# intuition for lattices in p-adic (or other non-Archimedean) vector spaces?

I could use some help to jumpstart my intuition for lattices in vector spaces over non-Archimedean fields, like $$\mathbb{Q}_p$$ and $$\mathbb{F}_q((t))$$.

I have some intuition for $$\mathbb{Z}$$-lattices in $$\mathbb{Q}$$- or $$\mathbb{R}$$-vector spaces, though perhaps it's not sophisticated enough. The question formed while watching some talks about affine Grassmannians and Witt-vector analogs, realizing that my prior intuition was leading me astray.

I'd like to develop some intuition, in particular, for $$R$$-lattices in $$K$$-vectors spaces, where $$R$$ is $$\mathbb{Z}_p$$ or $$\mathbb{F}_p[[t]]$$ and $$K$$ is $$\mathbb{Q}_p$$ or $$\mathbb{F}_p((t))$$.

As a starting point, I would like to have a good feel for 2-dimensional lattices $$R^2 \hookrightarrow K^2$$.

Question: What is a good way to visualize these objects? Or what is a minimal list of properties that I should aim to capture in a mental image, especially to reflect similarities and differences to the case of $$\mathbb{Z}$$-lattices?

: Details of current intuition:

My intuition for $$\mathbb{Z}$$-lattices mostly stems from the standard grid $$\mathbb{Z}^2 \hookrightarrow \mathbb{R}^2$$ and its images under $$GL_2(\mathbb{R})$$ action, moving the basis elements around arbitrarily, and then imagining higher dimensional analogs, to the extent possible.

I don't remember the full list of ways I noticed my intuition was leading me astray in the context of affine Grassmannians, but three insufficiencies with my current intuition are:

1) Unlike $$\mathbb{Z} \subset \mathbb{Q}$$, my mental image of $$\mathbb{Z}_p \subset \mathbb{Q}_p$$ already occupies two dimensions, rather than one, where I think about the $$p^{n+1}\mathbb{Z}_p$$ cosets as $$p$$ 2D blobs nested inside each $$p^n\mathbb{Z}_p$$ coset. So visualizing a rank-2 lattice here is akin to the difficulty of visualizing a plane curve over $$\mathbb{C}$$ using the intuition of plane curves over $$\mathbb{R}$$ - we've already got an ambient space of 4 dimensions.

2) $$\mathbb{Z} \subset \mathbb{R}$$ is cocompact, which is not the case for $$\mathbb{Z}_p \subset \mathbb{Q}_p$$. This is connected to the fact that if we move outward in $$\mathbb{R}$$, we keep encountering elements of $$\mathbb{Z}$$, whereas for my image of $$\mathbb{Q}_p$$, $$\mathbb{Z}_p$$ is located directly at the center, not occuring periodically.

3) My mental images for $$\mathbb{Q}_p$$ and $$\mathbb{F}_p((t))$$ are more or less the same, but I would like a mental image that somehow distinguishes them.

• Could you talk about what your intuition is for $\mathbb Z$-lattices, to get some feel for what's intuitive to you? (For example, "just picture a $\operatorname{GL}_2(K)$-translate of the standard embedding" probably isn't what you want, but what is?) Mar 26, 2020 at 16:50
• One distinction between the archimedean and non-archimedean situations is that $\mathbf Z$ in $\mathbf R$ is discrete with a compact quotient while $\mathbf Z_p$ inside $\mathbf Q_p$ is compact with a discrete quotient (i.e., $\mathbf Z_p$ is open in $\mathbf Q_p$). Mar 27, 2020 at 1:25
• @LSpice in my mind I know that is true, but not yet in my heart. The points I added are my best at putting into words why not. Mar 27, 2020 at 2:20
• @KConrad hmm, yes, an interesting reversal, which certainly does suggest the need for a different mental image. Is there a high-level explanation for why this is the case? My initial hunch is that it's connected to the fact that the real and p-adic expansions go off in different directions. Mar 27, 2020 at 2:23
• As a topological space you can embed $\Bbb{Q}_p$ into the real line, sending $\sum_{j=J}^\infty a_j p^j\in \Bbb{Q}_p, a_j\in 0 \ldots p-1$ to $\sum_{j=J}^\infty a_j p^{-3j}\in \Bbb{R}$. If $U\subset \Bbb{R}^n$ is open then $U\cap \Bbb{Q}_p^n$ is open in $\Bbb{Q}_p^n$, and a lattice in $\Bbb{Q}_p$ is a particular kind of open set closed under $p$-adic addition. The $p$-adic addition is obtained from the real addition by identifying $\sum_j (a_j+b_j p) p^{-3j}\sim \sum_j (a_j+b_{j-1}) p^{-3j},b_j\in 0,1$ Mar 27, 2020 at 4:41