# Equivalence of quadratic forms over $p$-adic integers vs over localisation at $p$

To discern whether two integral quadratic forms are equivalent over the $$p$$-adic integers, one can compute a Jordan decomposition at $$p$$ and read off some invariants. Restricting to $$p\ne2$$ for convenience, Conway and Sloane give an algorithm for this procedure (`$$p$$-adic diagonalisation'). But their algorithm in fact computes a diagonalisation over the localisation of $$\mathbb{Z}$$ at $$p$$, $$\mathbb{Z}_{(p)}:=\{\frac{a}{b}\text{ : }a,b\in\mathbb{Z},p\nmid b\}$$, rather than over the $$p$$-adic integers $$\mathbb{Z}_p$$ in generality. My question is: does equivalence over $$\mathbb{Z}_{(p)}$$ differ from equivalence over $$\mathbb{Z}_p$$? And if so, when do such differences occur?

Since $$\mathbb{Z}_{(p)}$$ can be thought of as a subring of $$\mathbb{Z}_{p}$$, if two quadratic forms $$Q_{1}$$ and $$Q_{2}$$ are equivalent over $$\mathbb{Z}_{(p)}$$, then they must be equivalence over $$\mathbb{Z}_{p}$$. (And this is the reason that it suffices for Conway and Sloane to compute a diagonalization over $$\mathbb{Z}_{(p)}$$.)

However, the converse is false. Equivalence over $$\mathbb{Z}_{p}$$ does not imply equivalence over $$\mathbb{Z}_{(p)}$$ and maybe a good way of explaining this is that $$\mathbb{Z}_{p}$$ has a lot fewer square classes. In particular, if $$p > 2$$ and $$x \in \mathbb{Z}_{p}$$, then $$x$$ is a square if and only if $$x = p^{2k} u$$ for some integer $$k \geq 0$$ and $$u \in \mathbb{Z}_{p}^{\times}$$ whose reduction $$\tilde{u} \in \mathbb{F}_{p}$$ is a square.

In particular, the quadratic forms $$Q_{1}(x,y) = x^{2} - y^{2}$$ and $$Q_{2}(x,y) = x^{2} - 7y^{2}$$ are equivalent in $$\mathbb{Z}_{3}$$, but are not equivalent in $$\mathbb{Z}_{(3)}$$. This follows from the observation that $$\sqrt{7} \in \mathbb{Z}_{3}$$, while $$\sqrt{7} \not\in \mathbb{Z}_{(3)}$$.

• Thanks, great answer! Do you a know a reference for studying equivalence over the localisation? Do we have invariants that classify equivalence over $\mathbb{Z}_{(p)}$ in the same way we do for equivalence over $\mathbb{Z}_p$ (and genera of forms)? Oct 27, 2022 at 8:20
• Yes. Chapter 9 of O'Meara's book "Introduction to Quadratic Forms" is titled "Integral Theory of Quadratic Forms over Local Fields", and Theorem 92.2 of that book (page 247 in the second printing) gives a classification (in the non-dyadic cases) in terms of the discriminants of the terms in the Jordan splitting. (Basically, a term in the Jordan splitting arises from diagonalising the quadratic form and then grouping together all terms where the $p$-adic valuation of the coefficients are the same.) Oct 27, 2022 at 14:51