## Section7.2Ramification of curves (John)

Two main references: P. A. Castillejo, Grothendieck-Ogg-Shafarevich formula for $\ell$-adic sheaves, Master Thesis (2016). http://www.mi.fu-berlin.de/users/castillejo/docs/160429_Master_GOS_formula_l-adic_sheaves.pdf. Lars Kindler, Kay Rülling Introductory course on $\ell$-adic sheaves and their ramification theory on curves. https://arxiv.org/pdf/1409.6899.pdf Also: Fundamental Groups in Characteristic $p$ - Pete L. Clark. Takashi Saito: Intro to wild ramification of schemes and sheaves.

Interested in ramification of $\aff^1$ in characteristic $p\text{.}$ This is interesting because of wild ramification, which we will talk about today.

Let $K$ be a complete local field.

###### Definition7.2.3.

Let $L/K$ be a finite Galois ext. let

\begin{equation*} G = \Gal L K \end{equation*}

and for $i\ge -1$

\begin{equation*} G_i = \{ \sigma \in G : \sigma \text{ acts trivially on }B/\ideal m L ^{i+1}\} \end{equation*}

where $B = \ints_L\text{,}$ and $\ideal m_L$ is the maximal ideal.

Problem: this numbering only behaves well w.r.t. subgroups not quotients.

###### Definition7.2.6.Herbrand function.

Let $G_u = G_{\lceil u \rceil}$ then we define

\begin{equation*} \phi _{L/K} \colon \lb -1,\infty ) \to \lb -1 , \infty ) \end{equation*}
\begin{equation*} \phi _{L/K}(u) = \int_0^u \frac{\diff t}{(G_0 : G_t)} \end{equation*}

if $t\in (-1,0)$ let $(G_0: G_t) = 1\text{,}$ if $t=-1$ let $(G_0: G_t) = 1/f\text{,}$ in particular $u \in \ZZ_{\ge 0}$

\begin{equation*} 1 + \phi _{L/K}(u) = \frac{1}{|G_0|} \sum_{i=0}^u |G_i| \end{equation*}

the formula arises from computing the image of $G_i$ in $G/H$ for

\begin{equation*} H \triangleleft G\text{.} \end{equation*}
###### Definition7.2.8.

Let $\psi _{L/K} = \phi \inv_{L/K}$ for $u \in \RR_{\ge -1}$

\begin{equation*} G^u = G_{\psi _{L/K}}(u)\text{.} \end{equation*}
###### Example7.2.9.Artin-Schreier.
\begin{equation*} L = K\lb t \rb /(t^{p^n} - t - x^{-m}) \end{equation*}

with $(m,p) = 1$ where $K = k ((x )) \text{,}$ lower numbering

\begin{equation*} \ZZ/p^n = G_0 = \cdots = G_m \supsetneq G_{m+1} = 0 \end{equation*}
\begin{equation*} \phi _{L/K} (u) = \begin{cases} u \amp \text{ if } 0 \le u \le m\\ m + \frac{u-m}p \amp \text{ if }u \gt m\end{cases} \end{equation*}

so

\begin{equation*} \psi _{L/K} (v) = \begin{cases} v \amp \text{ if } 0 \le v \le m\\ p(v-m) + m \amp \text{ if } v \gt m \end{cases}\text{.} \end{equation*}

The point being:

$L/K\text{,}$ $x$ a uniformizer

\begin{equation*} i_G \colon G \to \ZZ_{\ge 0} \cup\{\infty \} \end{equation*}
\begin{equation*} \sigma \mapsto v_L(\sigma (x) - x)\text{.} \end{equation*}
###### Definition7.2.13.
\begin{equation*} \mathrm{sw}_G = a_G - (r_G - r_{G/G_0}) \end{equation*}

where $r_G$ is the character of the regular representation of $G$ If $L/K$ is totally ramified then

\begin{equation*} \mathrm{sw}_G = a_G - u_G \end{equation*}

where

\begin{equation*} u_G = r_G - \mathrm{triv}_G\text{.} \end{equation*}

Fix $K^{sep} /K$ with $K$ having residue field $k$ and $\characteristic (k) = p \gt 0\text{.}$ $k$ perfect, $\ell \ne p$ prime.

$E/ \QQ_\ell$ finite extension and

\begin{equation*} \rho\colon G_K \to \GL(V) \end{equation*}

for $V$ fin. dim. vector space $/E\text{.}$

###### Definition7.2.15.
\begin{equation*} P_K \subset G_K = \Gal{K^{\sep}}{K} \end{equation*}

the wild ramification subgroup is the closed pro-$p$-group

\begin{equation*} \varprojlim_{L/K} \Gal L K _1\text{.} \end{equation*}
###### Definition7.2.16.

Let $R$ be a commutative ring and

\begin{equation*} \rho\colon G_K \to \GL_n(R) \end{equation*}

a group homomorphism, then we say:

1. $\rho$ is unramified if
\begin{equation*} \underbrace{G_K^0}_{\varprojlim \Gal L K ^0} \subseteq \ker \rho\text{.} \end{equation*}
2. $\rho$ is tamely ramified if
\begin{equation*} P_K\subseteq \ker \rho\text{.} \end{equation*}
3. $\rho$ is wildly ramified otherwise.
###### Definition7.2.18.

If $\lambda$ is a local parameter of $\ints_E$ and

\begin{equation*} \rho\colon G_K \to \GL(\mathcal V) \end{equation*}

is a representation over $\ints_E\text{.}$ The composition

\begin{equation*} \bar \rho \colon G_K \to \GL(\mathcal V) \to \GL(\overline V) \end{equation*}

where $\overline V =\mathcal V / \lambda\mathcal V$ is the reduction modulo $\lambda$ of $\rho\text{.}$ $\bar \rho$ is a $\FF_\lambda = \ints_E / \lambda$-rep.

##### Wild ramification.
###### Definition7.2.21.
\begin{equation*} \rho\colon G_K \to \GL(\mathcal V) \end{equation*}

be a continuous representation where $\mathcal V$ is a free $\ints _E$-module. Let $G = G_K/ \ker(\bar \rho)$ correspond to $L/K\text{.}$ Consider the swan representation over $\ZZ_\ell$ of $G$

\begin{equation*} b(\rho) = b(\mathcal V) = \dim_{\FF_\lambda} \Hom_{\FF_\lambda\lb G\rb }(\mathrm{Sw}_G \otimes_{\ZZ_\ell} \FF_\lambda , \bar \rho)\text{.} \end{equation*}
###### Remark7.2.22.

$b(\mathcal V)$ depends only on the class of $\mathcal V$ in the Grothendieck ring $R_{\FF_\lambda}(G)\text{.}$

###### Remark7.2.23.

If $\rho$ factors through a finite quotient then

\begin{equation*} b(\rho) = \dim_{\FF_\lambda} \Hom_{\FF_\lambda\lb G\rb }(\mathrm{Sw}_G \otimes_{\ZZ_\ell} \FF_\lambda , \bar \rho) \end{equation*}
\begin{equation*} = \dim_{E } \Hom_{E \lb G\rb }(\mathrm{Sw}_G \otimes_{\ZZ_\ell} E, \rho\otimes E) \end{equation*}

Swan conductor.

Break decomposition

###### Definition7.2.27.

$P_K$-modules are $\ZZ \lb 1/p\rb$-modules $M$ with a morphism

\begin{equation*} \rho\colon P_K \to \Aut_\ZZ(M) \end{equation*}

which factors through a finite discrete quotient. Morphisms are $\ZZ\lb1/p\rb$-module morphisms that respect the factoring.

###### Definition7.2.30.Swan conductors.

A local noetherian $\ZZ\lb 1/p\rb$-algebra $M$ as above. The Swan conductor is

\begin{equation*} \operatorname{Swan}(M) = \sum_{x\ge 0} x \rank_A(M(x)) \end{equation*}

for representations over fields

\begin{equation*} \operatorname{Swan}(V) = \sum_{x\ge 0} x \dim_A(M(x))\text{.} \end{equation*}