Polarized superspecial simple abelian surfaces with real Weil numbers : a survey (Theory and Applications of Supersingular Curves and Supersingular Abelian Varieties)

Abstract

In this survey article we describe recent results of explicit formulae for (a) the number of superspecial abelian surfaces over Fq with real Frobenius endomorphism √q equipped with a polarization module; (b) the type number of each genus of these polarized superspecial abelian surfaces; and (c) the refined class and type numbers of such abelian surfaces with a given automorphism group. These formulae suggest an interesting connection with arithmetic genera of Hilbert modular surfaces.