Almost regular quaternary quadratic forms

We investigate the almost regular positive definite integral quaternary quadratic forms. In particular, we show that every such form is p-anisotropic for at most one prime number p. Moreover, for a prime p there is an almost regular p-anisotropic quaternary quadratic form if and only if p≤37. We also study the genera containing some almost regular p-anisotropic quaternary form. We show several finiteness results concerning the families of these genera and give effective criteria for almost regularity.