An abelian variety over a field k of characteristic p>0 is called supersingular if it is isogenous over the algebraic closure of k to the self product of a supersingular elliptic curve. We explain how the classification of the reduced unit groups of maximal orders in certain totally definite quaternion algebras enables us to construct supersingular abelian surfaces whose endomorphism algebra is the real quadratic field Q(sqrt{p}).
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