Proposition 63.34.1. There exists a finite set $x_1, \ldots , x_ n$ of closed points of $X$ such that set of **all** frobenius elements corresponding to these points topologically generate $\pi _1(X)$.

## 63.34 How many primes decompose completely?

This section gives a second application of the Riemann Hypothesis for curves over a finite field. For number theorists it may be nice to look at the paper by Ihara, entitled “How many primes decompose completely in an infinite unramified Galois extension of a global field?”, see [Ihara]. Consider the fundamental exact sequence

Another way to state this is: There exist $x_1, \ldots , x_ n\in |X|$ such that the smallest normal closed subgroup $\Gamma $ of $\pi _1(X)$ containing $1$ frobenius element for each $x_ i$ is all of $\pi _1(X)$. i.e., $\Gamma = \pi _1(X)$.

**Proof.**
Pick $N\gg 0$ and let

Let $\Gamma \subset \pi _1(X)$ be as in the variant statement for these points. Assume $\Gamma \neq \pi _1(X)$. Then we can pick a normal open subgroup $U$ of $\pi _1(X)$ containing $\Gamma $ with $U \neq \pi _1(X)$. By R.H. for $X$ our set of points will have some $x_{i_1}$ of degree $N$, some $x_{i_2}$ of degree $N - 1$. This shows $\deg : \Gamma \to \widehat{\mathbf{Z}}$ is surjective and so the same holds for $U$. This exactly means if $Y \to X$ is the finite étale Galois covering corresponding to $U$, then $Y_{\overline{k}}$ irreducible. Set $G = \text{Aut}(Y/X)$. Picture

By construction all points of $X$ of degree $\leq N$, split completely in $Y$. So, in particular

Use R.H. on both sides. So you get

Since $2g_ Y-2 = (\# G)(2g_ X-2)$, this means

Thus we see that $G$ has to be the trivial group if $N$ is large enough. $\square$

**Weird Question.** Set $W_ X = \deg ^{-1}(\mathbf{Z})\subset \pi _1(X)$. Is it true that for some finite set of closed points $x_1, \ldots , x_ n$ of $X$ the set of all frobenii corresponding to these points *algebraically* generate $W_ X$?

By a Baire category argument this translates into the same question for all Frobenii.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)