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 $$\mathbb {Z}_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> , 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.