BUNTES Ramification of curves (John)

Section 7.2 Ramification 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.

Theorem 7.2.1. Grothendieck.

There exists a canonical surjection

\begin{equation*} sp\colon \pi _1(C_{\bar\eta}) ^{tame} \to \pi _1(C/k)^{tame} \end{equation*}

where

\begin{equation*} C_{\eta} \end{equation*}

is the generic fibre for a lift of \(C/k\) over a complete local noetherian ring with residue field \(k\text{.}\)

Theorem 7.2.2.

For \(\mathcal F\) a lisse \(\overline \QQ_\ell\)-sheaf on \(U\) (a curve over a perfect field of characteristic \(p \ne \ell\)).

\begin{equation*} \chi _c ( U, \mathcal F) = \rk (\mathcal F) \chi _c(\overline U, \overline \QQ_\ell) - \sum_{x \in C\smallsetminus U} \lb k(x) : k\rb ]\operatorname{Swan}_x(\mathcal F) \end{equation*}

\(C\) is the compactification of \(U\text{.}\)

\begin{equation*} \overline U = U \otimes_k \overline k \end{equation*}

\begin{equation*} \chi _c(\overline U, \mathcal F) = \sum_{i=0}^2 (-1)^i \dim H_c^i(\overline U_\et, \mathcal F)\text{.} \end{equation*}

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.

Proposition 7.2.4.

If \(H \subset G\text{,}\) then

\begin{equation*} H_i = H \cap G_i\text{.} \end{equation*}

Proposition 7.2.5.

\begin{equation*} G_{-1} = G \end{equation*}

\begin{equation*} G_{0} = \text{inertia} \end{equation*}

\begin{equation*} G_{0}/G_1 = \text{tamely ramified part} \end{equation*}

\begin{equation*} G_1 \ne 0 \iff L/K \text{ is wildly ramified} \end{equation*}

\begin{equation*} i\ge 1,\,G_i \text{ are }p\text{-groups} \end{equation*}

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*}

Proposition 7.2.7.

\begin{equation*} H \triangleleft G \end{equation*}

for all \(u \in \RR_{\ge -1}\)

\begin{equation*} G_uH/H = (G/H)_{\phi_{L/L^H}}(u)\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*}

Theorem 7.2.10. Hasse-Arf.

If \(G\) is abelian, then the jumps in the upper numbering filtration are all at integers.

The point being:

Proposition 7.2.11.

If \(H \triangleleft G\) we have

\begin{equation*} G^u / (H\cap G^u ) = (G/H)^u\text{.} \end{equation*}

\(L/K\text{,}\) \(x\) 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*}

Theorem 7.2.12.

\begin{equation*} a_G(g) = \begin{cases} - fi_G(g) \amp\text{ if } g \ne 1\\ f\sum_{g' \ne 1} i_G(g') \amp\text{ if } g = 1 \end{cases} \end{equation*}

is a character of a \(G\)-representation over \(\CC\text{.}\)

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*}

where

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

Theorem 7.2.14.

If \(\ell \) is a prime not equal to the residue characteristic of \(K\text{,}\) then

\(a_G\) and \(\mathrm{sw}_G\) are realisable over \(\QQ_\ell\)
There exists a f.g. projective left \(\ZZ_\ell \lb G\rb\)-module \(\mathrm{Sw}_G\) unique up to iso. such that
\begin{equation*} \mathrm{Sw}_G \otimes_{ \ZZ_\ell }\QQ_\ell \end{equation*}
is isomorphic to the Swan representation.

reference: Serre, Linear representations of finite groups.

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:

\(\rho\) is unramified if
\begin{equation*} \underbrace{G_K^0}_{\varprojlim \Gal L K ^0} \subseteq \ker \rho\text{.} \end{equation*}
\(\rho\) is tamely ramified if
\begin{equation*} P_K\subseteq \ker \rho\text{.} \end{equation*}
\(\rho\) is wildly ramified otherwise.

Lemma 7.2.17.

Let \(G\) be a compact group

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

a continuous representation over \(E\text{,}\) then there exists a free \(\ints_E\)-module \(\mathcal V \subseteq V\) s.t.

\begin{equation*} V = \mathcal V \otimes E \end{equation*}

and \(\rho\) factors as

\begin{equation*} \rho\colon G \to \GL(\mathcal V) \to \GL(V)\text{.} \end{equation*}

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.

Lemma 7.2.19.

If \(P\) is a pro-\(p\)-group and

\begin{equation*} \rho\colon P \to \GL_r(\ints_E) \end{equation*}

is a continuous representation. Then the image of \(\rho\) is finite and

\begin{equation*} \rho(P) \cap \ker(\GL_r(\ints_E) \twoheadrightarrow \GL_r(\FF_\lambda)) = \{1\}\text{.} \end{equation*}

Corollary 7.2.20.

\(\rho\) is tame if and only if \(\bar \rho \) is tame.

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*}

Proposition 7.2.24.

\begin{equation*} G = G_K/ \ker(\bar \rho) \end{equation*}

then

\begin{equation*} b(\mathcal V) = \sum_{i=1}^\infty \frac{|G_i|}{|G_0|} \dim_{\FF_\lambda} (\overline {\mathcal V} / \overline {\mathcal V}^{G_i})A\text{.} \end{equation*}

Proposition 7.2.25.

If \(\rho \colon G_K \to \GL_r(\ints_E)\) is a continuous representation then TFAE:

\(\rho \otimes E \colon G_K \to \GL_r(\ints_E) \hookrightarrow \GL_r(E)\)is tame
\(\rho\) is tame
\(\bar \rho\) is tame
\(\displaystyle b(\mathcal V) = 0\)

Swan conductor.

Break decomposition

Lemma 7.2.26.

\(\lambda \in \RR_{\ge 0 }\text{,}\)

\begin{equation*} G_K^{\lambda+} = \overline{\bigcup_{\lambda' \gt \lambda} G_K^{\lambda'}} \end{equation*}

then the upper numbering filtration satisfies

\begin{equation*} \bigcap_{\lambda\gt 0} G_K^\lambda = \{1\} \end{equation*}
\begin{equation*} \lambda\gt 0,\, G_K^\lambda = \bigcap_{0 \lt \lambda' \lt \lambda} G_K^{\lambda'} \end{equation*}
\begin{equation*} P_K = G_K ^{0+} \end{equation*}

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.

Proposition 7.2.28.

\(M\) is a \(P_K\)-module. Then There is a unique decomposition

\begin{equation*} M = \bigoplus _{x\in \RR_{\ge 0}} M(X) \end{equation*}

of \(P_K\)-modules s.t.

\begin{equation*} M(0) = M^{P_K} \end{equation*}

\begin{equation*} M(x)^{G^x}) = 0 \end{equation*}

for \(x \gt 0\text{,}\) and for \(x,y \in \RR_{\ge 0}\)

\begin{equation*} M(x) ^{G^y} = M(x) \end{equation*}

for \(x \gt y\text{,}\)

\begin{equation*} M(x) = 0 \end{equation*}

for all but fin many \(x\text{.}\)

\begin{equation*} M \to M(x) \end{equation*}

is an exact endofunctor on \(P_K\)-modules.

Corollary 7.2.29.

Let \(A \) be a \(\ZZ\)-algebra, \(M\) be an \(A\)-module, with a \(P_K\)-action, that factors through a finite quotient: Then

for a break decomposition of \(M\text{,}\) \(M(x)\) is an \(A\)-module.
for \(B\) an \(A\)-algebra a break decomposition of \(M\oplus B\) is \(\bigoplus (M(x) \oplus B)\text{.}\)
if \(A \) is local noetherian and \(M\) is a free \(A\)-module of finite rank, then \(M(x)\) is also free of finite rank for all \(x\text{.}\)

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*}

Proposition 7.2.31.

\begin{equation*} \operatorname{Swan}(M) = 0 \end{equation*}
iff \(P_K\) acts trivially on \(M\text{.}\)
For \(V = \mathcal V \otimes E\) we get \(V(x) = \mathcal V (x) \otimes E\) and \(\operatorname{Swan}(V) = \operatorname{Swan}(\mathcal V)\)
Similarly
\begin{equation*} \operatorname{Swan}(\mathcal V) = \operatorname{Swan}(\overline{\mathcal V})\text{.} \end{equation*}