How to geometrically interpret intertia of primes in field extensions?

Question

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?

Answer 1

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 $\mathbb{Z}[\sqrt{3}]$.

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