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In this paper, we add the information of level structure to supersingularelliptic curves and study these objects with the motivation of isogeny-basedcryptography. Supersingular elliptic curves with level structure map to Eichlerorders in a quaternion algebra, just as supersingular elliptic curves map tomaximal orders in a quaternion algebra. We study this map and the Eichlerorders themselves. We also look at isogeny graphs of supersingular ellipticcurves with level structure, and how they relate to supersingular isogenygraphs used in post-quantum cryptography.

Adding Level Structure to Supersingular Elliptic Curve Isogeny Graphs