Ramification in number fields

The local view

The phenomenon of ramification has an equivalent interpretation in terms of local rings. With $L/K$ as before, let $\P$ be a prime in $\O_L$ with $\p := \P \intersect \O_K$ . Then the induced map of localizations $(\O_K)_\p \hookrightarrow (\O_L)_\P$ is a local homomorphism of local rings (in fact, of discrete valuation rings), and the ramification index of $\P$ over $\p$ is the unique natural number $e$ such that $$ \p (\O_L)_\P = (\P (\O_L)_\P)^e \subset (\O_L)_\P. $$

An astute reader may notice that this formulation of ramification index does not require that $L$ and $K$ be number fields, or even that they play any role at all. We take advantage of this fact here to give a second, more general definition.

The reader who is not interested in local rings may assume that $A$ and $B$ are unique factorization domains, in which case $e(\P/\p)$ is the exponent of $\P$ in the factorization of the ideal $\iota(\p) B$ , just as in our first definition (but without the requirement that the rings $A$ and $B$ originate from number fields).

There is of course much more that can be said about ramification indices even in this purely algebraic setting, but we limit ourselves to the following remarks:

1. Suppose $A$ and $B$ are themselves discrete valuation rings, with respective maximal ideals $\p$ and $\P$ . Let $\hat{A} := \ilim A/\p^n$ and $\hat{B} := \ilim B/\P^n$ be the completions of $A$ and $B$ with respect to $\p$ and $\P$ . Then \begin{equation}\label{completion} e(\P/\p) = e(\P\hat{B}/\p\hat{A}). \end{equation}In other words, the ramification index of $\P$ over $\p$ in the $A$ -algebra $B$ equals the ramification index in the completions of $A$ and $B$ with respect to $\p$ and $\P$ .
2. Suppose $A$ and $B$ are Dedekind domains, with respective fraction fields $K$ and $L$ . If $B$ equals the integral closure of $A$ in $L$ , then \begin{equation}\label{sum-formula} \sum_{\P \div \p} e(\P/\p) f(\P/\p) \leq [L:K], \end{equation}where $\P$ ranges over all prime ideals in $B$ that divide $\p B$ , and $f(\P/\p) := \dim_{A/\p}(B/\P)$ is the inertial degree of $\P$ over $\p$ . Equality holds in Equation () whenever $B$ is finitely generated as an $A$ -module.

Ramification in algebraic geometry

The word ``ramify'' in English means ``to divide into two or more branches,'' and we will show in this section that the mathematical term lives up to its common English meaning.

Example 4   The picture in Figure 1 may be worth a thousand words. Let $k = \C$ and $C_1 = C_2 = \C = \A^1_\C$ . Take the map $f: \C \lra \C$ given by $f(y) = y^2$ . Then $f$ is plainly a map of degree 2, and every point in $C_2$ except for 0 has two preimages in $C_1$ . The point 0 is thus a ramification point of $f$ of index 2, and we have drawn the graph of $f$ near $0$ .

Figure 1: The function $f(y) = y^2$ near $y=0$ .

Note that we have only drawn the real locus of $f$ because that is all that can fit into two dimensions. We see from the figure that a typical point on $C_2$ such as the point $x = 1$ has two points in $C_1$ which map to it, but that the point $x = 0$ has only one corresponding point of $C_1$ which ``branches'' or ``ramifies'' into two distinct points of $C_1$ whenever one moves away from 0.

Relation to the number field case

The relationship between Definition 2 and Definition 3 is easiest to explain in the case where $f$ is a map between affine varieties. When $C_1$ and $C_2$ are affine, then their coordinate rings $k[C_1]$ and $k[C_2]$ are Dedekind domains, and the points of the curve $C_1$ (respectively, $C_2$ ) correspond naturally with the maximal ideals of the ring $k[C_1]$ (respectively, $k[C_2]$ ). The ramification points of the curve $C_1$ are then exactly the points of $C_1$ which correspond to maximal ideals of $k[C_1]$ that ramify in the algebraic sense, with respect to the map $f^*: k[C_2] \lra k[C_1]$ of coordinate rings.

Equation () in this case says $$ \sum_{P \in f^{-1}(p)} e(P/p) = n, $$ and we see that the well known formula () in number theory is simply the algebraic analogue of the geometric fact that the number of points in the fiber of $f$ , counting multiplicities, is always $n$ .

In the case where $f$ is a map between projective varieties, Definition 2 does not directly apply to the coordinate rings of $C_1$ and $C_2$ , but only to those of open covers of $C_1$ and $C_2$ by affine varieties. Thus we do have an instance of yet another new phenomenon here, and rather than keep the reader in suspense we jump straight to the final, most general definition of ramification that we will give.

Ramification in complex analysis

Ramification points or branch points in complex geometry are merely a special case of the high-flown terminology of Definition 6. However, they are important enough to merit a separate mention here.

Algebraic-analytic correspondence

Of course, the analytic notion of ramification given in Definition 9 can be couched in terms of locally ringed spaces as well. Any Riemann surface together with its sheaf of holomorphic functions is a locally ringed space. Furthermore the stalk at any point is always a discrete valuation ring, because germs of holomorphic functions have Taylor expansions making the stalk isomorphic to the power series ring $\C[[z]]$ . We can therefore apply Definition 6 to any holomorphic map of Riemann surfaces, and it is not surprising that this process yields the same results as Definition 9.

More generally, every map of algebraic varieties $f: V \lra W$ can be interpreted as a holomorphic map of Riemann surfaces in the usual way, and the ramification points on $V$ and $W$ under $f$ as algebraic varieties are identical to their ramification points as Riemann surfaces. It turns out that the analytic structure may be regarded in a certain sense as the ``completion'' of the algebraic structure, and in this sense the algebraic-analytic correspondence between the ramification points may be regarded as the geometric version of the equality () in number theory.

The algebraic-analytic correspondence of ramification points is itself only one manifestation of the wide ranging identification between algebraic geometry and analytic geometry which is explained to great effect in the seminal paper of Serre [6].

Bibliography

1

Robin Hartshorne, Algebraic Geometry, Springer-Verlag, 1977 (GTM 52).

2

Gerald Janusz, Algebraic Number Fields, Second Edition, American Mathematical Society, 1996 (GSM 7).

3

Jürgen Jost, Compact Riemann Surfaces, Springer-Verlag, 1997.

4

Dino Lorenzini, An Invitation to Arithmetic Geometry, American Mathematical Society, 1996 (GSM 9).

5

Jean-Pierre Serre, Local Fields, Springer-Verlag, 1979 (GTM 67).

6

Jean-Pierre Serre, ``Géométrie algébraique et géométrie analytique,'' Ann. de L'Inst. Fourier 6 pp. 1-42, 1955-56.

7

Joseph Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, 1986 (GTM 106).