What is an Archimedean prime?

Question

I need to learn some infinite ramification theory and I am stuck with understanding it. I understand that we consider the order of the inertia group as the ramification index, and if the inertia group is trivial, it is unramified. Mostly I am confused about what Archimedean vs non-Archimedean primes are. It says an Archimedean prime is an embedding $k \to R$ or a pair of conjugate embeddings $k \to C$. But how has this got anything to do with prime ideals, and why is it not sufficient to just consider normal 'non-Archimedean' primes. And why can't you fully understand ramification this way?

Answer 1

This has nothing to do with prime ideals. Think of it as reasoning by analogy if you want.

Or think of prime ideals in terms of extending non-archimedean absolute values if you want finite places to more closely resemble real and complex places. For example, instead of saying two primes lie over $5$ in $\mathbf Z[i]$, you can say the $5$-adic absolute value on $\mathbf Q$ extends in two ways to an absolute value on $\mathbf Q(i)$. This more closely resembles extending the archimedean place on $\mathbf Q$ to different real or complex places on a number field.

By making the three Archimedean conventions that

1. a real place extending to a complex place has $e = 2$,

2. a real place extending to a real place or a complex place extending to a complex place has $e = 1$

3. every extension of Archimedean places has $f = 1$,

a lot of formulas for Archimedean places resemble formulas for non-Archimedean places, such as $r_1 + 2r_2 = n$ turning into $\sum_{i=1}^{r_1+r_2} e_if_i = n$, which resembles $\sum_{\mathfrak p|p} e(\mathfrak p|p)f(\mathfrak p|p) = n$ for primes.

You ask why it's not sufficient to think only about non-Archimedean places. Here are a few instances where the Archimedean places are worth looking at alongside the non-Archimedean places (not only for ramification).

1. The product formula needs all places, including the Archimedean ones. There's no product formula on a number field that uses only the non-Archimedean places.

2. Consider results from class field theory, such as the maximal everywhere unramified abelian extension of a number field $K$ being an extension whose degree over $K$ is the class number $h(K)$. While the only finite abelian extension of $\mathbf Q$ that's unramified at all non-Archimedean places is $\mathbf Q$ (we can even drop the abelian assumption), it's simply incorrect for general number fields $K$ that the maximal abelian extension of $K$ unramified at all non-Archimedean places has degree $h(K)$ over $K$. For example, if $K = \mathbf Q(\sqrt{3})$ and $L = K(i)$, then the extension $L/K$ is abelian and it turns out to be unramified at all prime ideals of $K$, but $[L:K] = 2$ and $h(K) = 1$. Have we found a contradiction? No: the Archimedean places of $K$ are real, while the Archimedean places of $L$ are complex. So $L/K$ is ramified (only) at the Archimedean places of $K$. In fact, the only abelian extension of $K$ that's unramified at all places of $K$, including the Archimedean ones, is $K$ itself since $h(K) = 1$. If you refuse to consider behavior at Archimedean places because you think it's irrelevant, then the theorems of class field theory won't be correct anymore as written.

3. The $S$-unit theorem. Let $S$ be a finite set of places on a number field $K$ that includes all Archimedean places. An $S$-integer in $K$ is an element of $K$ that is integral at all places ouside of $S$. So if $S = S_\infty$, then the $S$-integers of $K$ are the ordinary algebraic integers of $K$. If $S = \{\infty,2\}$ on $\mathbf Q$ then the set of all $S$-integers of $\mathbf Q$ is $\mathbf Z[1/2]$. In general, the set $\mathcal O_{K,S}$ of all $S$-integers of $K$ is a Dedekind domain containing $\mathcal O_K$ and the $S$-unit theorem says the unit group $\mathcal O_{K,S}^\times$ is finitely generated with rank $|S| - 1$. When $S = S_{\infty}$, so $|S| = r_1 + r_2$ since $K$ has $r_1+r_2$ Archimedean places, this becomes the usual Dirichlet unit theorem about the rank of the unit group of $\mathcal O_{K} = \mathcal O_{K,S_\infty}$. And if $K = \mathbf Q$ and $S = \{\infty,p_1,\ldots,p_m\}$ contains $m$ primes together with the Archimedean place, then $\mathcal O_{\mathbf Q,S} = \mathbf Z[1/p_1,\ldots,1/p_m]$ and its unit group is $\pm p_1^\mathbf Z\cdots p_m^\mathbf Z$, which has rank $m = |S| - 1$. We can also introduce the $S$-zeta-function $$ \zeta_{K,S}(s) = \prod_{v \not\in S} \frac{1}{1 - 1/{\rm N}v^s} $$ for ${\rm Re}(s) > 1$, where ${\rm N}v$ is the size of the residue field at the (non-Archimedean) place $v$. When $S = S_\infty$, $\zeta_{K,S}(s)$ is the Dedekind zeta-function $\zeta_K(s)$, which turns out to have a meromorphic continuation to $\mathbf C$ with ${\rm ord}_{s=0} \zeta_K(s) = r_1 + r_2 - 1 = |S_\infty| - 1$. Since $$ \zeta_{K,S}(s) = \zeta_K(s)\prod_{\mathfrak p \in S - S_\infty} \left(1 - \frac{1}{\rm N\mathfrak p^s}\right), $$ each prime ideal in $S$ contributes an additional simple zero at $s = 0$, so $$ {\rm ord}_{s=0} \zeta_{K,S}(s) = r_1 + r_2 - 1 + |S| - |S_\infty| = |S| - 1. $$

4. The Riemann zeta-function $\zeta(s)$ has an ugly functional equation relating $\zeta(s)$ to $\zeta(1-s)$, but $Z(s) := \pi^{-s/2}\Gamma(s/2)\zeta(s)$ has a nice functional equation $Z(s) = Z(1-s)$. Likewise, the Dedekind zeta-function $\zeta_K(s)$ has an ugly functional equation but $$ Z_K(s) = (2^{r_2}\pi^{N/2})^s\Gamma(s/2)^{r_1}\Gamma(s)^{r_2}\zeta_K(s) $$ has a nice functional equation $Z_K(s) = |d_K|^{1/2-s}Z_K(1-s)$, where $N = [K:\mathbf Q]$ and $|d_K|$ is the absolute value of the discriminant of $K$. We can clean that up by writing $N = r_1 + 2r_2$, which allows us to rewrite the above definition of $Z_K(s)$ as $$ Z_K(s) = (\pi^{-s/2}\Gamma(s/2))^{r_1}((2\pi)^{-s}\Gamma(s))^{r_2}\zeta_K(s). $$ From a 19th and early 20th century point of view those exponential and Gamma-functions seem like a peculiar additional factor to multiply by $\zeta(s)$ or $\zeta_K(s)$, but if we write those zeta-functions as Euler products over primes, then the other exponential and Gamma pieces can be regarded as factors associated to the Archimedean places: $r_1$ real places and $r_2$ complex places: for ${\rm Re}(s) > 1$, $$ Z(s) = \pi^{-s/2}\Gamma(s/2)\prod_p \frac{1}{1-1/p^s} $$ and $$ Z_K(s) = (\pi^{-s/2}\Gamma(s/2))^{r_1}((2\pi)^{-s}\Gamma(s))^{r_2}\prod_{\mathfrak p} \frac{1}{1 - 1/{\rm N}\mathfrak p^s}. $$ In Tate's thesis, the factor in the Euler product at $\mathfrak p$ can be written as an integral of a certain complex-valued function on $K_\mathfrak p^\times$, each $\pi^{-s/2}\Gamma(s/2)$ is an integral over $\mathbf R^\times$ ($K$ has $r_1$ completions equal to $\mathbf R$), and each $(2\pi)^{-s}\Gamma(s)$ is an integral over $\mathbf C^\times$ ($K$ has $r_2$ completions equal to $\mathbf C$). This allows the functional equation for $Z(s)$ and $Z_K(s)$ to be derived using harmonic analysis with the adeles and ideles of $K$, which are constructions that bring together the completions of $K$ at all of its places (Archimedean and non-Archimedean). Ignoring the Archimedean places ruins the overall symmetry, e.g., $K$ is discrete and cocompact in its adele ring, but it becomes dense if you drop the part of the adele ring coming from Archimedean places. That is ultimately due to the role of Archimedean places in the product formula.

Answer 2

In many of these situations, you're really looking at completions of $k$. By Ostrowski's theorem, there are only two possibilities: the completion at the $\mathfrak{p}$-adic absolute value $|x| = p^{-\nu_{\mathfrak{p}}(x)}$ for some prime $\mathfrak{p}\in \mathcal{O}_k$, and the completion at the ordinary absolute value $|\sigma(x)|$ for some embedding $\sigma:k \to \mathbb{C}$. The former is called non-Archimedean becaus it satisfies a stronger version of the triangle inequality: $|x + y|_p \leq \max(|x|_p, |y|_p)$. Because each non-Archimedean absolute value is associated with a prime $\mathfrak{p}$, those are called the finite primes or places of $k$; the other absolute values are referred to as infinite.

As for why this distinction is worth making, I'd recommend taking a look at some of the results in Cassels and Froehlich or Silverman's book on elliptic curves. At the very least, there's the product formula that KCd refers to above: $\prod_{\nu\in M_k} |x|_\nu = 1$ for all nonzero $x\in k$, where the product runs over the set of (non-Archimedean and Archimedean) places $M_k$ of $k$.