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The geometry of lines in $\mathbb{R}^2$ is fundamental to mathematics and likewise for lines in $\mathbb{C}^2$ since $\mathbb{C}^2 \cong \mathbb{R}^4$. But is there a good treatment of lines in $\mathbb{Q}_p^2$, i.e. the $p$-adic plane?

For example, suppose $y= \zeta x$ is a $p$-adic line. How does it interact with the lattice $\mathbb{Z}^2 \subset \mathbb{Q}_p^2$? Certainly the line will pass through a well-determined set of lattice points if and only if $\zeta \in \mathbb{Q} \subset \mathbb{Q}_p$. So what if $\zeta \not\in \mathbb{Q}$? Can we make a statement (similar to irrational slopes in the reals) that the line passes arbitrarily close to lattice points?

Perhaps more importantly, is there any value added by thinking geometrically since most of these questions could be answered algebraically and the non-archimedean space is difficult to imagine?

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    $\begingroup$ The most fundamental reason to be 'less interested' in lines in $\mathbb{Q}_p^2$, for me, is less the non-archimedean aspects specifically and more the lack of a meaningful order relation on elements - for instance, lines no longer define a half-plane. This is also the case over $\mathbb{C}^2$, certainly, but as you note the isomorphism to $\mathbb{R}^4$ - a vector space over a base field that is ordered - provides a lot of its structure; for the $p$-adics that's no longer available. $\endgroup$
    – Steven Stadnicki
    Dec 5, 2013 at 20:41
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    $\begingroup$ Why is the geometry of lines in the plane fundamental to mathematics? (I'm not arguing otherwise, I'm just a bit surprised by the claim.) $\endgroup$
    – Bruno Joyal
    Dec 5, 2013 at 21:06

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Unfortunately, there are no lattices in $p$-adic vector spaces. The reason is that if $v$ is a nonzero vector, then $p^nv$ approaches $0$ in the $p$-adic topology, so $0$ is an accumulation point of every subgroup.

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