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).
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
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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