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64.34 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

\[ 1 \to \pi _1(X_{\overline{k}}) \to \pi _1(X) \xrightarrow {\deg } \widehat{\mathbf{Z}} \to 1 \]

Proposition 64.34.1. 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 $\pi _1(X)$.

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

\[ \{ x_1, \ldots , x_ n\} = {\text{ set of all closed points of} \atop X \text{ of degree} \leq N\text{ over } k} \]

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

\[ Y \to ^ G X,\quad G = \pi _1(X)/U \]

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

\[ \# Y(k_ N)\geq (\# G)\# X(k_ N) \]

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

\[ q^ N+1+2g_ Yq^{N/2}\geq \# G\# X(k_ N)\geq \# G(q^ N+1-2g_ Xq^{N/2}) \]

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

\[ q^ N + 1 + (\# G)(2g_ X - 1) + 1)q^{N/2}\geq \# G (q^ N + 1 - 2g_ Xq^{N/2}) \]

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.

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