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Section 7.2 Ramification of curves (John)

Two main references: P. A. Castillejo, Grothendieck-Ogg-Shafarevich formula for \(\ell\)-adic sheaves, Master Thesis (2016). Lars Kindler, Kay Rülling Introductory course on \(\ell\)-adic sheaves and their ramification theory on curves. 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.

Definition 7.2.3.

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.

Definition 7.2.6. 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*}
Definition 7.2.8.

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*}
Example 7.2.9. 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*}


\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*}
Definition 7.2.13.
\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*}


\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 \(/E\text{.}\)

Definition 7.2.15.
\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*}
Definition 7.2.16.

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.
Definition 7.2.18.

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.
Definition 7.2.21.
\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*}
Remark 7.2.22.

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

Remark 7.2.23.

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

Definition 7.2.27.

\(P_K\)-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.

Definition 7.2.30. 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*}