Theorem 11 (*Conway & Sloane, Sphere Packings, Lattices and Groups, 3rd Edition, pp 383, Ch 15*). If a system of putative $p$-adic symbols for each $p$ satisfies the determinant, oddity, and $p$-adic existence conditions, then there exists an integral quadratic form with these $p$-adic symbols.


The book does not provide any references or pointers on where to find a proof. We checked the obvious reference i.e., B. W. Jones, Cassels, O'Meara, and could not find the relevant theorem. Unfortunately, we find it far from obvious and hence are unable to prove it ourselves.


We have two quick questions.


1. Can someone please recommend a reference where a proof appears ?
2. Is the Theorem if and only if ?


Thank you for your time.



Proving the existence of an integral quadratic form
The Conway–Sloane method is getting popular, partly because they gave the version of The Mass Formula that everyone uses. Some students of [Gabriele Nebe](http://www.math.rwth-aachen.de/homes/Gabriele.Nebe/), at Aachen, have begun doing calculations with it, see [Lorch and Kirschmer - Single-Class Genera of Positive Integral Lattices](http://arxiv.org/abs/1208.5638). Meanwhile, [Daniel Allcock](http://www.ma.utexas.edu/users/allcock/) taught a course on this last year and all the [homework](http://www.ma.utexas.edu/users/allcock/qf/index.html) was in this C–S slang. Now, Allcock's adviser was Borcherds, and Borcherds' adviser was Conway. You can't tell me that's a coincidence.


Meanwhile, given a description of a genus by discriminant, signature, and a representative in each of the $p$-adic integers, why is the genus not empty? It seems magic sometimes. For positive ternary forms, the careful article is [Levels of positive definite ternary quadratic forms](https://doi.org/10.2307/2153043) by Larry Lehman (1992), I put a pdf at [ternary](http://zakuski.math.utsa.edu/%7Ekap/forms.html). I wrote an article using that, it said there was a genus, I looked for it in my huge computer list, there they were. Amazing.


I think the most reassuring thing should be Siegel's weighted representation. Along with the mass formula, Siegel also showed how to calculate the average number of times a given integer was represented by a genus. For positive forms, the actual number of representations by an individual form is just divided by the number of integer automorphs of the form. For indefinite forms, the automorphism groups are infinite, so instead, the count is taken of distinct orbits. Here is the magic part: if an integer is $p$-adically eligible for each prime (including $\infty,$ which Conway has taken to calling $-1$), then the average number of representations, over the genus, is nonzero. Meaning the genus is not empty.


If you have not looked at it, Watson's book gives the beginnings of this argument. Remember that we create genera of binary forms by taking an eligible prime $p,$ solving $ \beta^2 \equiv \Delta \pmod {4p},$ finally arriving at the form $$ \left\langle p, \beta, \frac{\beta^2 - \Delta}{4p} \right\rangle. $$


