Gram matrix modulo 4

Question

Suppose we have a full rank, integer sublattice $L$ of the integer lattice $\mathbb Z^d$, where we fix the dimension $d$. Consider the Gram matrix $M$ of $L$, relative to some basis for $L$, and reduce all the entries of $M$ mod $4$. Is there a nice clean description of all the finite types of such mod $4$ reductions of Gram matrices, as we vary over all full rank integer sublattices $L \subset \mathbb Z^d$ while keeping the dimension $d$ fixed?

There is a theorem that attempts to describe the Gram matrix of an integer lattice mod powers of $2$, in J.W.S. Cassells' book ``Rational quadratic forms", Section VIII.4, p. 117 in this book. But in the beginning of that section he writes "...This section is only for the masochistic". I would be very grateful if anyone has found a cleaner description and/or proof, at least in the mod $4$ case.

Answer 1

I use the 2-adic decompositions for various tasks. I can't say I know what would be useful for you, but let me call your attention to page 141, Lemma 4.3. This refers back to Lemma 5.2 on page 123, (paraphrase) if $d(g) = u^2 d(f)$ for some unit $u$ and $$ g_{ij} \equiv f_{ij} \pmod {2^{ v_2(d(f)) + 2}}, $$ then $f,g$ are $\mathbb Z_2$-equivalent. Here $v_2$ is the 2-adic valuation, the highest power of 2 dividing $d(f)$ in this case.

Meanwhile, I can point you to A Canonical quadratic form for the ring of 2-adic integers, Burton W. Jones, Duke Math. J. 11 (1944), 715-727. This and two others are at the top of page 120 in Cassels, look for ``parliamentary draftsman.'' Note that Watson (1976), Mathematika 23 94-106, The 2-adic density of a quadratic form, is the source for the 2-adic terms in Conway and Sloane on the Mass Formula. In turn Watson's own 1960 book is the main source for the 1976 article.