Abstract : The theme of the survey is how arithmetic theory of quatenion hermitian lattices can be applied to the theory of supersingular abelian varieties. Here the following geometric objects will be explained by arithmetics. Principal polarizations of superspecial abelian varieties, of supersingular surfaces, components of supersingular moduli and their configuration for small dimensions, their automorphims, fields of definition, and existence of maximal curves of genus three over $F_{p}^{2}$ for all odd primes $p$. Corresponding arithmetics are class numbers, type numbers, lattice automorphisms, and parahoric subgroups of quaternion hermitian groups, mass or trace formulas.