Quinary lattices and binary quaternion hermitian lattices

Abstract

In our previous papers, we defined the $G$-type number of any genera of quaternion hermitian lattices as a generalization of the type number of a quaternion algebra. Now we prove in this paper that the $G$-type number of any genus of positive definite binary quaternion hermitian maximal lattices in $B^2$ for a definite quaternion algebra $B$ over $\mathbb Q$ is equal to the class number of some explicitly defined genus of positive definite quinary quadratic lattices. This is a generalization of a part of the results in 1982, where only the principal genus was treated. Explicit formulas for this type number can be obtained by using Asai's class number formula. In particular, in case when the discriminant of $B$ is a prime, we will write down an explicit formula for $T$, $H$ and $2T-H$ for the non-principal genus, where $T$ and $H$ are the type number and the class number. This number was known for the principal genus before. In another paper, our new result is applied to polarized superspecial varieties and irreducible components of supersingular locus in the moduli of principally polarized abelian varieties having a model over a finite prime field, where $2T-H$ plays an important role.