I'm studying a geometrical problem where an (even, negative-definite) lattice $L$ arises. Roughly, as an intersection pairing for curves on a surface. In fact, the problem naturally leads me to consider the shifted lattices $L + a$. In principle, you can certainly shift by any fractional vector $a \in L \otimes_{\mathbb{Z}} \mathbb{Q}$. But it turns out that I only ever have to shift by elements in the discriminant group, i.e. $a \in L^{*}/L$. I didn't require this, it came naturally from the geometry.

Moreover, the geometry tells me to consider a shifted lattice for each orbit of $L^{*}/L$ under the action of the isometry group of $L$. So the problem somehow spans all possible shifted lattices of this type, up to isometry.

Basically, I am looking for some context. Is there a structural or theoretical reason why one "should" shift by elements of the discriminant group as opposed to some general fractional vector? Like might this be hinting at something, or are these special kinds of shifted lattices that show up in other areas of math? One idea is that the discriminant group somehow measures non-unimodularity. Unimodular lattices show up in a closely related problem, so maybe this family of shifted lattices is naturally what you should consider in the non-unimodular case.