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The Geometry of $\mathbb Q_p$ and $\mathbb Z_p$

The $p$-adics and Related Concepts

Once we define distance appropriately, we’ll find a distance preserving embedding of $\mathbb Z_p$ into $\mathbb R^2$—that is we’ll be able to draw a picture of $\mathbb Z_p$ and determine the distance between points by looking at that picture. This in turn will provide a related geometric picture for $\mathbb Q_p$.

A graphical representation of the 5-adic integers.

The $5$-adic integers in this picture are points on the fractal boundary. The naturally appearing pentagons in the picture (or rather their fractal boundaries) are closed balls in $\mathbb Z_5$. The large gray pentagon is the maximal ideal $5 \mathbb Z_5$ and the other labeled large pentagons are the cosets (balls of radius $p$). The ball of radius $1/25$ is labeled $25 \mathbb Z_5$. All other balls of radius $1/25$ are given by the fractal boundaries of like-sized pentagons.

We can describe a point on the boundary of the fractal by systematically labelling the cosets of each closed ball (counterclockwise if we want to do this in a uniform fashion) from 0 to 4, and then providing an infinite tuple of integers in $\{0, 1, 2, 3, 4\}$ describing the smaller and smaller balls containing the point. This infinite tuple gives the coefficients for the canonical series representation of the point.

We may visualize the Haar measure of closed balls, by recalling that $\lambda(\mathbb Z_5) = 1$ and then assigning to other closed balls the proportion of the graph they occupy. For instance $5 \mathbb Z_5$ occupies 1/5 of the graph, and hence $\lambda(5 \mathbb Z_5) = 1/5$, and $25 \mathbb Z^5$ occupies 1/25 of the graph and hence $\lambda(25 \mathbb Z_5) = 1/25$. Note that the open ball of radius $p^{-n}$ is equal to the closed ball of radius $p^{-n-1}$ and hence the measures of open balls can likewise be determined graphically.

We may also visualize distance between points in $\mathbb Z_5$, though not by using the normal distance in $\mathbb R^2$. Our notion of distance is connected with the notion of Haar measure of closed balls.

Theorem: The $p$-adic distance between two points is given by the measure of the smallest closed ball containing them.

This theorem means we can find the distance between two points on the graph by computing the measure of the smallest closed ball containing both point, and this measure can be discerned by noting the number of cosets of that ball.

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