Local description of primitive representations by ternary quadratic forms

Abstract In this paper, for each integral quadratic ternary form Q with prescribed Jordan decomposition over the ring Zₚ Z p, where p is odd, we give criteria for determining whether there exists a primitive representation of integers A with p -valuation at most three by the form Q. As an application of the local criteria, we prove a result related to primitive prime-universality and primitive odd-universality.