Terminology: what is the "generic character" of a ternary quadratic form?

Question

The title says it all:

What is the "generic character" of a ternary quadratic form?

Motivation: I'm reading a really old paper, and the author refers to this terminology without any further definition.

Answer 1

Arnold E. Ross and Alexander V. Oppenheim were students of Leonard Eugene Dickson. You mostly want to borrow two of Dickson's many books;

Modern Elementary Theory of Numbers (1939), especially page 161 where he gives references about universal indefinite ternary forms;

Studies in the Theory of Numbers (1929) which has more but is harder reading.

For an intermediate window on that type of writing, Burton Wadsworth Jones, The Arithmetical Theory of Quadratic Forms (1950). He does not seem to do characters for ternaries...

The upshot, by the way is that there are four slightly different types (up to integral equivalence) of indefinite ternary forms that integrally represent all integers:

Let $N$ be odd and $M$ be any integer, allowed positive or negative but not $0,$

$$ xy - M z^2 $$ $$ 2xy - N z^2 $$ $$ 2xy + y^2 - N z^2 $$ $$ 2xy + y^2 - 2N z^2 $$

I think the first type is easiest to see in Oppenheim (1930), I'm not sure if it is included in Dickson or in the Ross article.

Looking again, you need the earlier Studies, Dickson does not mention generic characters in the later (and intentionally easier) book.

Hmmm. Well, I guess what I should really recommend is my own, see the "very long manuscript" at http://zakuski.utsa.edu/~jagy/ which is about as easy as it is possible to get.

I put many related documents at http://zakuski.math.utsa.edu/~kap/forms.html but mostly about positive forms.

EEDDIITT: given a quadratic form, we can write it as $$ (1/2) X' H X, $$ where $H$ is the Hessian matrix of second partials, for us all entries integers, $X$ is a column vector and $X'$ is its transpose, a row vector. Two forms are equivalent if there is a matrix $P$ of integers with $\det P = 1$ such that $$ P' H_1 P = H_2. $$

Weaker, two matrices are in the same genus if the have the same determinant $h,$ and there are two matrices, $\det Q = q, \det R = r,$ with $\gcd(q,2h) = 1,\gcd(r,2h) = 1, $ such that $$ Q' H_1 Q = q^2 H_2, \; \; \mbox{and} \; \; R' H_2 R = r^2 H_1. $$ This is the definition of Siegel, he called it "rationally equivalent without essential denominator."