How to geometrically interpret intertia of primes in field extensions?

I am trying to understand the intuition of thinking about number theoretic ideas in terms of geometric ones. For example, ramification is something that happens when a “covering” space of a Riemann surfaces collapses down to a point. I can sort of see the analogy with ramification of primes like this:

Let $L/K$ be an extension of number fields, and $P$ a prime of $\mathcal{O}_K$. If $P$ splits in $\mathcal{O}_L$, then I imagine each of the $n = [L:K]$ primes as lying in $n$ distinct “layers” of some “covering” of the primes of $K$. Then when we have ramification, say $P = Q^e \prod Q_i$, the $Q_i$ remain as expected in the covering, but we have a degeneration at $Q$ giving us $e$ layers collapsing into just one above $P$, like in the case of Riemann surfaces.

So my question is: using this type of geometric analogy, where is the inertia hiding? I don’t yet have enough background to understand etale morphisms yet, but I’ve heard that these give a nice analogue of covering maps in this kind of setting. Is there some concrete way I can try to build intuition about first? Maybe is there a way to interpret the residue field extension $k_Q/k_P$ using a corresponding unramified extension of the local fields $L_Q/K_P$ and give a uniform way of looking at this “covering” interpretation so that only ramification really feels like the “bad” points where we have less than expected cardinality of the fibers?

Solutions Collecting From Web of "How to geometrically interpret intertia of primes in field extensions?"

For a geometric picture, I would say that inertia means there’s fewer points $Q_i$ above a given point $P$ in the covering because some of the points got “fat”, in the sense that
$$|\mathcal{O}_L/Q_i|>|\mathcal{O}_K/P|$$
(Indeed, the inertial degree is the degree of the extension $[\mathcal{O}_L/Q_i:\mathcal{O}_K/P]$.)

I like to think of an analogy with conservation of momentum. The extension $L/K$ has “energy” $n$, where $n=[L:K]$. At any given point (prime) $P$ of $K$, the energy can go into

  • pushing $n$ different primes of the same size at the same speed (totally split)
  • pushing a single prime of size $n$ at the same speed (totally inert)
  • pushing a single prime of size $1$ at $n$ times the speed (totally ramified)

or some mixture of these effects. The fundamental relation $n=\sum_{i=1}^r e_if_i$ just expresses conservation of momentum.

Here’s an image from Eisenbud and Harris’ The Geometry of Schemes of $\mathrm{Spec}(\mathbb{Z}[\sqrt{3}])$. Note that $(5)$ and $(7)$ are inert in $\mathbb{Z}[\sqrt{3}]$.

enter image description here

Here’s a long excerpt explaining what’s going on in the image.