Polarized superspecial simple abelian surfaces with real Weil numbers

Abstract Let be an odd power of a prime number , and be the finite set of isomorphism classes of principally polarized superspecial abelian surfaces in the simple isogeny class over corresponding to the real Weil ‐numbers . The main contribution provides explicit formulae for of the following kinds: (i) the class number formula, that is, the cardinality of ; (ii) the type number formula, that is, the number of endomorphism rings up to isomorphism of the underlying abelian surfaces of . Similar formulae are obtained for other collections of polarized superspecial members of this isogeny class grouped together according to their polarization modules. The main tools are the notion of genera, the trace formula for norm‐one groups of totally definite quaternion algebras built on the optimal spinor selectivity theory, and analyzing algebraic structures for terms in the trace formula. Using these explicit formulae, we observe several surprising identities involving the arithmetic genus of certain Hilbert modular surface on one side and the class number or type number of polarized superspecial abelian surfaces in this isogeny class on the other side.