Automorphisms and monodromy C. L

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Automorphisms and monodromy

C. LEHR¹ M. MATIGNON²

1MPI Bonn ²Laboratoire A2X Bordeaux

Workshop Chuo University April 2006

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Plan

Monodromy and automorphism groups

Automorphism groups of curves in char.p>0 p -cyclic covers of the affine line

Structure of G_∞,1(f) Bounds for|G_∞,1(f)|

Characterization of G_∞,1(f)

Actions of p-groups over a curve C with g(C)≥2 Big actions (I)

Condition (N) and G₂ Riemann surfaces Ray class fields Big actions (II) Maximal curves Monodromy polynomial

Marked stable model Potentially good reduction Genus 2

References

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Plan

Monodromy and automorphism groups

Automorphism groups of curves in char.p>0 p -cyclic covers of the affine line

Structure of G_∞,1(f) Bounds for|G_∞,1(f)|

Characterization of G_∞,1(f)

Actions of p-groups over a curve C with g(C)≥2 Big actions (I)

Condition (N) and G₂ Riemann surfaces Ray class fields Big actions (II) Maximal curves Monodromy polynomial

Marked stable model Potentially good reduction Genus 2

References

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Plan

Monodromy and automorphism groups

Automorphism groups of curves in char.p>0 p -cyclic covers of the affine line

Structure of G_∞,1(f) Bounds for|G_∞,1(f)|

Characterization of G_∞,1(f)

Actions of p-groups over a curve C with g(C)≥2 Big actions (I)

Condition (N) and G₂ Riemann surfaces Ray class fields Big actions (II) Maximal curves Monodromy polynomial

Marked stable model Potentially good reduction Genus 2

References

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Plan

Monodromy and automorphism groups

Automorphism groups of curves in char.p>0 p -cyclic covers of the affine line

Structure of G_∞,1(f) Bounds for|G_∞,1(f)|

Characterization of G_∞,1(f)

Actions of p-groups over a curve C with g(C)≥2 Big actions (I)

Condition (N) and G₂ Riemann surfaces Ray class fields Big actions (II) Maximal curves Monodromy polynomial

Marked stable model Potentially good reduction Genus 2

References

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Plan

Monodromy and automorphism groups

Automorphism groups of curves in char.p>0 p -cyclic covers of the affine line

Structure of G_∞,1(f) Bounds for|G_∞,1(f)|

Characterization of G_∞,1(f)

Actions of p-groups over a curve C with g(C)≥2 Big actions (I)

Condition (N) and G₂ Riemann surfaces Ray class fields Big actions (II) Maximal curves Monodromy polynomial

Marked stable model Potentially good reduction Genus 2

References

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Monodromy and automorphism groups.

I R is a strictly henselian DVR of inequal characteristic(0,p).

K :=FrR ; for example K/Q^urp finite.

πa uniformizing parameter.

k :=R_K/πR_K.

C/K smooth projective curve, g(C)≥1.

I C has potentially good reduction over K if there is L/K (finite) such that C×K L has a smooth model over R_L. Then:

I There is a minimal extension L/K with this property ; it is Galois and called the monodromy extension.

I Gal(L/K)is the monodromy group.

I Its p-Sylow subgroup is the wild monodromy group .

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Monodromy and automorphism groups.

I R is a strictly henselian DVR of inequal characteristic(0,p).

K :=FrR ; for example K/Q^urp finite.

πa uniformizing parameter.

k :=R_K/πR_K.

C/K smooth projective curve, g(C)≥1.

I C has potentially good reduction over K if there is L/K (finite) such that C×K L has a smooth model over R_L. Then:

I There is a minimal extension L/K with this property ; it is Galois and called the monodromy extension.

I Gal(L/K)is the monodromy group.

I Its p-Sylow subgroup is the wild monodromy group .

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Monodromy and automorphism groups.

I R is a strictly henselian DVR of inequal characteristic(0,p).

K :=FrR ; for example K/Q^urp finite.

πa uniformizing parameter.

k :=R_K/πR_K.

C/K smooth projective curve, g(C)≥1.

I C has potentially good reduction over K if there is L/K (finite) such that C×K L has a smooth model over R_L. Then:

I There is a minimal extension L/K with this property ; it is Galois and called the monodromy extension.

I Gal(L/K)is the monodromy group.

I Its p-Sylow subgroup is the wild monodromy group .

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Monodromy and automorphism groups.

I R is a strictly henselian DVR of inequal characteristic(0,p).

K :=FrR ; for example K/Q^urp finite.

πa uniformizing parameter.

k :=R_K/πR_K.

C/K smooth projective curve, g(C)≥1.

I C has potentially good reduction over K if there is L/K (finite) such that C×K L has a smooth model over R_L. Then:

I There is a minimal extension L/K with this property ; it is Galois and called the monodromy extension.

I Gal(L/K)is the monodromy group.

I Its p-Sylow subgroup is the wild monodromy group .

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Monodromy and automorphism groups.

I R is a strictly henselian DVR of inequal characteristic(0,p).

K :=FrR ; for example K/Q^urp finite.

πa uniformizing parameter.

k :=R_K/πR_K.

C/K smooth projective curve, g(C)≥1.

I C has potentially good reduction over K if there is L/K (finite) such that C×K L has a smooth model over R_L. Then:

I There is a minimal extension L/K with this property ; it is Galois and called the monodromy extension.

I Gal(L/K)is the monodromy group.

I Its p-Sylow subgroup is the wild monodromy group .

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

I The base change C×K K^alginduces an

homomorphismϕ:Gal(K^alg/K)→Aut_kCs, where C_sis the special fiber of the smooth model over R_Land L= (K^alg)^kerϕ.

I Let`be a prime number, then, n<sub>`</sub> :=v_{`</sub>(|Gal(L/K)|)≤v<sub>`}(|AutkC_s|).

I If` /∈ {2,p},then`<sup>n`</sup>is bounded by the maximal order of an`-cyclic subgroup ofGL_2g(Z/`Z)i.e.

`<sup>n`</sup> ≤O(g).

I If p>2, then

n_p≤inf<sub>`6=2,p</sub>v<sub>p</sub>(|GL<sub>2g</sub>(Z/`Z)|) =a+ [a/p] +..., where a= [_p−1^2g ].

This gives an exponential type bound in g for

|Aut_kC_s|. This justifies our interest in looking at Stichtenoth ([St 73]) and Singh ([Si 73]).

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

I The base change C×K K^alginduces an

homomorphismϕ:Gal(K^alg/K)→Aut_kCs, where C_sis the special fiber of the smooth model over R_Land L= (K^alg)^kerϕ.

I Let`be a prime number, then, n<sub>`</sub> :=v_{`</sub>(|Gal(L/K)|)≤v<sub>`}(|AutkC_s|).

I If` /∈ {2,p},then`<sup>n`</sup>is bounded by the maximal order of an`-cyclic subgroup ofGL_2g(Z/`Z)i.e.

`<sup>n`</sup> ≤O(g).

I If p>2, then

n_p≤inf<sub>`6=2,p</sub>v<sub>p</sub>(|GL<sub>2g</sub>(Z/`Z)|) =a+ [a/p] +..., where a= [_p−1^2g ].

This gives an exponential type bound in g for

|Aut_kC_s|. This justifies our interest in looking at Stichtenoth ([St 73]) and Singh ([Si 73]).

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

I The base change C×K K^alginduces an

homomorphismϕ:Gal(K^alg/K)→Aut_kCs, where C_sis the special fiber of the smooth model over R_Land L= (K^alg)^kerϕ.

I Let`be a prime number, then, n<sub>`</sub> :=v_{`</sub>(|Gal(L/K)|)≤v<sub>`}(|AutkC_s|).

I If` /∈ {2,p},then`<sup>n`</sup>is bounded by the maximal order of an`-cyclic subgroup ofGL_2g(Z/`Z)i.e.

`<sup>n`</sup> ≤O(g).

I If p>2, then

n_p≤inf<sub>`6=2,p</sub>v<sub>p</sub>(|GL<sub>2g</sub>(Z/`Z)|) =a+ [a/p] +..., where a= [_p−1^2g ].

This gives an exponential type bound in g for

|Aut_kC_s|. This justifies our interest in looking at Stichtenoth ([St 73]) and Singh ([Si 73]).

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

I The base change C×K K^alginduces an

homomorphismϕ:Gal(K^alg/K)→Aut_kCs, where C_sis the special fiber of the smooth model over R_Land L= (K^alg)^kerϕ.

I Let`be a prime number, then, n<sub>`</sub> :=v_{`</sub>(|Gal(L/K)|)≤v<sub>`}(|AutkC_s|).

I If` /∈ {2,p},then`<sup>n`</sup>is bounded by the maximal order of an`-cyclic subgroup ofGL_2g(Z/`Z)i.e.

`<sup>n`</sup> ≤O(g).

I If p>2, then

n_p≤inf<sub>`6=2,p</sub>v<sub>p</sub>(|GL<sub>2g</sub>(Z/`Z)|) =a+ [a/p] +..., where a= [_p−1^2g ].

This gives an exponential type bound in g for

|Aut_kC_s|. This justifies our interest in looking at Stichtenoth ([St 73]) and Singh ([Si 73]).

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Theorem

([Ra 90]). Let Y_K →X_K be a Galois cover with group G. Let us assume that:

I G is nilpotent.

I X_K has a smooth model X .

I The Zariski closure B of the branch locus B_K in X is étale over R_K.

Then,

I the special fiber of the stable model Y_K is tree-like, i.e. the Jacobian of Y_K has potentially good reduction.

I Raynaud’s proof is qualitative and it seems difficult to give a constructive one in the simplest cases.

I We have given in [Le-Ma3] such a proof in the case of p-cyclic covers of the projective line.

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Theorem

([Ra 90]). Let Y_K →X_K be a Galois cover with group G. Let us assume that:

I G is nilpotent.

I X_K has a smooth model X .

I The Zariski closure B of the branch locus B_K in X is étale over R_K.

Then,

I the special fiber of the stable model Y_K is tree-like, i.e. the Jacobian of Y_K has potentially good reduction.

I Raynaud’s proof is qualitative and it seems difficult to give a constructive one in the simplest cases.

I We have given in [Le-Ma3] such a proof in the case of p-cyclic covers of the projective line.

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Theorem

([Ra 90]). Let Y_K →X_K be a Galois cover with group G. Let us assume that:

I G is nilpotent.

I X_K has a smooth model X .

I The Zariski closure B of the branch locus B_K in X is étale over R_K.

Then,

I the special fiber of the stable model Y_K is tree-like, i.e. the Jacobian of Y_K has potentially good reduction.

I Raynaud’s proof is qualitative and it seems difficult to give a constructive one in the simplest cases.

I We have given in [Le-Ma3] such a proof in the case of p-cyclic covers of the projective line.

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Theorem

([Ra 90]). Let Y_K →X_K be a Galois cover with group G. Let us assume that:

I G is nilpotent.

I X_K has a smooth model X .

I The Zariski closure B of the branch locus B_K in X is étale over R_K.

Then,

I the special fiber of the stable model Y_K is tree-like, i.e. the Jacobian of Y_K has potentially good reduction.

I Raynaud’s proof is qualitative and it seems difficult to give a constructive one in the simplest cases.

I We have given in [Le-Ma3] such a proof in the case of p-cyclic covers of the projective line.

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Theorem

([Ra 90]). Let Y_K →X_K be a Galois cover with group G. Let us assume that:

I G is nilpotent.

I X_K has a smooth model X .

I The Zariski closure B of the branch locus B_K in X is étale over R_K.

Then,

I the special fiber of the stable model Y_K is tree-like, i.e. the Jacobian of Y_K has potentially good reduction.

I Raynaud’s proof is qualitative and it seems difficult to give a constructive one in the simplest cases.

I We have given in [Le-Ma3] such a proof in the case of p-cyclic covers of the projective line.

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Theorem

([Ra 90]). Let Y_K →X_K be a Galois cover with group G. Let us assume that:

I G is nilpotent.

I X_K has a smooth model X .

I The Zariski closure B of the branch locus B_K in X is étale over R_K.

Then,

I the special fiber of the stable model Y_K is tree-like, i.e. the Jacobian of Y_K has potentially good reduction.

I Raynaud’s proof is qualitative and it seems difficult to give a constructive one in the simplest cases.

I We have given in [Le-Ma3] such a proof in the case of p-cyclic covers of the projective line.

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

p-cyclic covers of the affine line

k is an algebraically closed of char. p>0.

I f(X)∈Xk[X]monic,deg f =m>1 prime to p, monic.

I C_f :W^p−W =f(X). Let∞be the point of C_f above X =∞and z a local parameter. Then, g :=g(Cf) = ^p−1₂ (m−1)>0.

I G_∞(f) :={σ ∈Aut_kC_f |σ(∞) =∞}.

I G_∞,1(f) :={σ ∈Aut_kC_f |v_∞(σ(z)−z)≥2}, the p-Sylow.

I ([St 73]) Let g(C_f)≥2, then G_∞,1(f)is a p-Sylow ofAutkC_f.

I It is normal except for f(X) =X^mwhere m|1+p.

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

p-cyclic covers of the affine line

k is an algebraically closed of char. p>0.

I f(X)∈Xk[X]monic,deg f =m>1 prime to p, monic.

I C_f :W^p−W =f(X). Let∞be the point of C_f above X =∞and z a local parameter. Then, g :=g(Cf) = ^p−1₂ (m−1)>0.

I G_∞(f) :={σ ∈Aut_kC_f |σ(∞) =∞}.

I G_∞,1(f) :={σ ∈Aut_kC_f |v_∞(σ(z)−z)≥2}, the p-Sylow.

I ([St 73]) Let g(C_f)≥2, then G_∞,1(f)is a p-Sylow ofAutkC_f.

I It is normal except for f(X) =X^mwhere m|1+p.

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

p-cyclic covers of the affine line

k is an algebraically closed of char. p>0.

I f(X)∈Xk[X]monic,deg f =m>1 prime to p, monic.

I C_f :W^p−W =f(X). Let∞be the point of C_f above X =∞and z a local parameter. Then, g :=g(Cf) = ^p−1₂ (m−1)>0.

I G_∞(f) :={σ ∈Aut_kC_f |σ(∞) =∞}.

I G_∞,1(f) :={σ ∈Aut_kC_f |v_∞(σ(z)−z)≥2}, the p-Sylow.

I ([St 73]) Let g(C_f)≥2, then G_∞,1(f)is a p-Sylow ofAutkC_f.

I It is normal except for f(X) =X^mwhere m|1+p.

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

p-cyclic covers of the affine line

k is an algebraically closed of char. p>0.

I f(X)∈Xk[X]monic,deg f =m>1 prime to p, monic.

I C_f :W^p−W =f(X). Let∞be the point of C_f above X =∞and z a local parameter. Then, g :=g(Cf) = ^p−1₂ (m−1)>0.

I G_∞(f) :={σ ∈Aut_kC_f |σ(∞) =∞}.

I G_∞,1(f) :={σ ∈Aut_kC_f |v_∞(σ(z)−z)≥2}, the p-Sylow.

I ([St 73]) Let g(C_f)≥2, then G_∞,1(f)is a p-Sylow ofAutkC_f.

I It is normal except for f(X) =X^mwhere m|1+p.

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

p-cyclic covers of the affine line

k is an algebraically closed of char. p>0.

I f(X)∈Xk[X]monic,deg f =m>1 prime to p, monic.

I C_f :W^p−W =f(X). Let∞be the point of C_f above X =∞and z a local parameter. Then, g :=g(Cf) = ^p−1₂ (m−1)>0.

I G_∞(f) :={σ ∈Aut_kC_f |σ(∞) =∞}.

I G_∞,1(f) :={σ ∈Aut_kC_f |v_∞(σ(z)−z)≥2}, the p-Sylow.

I ([St 73]) Let g(C_f)≥2, then G_∞,1(f)is a p-Sylow ofAutkC_f.

I It is normal except for f(X) =X^mwhere m|1+p.

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

p-cyclic covers of the affine line

k is an algebraically closed of char. p>0.

I f(X)∈Xk[X]monic,deg f =m>1 prime to p, monic.

I C_f :W^p−W =f(X). Let∞be the point of C_f above X =∞and z a local parameter. Then, g :=g(Cf) = ^p−1₂ (m−1)>0.

I G_∞(f) :={σ ∈Aut_kC_f |σ(∞) =∞}.

I G_∞,1(f) :={σ ∈Aut_kC_f |v_∞(σ(z)−z)≥2}, the p-Sylow.

I ([St 73]) Let g(C_f)≥2, then G_∞,1(f)is a p-Sylow ofAutkC_f.

I It is normal except for f(X) =X^mwhere m|1+p.

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Structure of G

(f )

I Letρ(X) =X ,ρ(W) =W +1, then

< ρ >=G_∞,2⊂Z(G_∞,1)

I 0→< ρ >→G_∞,1→V →0, V ={τy|τy(X) =X+y, y ∈k}.

f(X +y) =f(X) +f(y) + (F −Id)(P(X,y)), P(X,y)∈Xk[X].

V '(Z/pZ)^v as a subgroup of k .

I Letτy(W) :=W +a_y+P(X,y), ay ∈Fp, then [τy, τz] =ρ^(y,z), where: V×V →F^pis an alternating form.

I is non degenerated iff< ρ >=Z(G_∞,1).

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Structure of G

(f )

I Letρ(X) =X ,ρ(W) =W +1, then

< ρ >=G_∞,2⊂Z(G_∞,1)

I 0→< ρ >→G_∞,1→V →0, V ={τy|τy(X) =X+y, y ∈k}.

f(X +y) =f(X) +f(y) + (F −Id)(P(X,y)), P(X,y)∈Xk[X].

V '(Z/pZ)^v as a subgroup of k .

I Letτy(W) :=W +a_y+P(X,y), ay ∈Fp, then [τy, τz] =ρ^(y,z), where: V×V →F^pis an alternating form.

I is non degenerated iff< ρ >=Z(G_∞,1).

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Structure of G

(f )

I Letρ(X) =X ,ρ(W) =W +1, then

< ρ >=G_∞,2⊂Z(G_∞,1)

I 0→< ρ >→G_∞,1→V →0, V ={τy|τy(X) =X+y, y ∈k}.

f(X +y) =f(X) +f(y) + (F −Id)(P(X,y)), P(X,y)∈Xk[X].

V '(Z/pZ)^v as a subgroup of k .

I Letτy(W) :=W +a_y+P(X,y), ay ∈Fp, then [τy, τz] =ρ^(y,z), where: V×V →F^pis an alternating form.

I is non degenerated iff< ρ >=Z(G_∞,1).

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Structure of G

(f )

I Letρ(X) =X ,ρ(W) =W +1, then

< ρ >=G_∞,2⊂Z(G_∞,1)

I 0→< ρ >→G_∞,1→V →0, V ={τy|τy(X) =X+y, y ∈k}.

f(X +y) =f(X) +f(y) + (F −Id)(P(X,y)), P(X,y)∈Xk[X].

V '(Z/pZ)^v as a subgroup of k .

I Letτy(W) :=W +a_y+P(X,y), ay ∈Fp, then [τy, τz] =ρ^(y,z), where: V×V →F^pis an alternating form.

I is non degenerated iff< ρ >=Z(G_∞,1).

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Bounds for |G

(f )|

Lemma If f(X) =P

1≤i≤mt_iXⁱ ∈k[X]is monic, then:

I ∆(f)(X,Y) :=f(X+Y)−f(X)−f(Y) = R(X,Y) + (F −Id)(P_f(X,Y)),

where R∈L

b^m_pc≤ipⁿ⁽ⁱ⁾<m,(i,p)=1k[Y]X^ipn(i) and P_f ∈Xk[X,Y].

I P_f = (Id+F +...+Fⁿ⁻¹)(∆(f)) mod X^{[m−1p ]+1}.

I Let us denote byAdf(Y)the content of R(X,Y)∈k[Y][X].

I Ad_f(Y)is an additive and separable polynomial.

I Z(Adf(Y))'V .

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Bounds for |G

(f )|

Lemma If f(X) =P

1≤i≤mt_iXⁱ ∈k[X]is monic, then:

I ∆(f)(X,Y) :=f(X+Y)−f(X)−f(Y) = R(X,Y) + (F −Id)(P_f(X,Y)),

where R∈L

b^m_pc≤ipⁿ⁽ⁱ⁾<m,(i,p)=1k[Y]X^ipn(i) and P_f ∈Xk[X,Y].

I P_f = (Id+F +...+Fⁿ⁻¹)(∆(f)) mod X^{[m−1p ]+1}.

I Let us denote byAdf(Y)the content of R(X,Y)∈k[Y][X].

I Ad_f(Y)is an additive and separable polynomial.

I Z(Adf(Y))'V .

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Bounds for |G

(f )|

Lemma If f(X) =P

1≤i≤mt_iXⁱ ∈k[X]is monic, then:

I ∆(f)(X,Y) :=f(X+Y)−f(X)−f(Y) = R(X,Y) + (F −Id)(P_f(X,Y)),

where R∈L

b^m_pc≤ipⁿ⁽ⁱ⁾<m,(i,p)=1k[Y]X^ipn(i) and P_f ∈Xk[X,Y].

I P_f = (Id+F +...+Fⁿ⁻¹)(∆(f)) mod X^{[m−1p ]+1}.

I Let us denote byAdf(Y)the content of R(X,Y)∈k[Y][X].

I Ad_f(Y)is an additive and separable polynomial.

I Z(Adf(Y))'V .

Automorphisms C. LEHR, M. MATIGNON

Plan Introduction

Monodromy and automorphism groups Automorphism groups

Covers of the affine line Structure of G_∞,1(f) Bounds for|G∞,1(f)|

Characterization of G_∞,1(f) Actions of p-groups

Nakajima condition More about G₂ Riemann surfaces Ray class fields Big actions Maximal curves Monodromy

Marked stable model Potentially good reduction Genus 2

Références

Bounds for |G

(f )|

Lemma If f(X) =P

1≤i≤mt_iXⁱ ∈k[X]is monic, then:

I ∆(f)(X,Y) :=f(X+Y)−f(X)−f(Y) = R(X,Y) + (F −Id)(P_f(X,Y)),

where R∈L

b^m_pc≤ipⁿ⁽ⁱ⁾<m,(i,p)=1k[Y]X^ipn(i) and P_f ∈Xk[X,Y].

I P_f = (Id+F +...+Fⁿ⁻¹)(∆(f)) mod X^{[m−1p ]+1}.

I Let us denote byAdf(Y)the content of R(X,Y)∈k[Y][X].

I Ad_f(Y)is an additive and separable polynomial.

I Z(Adf(Y))'V .