## Section7.4Serre's proof in the solvable case (Angus)

$k = \bar k$ field of char $p \gt 0\text{.}$

###### Definition7.4.1.

A quasi $p$-group is a group that is generated by its Sylow-$p$ subgroups.

Let $\Pi(\aff^1)$ be the set of groups occurring as Galois groups of covers

\begin{equation*} X \to \aff^1\text{.} \end{equation*}

Today we will prove this when $G$ is solvable.

###### Definition7.4.4.Solvable groups.

A group $G$ is called solvable if there exists a series

\begin{equation*} G = G_{k} \triangleright G_{k-1}\triangleright \cdots \triangleright G_0 = 1 \end{equation*}

such that each $G_k/G_{k-1}$ is abelian.

Reminders/background:

\begin{equation*} \{\text{covers } X\to \aff^1 \text{ w/ gal. gp. }G\} \end{equation*}
\begin{equation*} \updownarrow \end{equation*}
\begin{equation*} \{L/k(T) : \text{Gal. extn. w/ gp. }G\text{ unram. outside }\infty \} \end{equation*}
\begin{equation*} \updownarrow \end{equation*}
\begin{equation*} \{\text{surjections }\pi _1^\et(aff^1) \twoheadrightarrow G\}\text{.} \end{equation*}

Fixing $\bar x\in X$ we have an equivalence of categories

\begin{equation*} \{\text{loc. const. }\FF_\ell\text{-sheaves on }X\text{ w/ finite stalks}\} \end{equation*}
\begin{equation*} \updownarrow \end{equation*}
\begin{equation*} \{\text{finite dim. }\pi _1^\et(X, \bar x)\text{ reps over }\FF_\ell\} \end{equation*}

given by

\begin{equation*} \sheaf F \mapsto \sheaf F_{\bar x}\text{.} \end{equation*}

Etale cohomology satisfies:

Exactness axiom: Let $Z\subseteq X$ a closed subscheme with $U = X\smallsetminus Z\text{.}$ Let

\begin{equation*} \Gamma _Z (X, \sheaf F) = \ker(\Gamma (X, \sheaf F) \to \Gamma (U, \sheaf F)) \end{equation*}

the right derived functor of

\begin{equation*} \Gamma _Z(X, -) \end{equation*}

is

\begin{equation*} H^* _Z(X, -) \end{equation*}

the cohomology with support on $Z\text{.}$ Then we have a LES

\begin{equation*} \cdots \to H_Z^i(X, \sheaf F) \to H^i(X, \sheaf F) \to H^i(U, \sheaf F) \to H_Z^{i+1}(X, \sheaf F)\to \cdots\text{.} \end{equation*}

For $X$ an affine curve, $\pi _1(X)$ has cohomological dimension $\le 1\text{.}$ In particular given a surjection

\begin{equation*} \pi _1(X) \twoheadrightarrow G/H \end{equation*}

we can lift to a map

\begin{equation*} \pi _1(X) \to G\text{.} \end{equation*}

Let $\widetilde G = N\text{.}$

The advantage of this is the following:

Consider a SES

\begin{equation*} 1 \to K \to N \to H \to 1 \end{equation*}

where

\begin{equation*} G = \widetilde G/N = (\widetilde G/ K) / (N / K) = (\widetilde G/K)/H\text{.} \end{equation*}

Since $N$ is solvable, given a sequence of subgroups with abelian quotients we can reduce to the abelian case, which can then be reduced to $(\ZZ/\ell)^n\text{.}$ Further can be reduced to the irreducible $G$-module case.

$\widetilde G$ is an extension of $G$ by $N$ which gives a class

\begin{equation*} e \in H^2(G, N) \end{equation*}

we have cases

1. \begin{equation*} e\ne 0\text{ essential extension} \end{equation*}
2. \begin{equation*} e= 0,\,\widetilde G = N\rtimes G \end{equation*}

Proof of the theorem in case 1:

\begin{equation*} G \in \Pi(\aff^1) \leadsto \phi \colon \pi \to G \end{equation*}

by the cohomological dimension argument there exists a lift

\begin{equation*} \tilde \phi \colon \pi \to \widetilde G \end{equation*}

with $H = \im(\tilde \phi)$ so $NH = \widetilde G$ and $N\cap H$ is a sub-$G$-module of $N\text{.}$ If $N\cap H = 1$ then $\widetilde G = N\rtimes G\text{,}$ a contradiction with the fact we are in case 1.

Then by irreducibility of $N\text{,}$ $N \cap H = N$ and

\begin{equation*} N \subseteq H \implies H = NH = \widetilde G\text{.} \end{equation*}

In case 2. Choose a surjection

\begin{equation*} \phi \colon \pi \twoheadrightarrow G \end{equation*}

this endows $N$ with a $\pi$-module structure, $N_\phi$ we get a corresponding sheaf $\sheaf N_\phi$ on $\aff^1\text{.}$ We have

\begin{equation*} H^1(G, N) \hookrightarrow H^1(\pi , N_\phi) \xrightarrow\sim H^1(\aff^1, \sheaf N_\phi )\text{.} \end{equation*}

We only need ($\Leftarrow$) today. Let $(a \colon \pi \to N_\phi ) \in H^1(\pi , N_\phi ) \smallsetminus H^1(G, N)\text{.}$ Then, combined with $\phi$ with we construct a morphism

\begin{equation*} \tilde \phi \colon \pi \to N\cdot G = \widetilde G\text{.} \end{equation*}

Assume that

\begin{equation*} \im(\tilde \phi ) = H \subsetneq \widetilde G \end{equation*}

then

\begin{equation*} N\cap H = 1,\,NH = \widetilde G\text{.} \end{equation*}

Given this, $a$ arises from a cocycle in $H^1(G, N)$ a contradiction.

We are reduced to finding

\begin{equation*} \phi \colon \pi \to G \end{equation*}

such that

\begin{equation*} \dim_{\FF_\ell} H^1(G, N) \lt \dim _{\FF_\ell} H^1(\pi , N_\phi ) \end{equation*}

two cases, $\ell \ne p$ and $\ell = p\text{.}$

In the $\ell \ne p$ case we must have

\begin{equation*} G \acts N \end{equation*}

non-trivial else

\begin{equation*} \widetilde G = N\times G \end{equation*}

is not quasi-$p\text{.}$

Let $I \subseteq G$ be the inertia group at $\infty \text{,}$ consider the ramification groups

\begin{equation*} I \supseteq I_1\supseteq I_2 \supseteq I_3 \supseteq \cdots \end{equation*}

we have the swan conductor of $N_\phi$ is

\begin{equation*} \operatorname{Swan}_{\infty } (N_\phi ) = \sum_{n \ge 1 } \frac{1}{\lb I:I_n\rb } \dim (N/N^{I_n})\text{.} \end{equation*}

Note

\begin{equation*} H^0( \pi , N_\phi ) = 0 \end{equation*}

since nontrivial irreducible

\begin{equation*} H^{i\ge 2}( \pi , N_\phi ) = 0 \end{equation*}

by cohomological dimension. Then

\begin{equation*} \dim H^1(\pi , N_\phi ) = - \chi ( H^*(\pi , N_\phi ))= - \chi ( H^*(\aff^1 , \sheaf N_\phi )) \end{equation*}

let

\begin{equation*} i\colon \aff^1 \hookrightarrow \PP^1 \end{equation*}

then exactness gives

\begin{equation*} \chi (H^*(\aff^1, \sheaf N_\phi )) = \chi (H^*(\PP^1, i_* \sheaf N_\phi )) - \chi (H^*_\infty (\PP^1,i_* \sheaf N_\phi )) \end{equation*}

now

\begin{equation*} \chi (H^*_\infty (\PP^1,i_* \sheaf N_\phi )) = \dim N^I\text{.} \end{equation*}

Grothendieck-Ogg-Shafarevich gives

\begin{equation*} \chi (H^*(\PP^1, i_* \sheaf N_\phi )) = \dim N + \dim N^I - \operatorname{Swan}_\infty (N_\phi )\text{.} \end{equation*}

We are reduced to

\begin{equation*} \dim_{\FF_\ell} H^1(G, N) \lt \operatorname{Swan}_\infty (N_\phi ) - \dim _{\aff_\ell} N \end{equation*}

there exists $\phi$ for which this can be an equality (Artin-Schreier). We can always introduce extra ramification. Consider

\begin{equation*} (m)\colon \aff^1 \to \aff^1 \end{equation*}
\begin{equation*} T \mapsto T^m \end{equation*}

and write $\psi \colon Y \to \aff^1$ the cover corresponding to $\phi \text{.}$ Take the pullback to get $\psi _m\colon Y_m \to \aff^1$ a Galois cover with group $G\text{.}$ $\leadsto \phi _m \colon \pi \twoheadrightarrow G\text{.}$ One can show that

\begin{equation*} \operatorname{Swan}_\infty (N_{\phi _m}) = m \operatorname{Swan}_\infty (N_\phi ) \end{equation*}

so choosing $m \gt 1$ forces the inequality to be strict.

In this case we show

\begin{equation*} \dim H^1(\pi , N_\phi ) = \infty \end{equation*}

exactness gives

\begin{equation*} H^1(\pi , N_\phi ) = H^1(\aff^1, \sheaf N_\phi ) \to H^2_\infty (\PP^1, i_*\sheaf N_\phi ) \to H^2(\PP^1, i_* \sheaf N_\phi ) = 0 \end{equation*}
\begin{equation*} H^2_\infty (\PP^1, i_*\sheaf N_\phi ) = H^2_\infty (\Spec k \lb \lb t\inv \rb \rb , i_*\sheaf N_\phi ) = H^1 (k ((T\inv )) , N_\phi )\text{.} \end{equation*}

We can take

\begin{equation*} G_F \acts V \end{equation*}

irreducible, then if $I_1$ is the pro-$p$-Sylow subgroup of $G_F$ then the action of

\begin{equation*} I_1\acts V \end{equation*}

is trivial so the action factors through the tame quotient

\begin{equation*} I_t = G_F/I_1\text{.} \end{equation*}

Choosing an identification of $V$ with

\begin{equation*} \FF_q / \FF_p \end{equation*}

then

\begin{equation*} I_t \acts V \end{equation*}

is determined by a character

\begin{equation*} \psi \colon I_t \to \FF_q^\times \end{equation*}

let $m = \operatorname{order}(\psi)\text{,}$ $t_m = t^{1/m}$ and $F_m = k((t_m))\text{.}$ The Galois group

\begin{equation*} C_m = \Gal{F_m}F \end{equation*}

is identified with the group of $m$-th roots of unity by a character

\begin{equation*} \chi \colon C_m \to k^\times\text{.} \end{equation*}

Choosing $\FF_q \hookrightarrow k$ gives

\begin{equation*} \psi = \chi ^i \end{equation*}

for some $i \in (\ZZ/m)^\times\text{,}$ then

\begin{equation*} H^1(G_F, V)\simeq H^0(C_m, H^1(I_m, V)) = H^0(C_m, \Hom(I_m, \FF_p) \otimes V \end{equation*}

when $I_m = \Gal {F^\sep}{F_m}\text{.}$ We have

\begin{equation*} \Hom(I_m, \FF_p) = F_m / \wp F_m \end{equation*}

for $\wp$ the Artin-Schreier map, so it is sufficient to show that any character of $C_m$ occurs in the $C_m$-representation

\begin{equation*} F_m/ \wp F_m) \otimes \FF_p \end{equation*}

infinitely often. The group $F_m / \wp F_m$ has representatives Laurent series

\begin{equation*} \sum a_j t_m^j \end{equation*}

for $a_j \in k,\,j \gt 0,(j,p) = 1\text{.}$ Consider the subgroup

\begin{equation*} k \{t_m^{-j}\} \end{equation*}

on which $C_m$ acts by $\chi ^{-j}\text{.}$ Since $\lb k : \FF_q \rb = \infty \text{,}$ $\chi ^{-j}$ occurs infinitely often.

So

\begin{equation*} \dim H^1(\pi , N_\phi ) = \infty \end{equation*}

and the desired inequality is satisfied and we have a surjective lift

\begin{equation*} \pi \to \widetilde G \end{equation*}

in all cases giving the original theorem.