Number of representations of local $p$-one-dimensional forms by the genus

We study branching of representations of a locally $p$-one-dimensional form by a genus of positive definite integral quadratic forms. We give a complete list of minimal representations by a genus for forms of square level. Gauss–Minkowski formulas are obtained for heights of representations over the ring of integers. As an application, we obtain formulas for heights of primitive representations by genera for specific forms constructed by the method of orthogonal complement.