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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?

[Edit]: 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.

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    $\begingroup$ 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?) $\endgroup$
    – LSpice
    Mar 26, 2020 at 16:50
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    $\begingroup$ 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 $\mathbf Q_p$). $\endgroup$
    – KConrad
    Mar 27, 2020 at 1:25
  • $\begingroup$ @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. $\endgroup$
    – PrimeRibeyeDeal
    Mar 27, 2020 at 2:20
  • $\begingroup$ @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. $\endgroup$
    – PrimeRibeyeDeal
    Mar 27, 2020 at 2:23
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    $\begingroup$ 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 $\sum_j (a_j+b_j p) p^{-3j}\sim \sum_j (a_j+b_{j-1}) p^{-3j},b_j\in 0,1$ $\endgroup$
    – reuns
    Mar 27, 2020 at 4:41

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