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I am currently reading Serre's "Topics in Galois Theory". Specifically, I am looking at the proof of Theorem 3.6.2, and there is one aspect of the proof that is unclear to me. I'll describe the setting, and state my questions below.

The setting is as follows. Let $K$ be a number field, $W$ and $V$ absolutely irreducible varieties of dimension $n$ over $K$, and $\pi: W \to V$ a generically surjective morphism. The map $\pi$ induces an extension of function fields $K(W) / K(V)$. Let $K(W)^{gal}$ be the Galois closure of $K(W)$ in this extension, $W^{gal}$ the normalization of $W$ in $K(W)^{gal}$, and $\pi ' : W^{gal} \to V$ the composition of $\pi$ with the projection from $W^{gal}$ to $W$.

Let $v$ be a prime of $K$ that splits completely in the algebraic closure of $K$ in $K(W)^{gal}$. Serre states that "the connected components of the fiber [of $W^{gal}$] at $v$ are absolutely irreducible" (bracketed text mine), and uses this along with the Lang-Weil estimates to compute $\# W^{gal}(\mathbb F _v)$.

Questions:

  1. What does the language of schemes contribute to this statement? Can the statement be made precise without the language of schemes?

  2. Where could one find a proof of this statement? Alternately, is it easy enough to prove directly?

I have looked at "small" examples and I can see how the point-count of the image of a map modulo $v$ varies with the splitting behavior of the prime $v$ as described above. I am not uncomfortable with the statement, in that it makes sense with my understanding of the "yoga" of varieties over a number field. Sadly, neither of these translate into an understanding of how one would prove such a fact.

Insights into this, as well as corrections to any of my misunderstandings of the situation that I make clear in this exposition, are all welcome. Ultimately, I am interested in what other choices I can make for $K$ other than "number field".

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The question is intrinsic to $W^{\rm{Gal}}$, so forget $V$ and $W$. Your question concerns a (separated) integral finite type normal scheme $W'$ over $K$ and the $K$-finite algebraic closure $K'$ of $K$ in $K(W')$; $W'$ is a $K'$-scheme by normality. To define "fiber of $W'$ at $v$" you must specify an $O_K$-model of $W'$. You want to allow ignoring a finite set of places. To relate algebraic geometry over $K$ and $\mathbf{F}_v$, it is foolish to avoid schemes. The key is to apply EGA IV$_3$, 9.7.7(ii) to an $O_{K'}$-model of $W'$. With good technique, $O_K$ can be replaced with any domain. –  user36938 Jul 16 '13 at 5:46

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