The quaternion $\ell$-isogeny path problem {#sec:quaternions}
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In this section, we first motivate and define the quaternion $\ell$-isogeny path problem. We then recall basic facts on quaternion algebras. We introduce *$p$-extremal* maximal orders, which will play an important role in our solution of the quaternion $\ell$-isogeny problem. We finally discuss properties of reduced norms and ideal morphisms.

“Hard” isogeny problems {#sec:intro:hard}
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The motivation for studying the quaternion $\ell$-isogeny problem is based on the analogous (indeed categorically equivalent) problem for supersingular elliptic curves. The difficulty of this problem for elliptic curves underlies the security of the Charles, Goren and Lauter hash function [@Charles2009].

As an example, finding a preimage (inverting the 