I am trying to understand the definition of the genus equivalence (semi-equivalence) for positive definite ternary forms. I recognized that there are some (serious) issues in the literature, and I was confused. To ask my question, I need some conventions:
Let $q(x,y,z)=ax^2+by^2+cz^2+dyz+exz+fxy$ be a positive definite, primitive ternary form. Here $q$ is defined Watson's sense. So by a primitive form, I mean the $\gcd(a,b,c,d,e,f)=1.$
In Gauss, Smith, Dickson's sense, it is a ternary form if $d,e,f$ are even. So, the definition of a ternary form and a primitive form are different from Watson's sense.
Since Smith assumed that $d,e,f$ are even number, we cannot directly use his results to check when two (primitive) ternary forms in Watson's sense are genus equivalence.
My question is as follows: Is there a way to extend his results to any ternary forms in Watson's sense? More explicitly, given two primitive ternary forms (in Watson's sense) can we decide whether they are genus equivalent or not by checking their assigned characters?
For example, there is a very nice (unproven!) result in Lehman's paper (https://www.jstor.org/stable/2153043?seq=1), which is Proposition 4 in page 410. He used Watson's definition for ternary forms, and stated this result in Watson's sense. However, he does not prove this very nontrivial fact, and he only gave references. One of the references he gave there is Dickson's book. But Dickson's definition for ternary forms is different as I said above. So it requires many arguments to use this reference (in my opinion). If Lehman's result was true, it would be great. I tried to prove Lehman's result, but it seems extremely hard. I hope I could explain my issue clearly. My question is again that are there any (proven) results as Lehman's one related to my question? Or can anyone see how we can prove Lehman's result?
As a note, I stated there are some issues in the literature because I heart from a Professor (who is expert about the theory of the quadratic forms) that he has counterexamples for both Lehman's the aforementioned result and Dickson's result related to this topic.