Abstract

Let be a totally definite quaternion algebra over a totally real field , and an -order in . The quotient is a finite group, called the reduced unit group of . Assume that is real quadratic.  We classify the reduced unit groups of all maximal orders, and count the number of conjugacy classes of maximal orders with a fixed reduced unit group for each possible non-cyclic group. When applied to the case and   splitting at all finite places of , this result produces the number of isomorphism classes of certain supersingular abelian surfaces with a specific reduced automorphism group within the isogeny class corresponding to the Weil numbers . This is a joint work with Qun Li and Prof. Chia-Fu Yu.

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