Automorphisms and monodromy - u-bordeaux.frmmatigno/chuoslides.pdfAutomorphisms C. LEHR, M. MATIGNON...
Transcript of Automorphisms and monodromy - u-bordeaux.frmmatigno/chuoslides.pdfAutomorphisms C. LEHR, M. MATIGNON...
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Automorphisms and monodromy
C. LEHR 1 M. MATIGNON2
1MPI Bonn 2Laboratoire A2X Bordeaux
Workshop Chuo University April 2006
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
PlanIntroduction
Monodromy and automorphism groupsAutomorphism groups of curves in char.p > 0
p -cyclic covers of the affine lineStructure 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) ≥ 2Big actions (I)Condition (N) and G2
Riemann surfacesRay class fieldsBig actions (II)Maximal curves
Monodromy polynomialMarked stable modelPotentially good reductionGenus 2
References
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
PlanIntroduction
Monodromy and automorphism groupsAutomorphism groups of curves in char.p > 0
p -cyclic covers of the affine lineStructure 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) ≥ 2Big actions (I)Condition (N) and G2
Riemann surfacesRay class fieldsBig actions (II)Maximal curves
Monodromy polynomialMarked stable modelPotentially good reductionGenus 2
References
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
PlanIntroduction
Monodromy and automorphism groupsAutomorphism groups of curves in char.p > 0
p -cyclic covers of the affine lineStructure 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) ≥ 2Big actions (I)Condition (N) and G2
Riemann surfacesRay class fieldsBig actions (II)Maximal curves
Monodromy polynomialMarked stable modelPotentially good reductionGenus 2
References
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
PlanIntroduction
Monodromy and automorphism groupsAutomorphism groups of curves in char.p > 0
p -cyclic covers of the affine lineStructure 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) ≥ 2Big actions (I)Condition (N) and G2
Riemann surfacesRay class fieldsBig actions (II)Maximal curves
Monodromy polynomialMarked stable modelPotentially good reductionGenus 2
References
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
PlanIntroduction
Monodromy and automorphism groupsAutomorphism groups of curves in char.p > 0
p -cyclic covers of the affine lineStructure 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) ≥ 2Big actions (I)Condition (N) and G2
Riemann surfacesRay class fieldsBig actions (II)Maximal curves
Monodromy polynomialMarked stable modelPotentially good reductionGenus 2
References
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Monodromy and automorphism groups.
I R is a strictly henselian DVR of inequalcharacteristic (0, p).K := FrR ; for example K/Qur
p finite.π a uniformizing parameter.k := RK /πRK .C/K smooth projective curve, g(C) ≥ 1.
I C has potentially good reduction over K if there isL/K (finite) such that C ×K L has a smooth modelover RL. Then:
I There is a minimal extension L/K with thisproperty ; it is Galois and called the monodromyextension.
I Gal(L/K ) is the monodromy group.I Its p-Sylow subgroup is the wild monodromy
group .
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Monodromy and automorphism groups.
I R is a strictly henselian DVR of inequalcharacteristic (0, p).K := FrR ; for example K/Qur
p finite.π a uniformizing parameter.k := RK /πRK .C/K smooth projective curve, g(C) ≥ 1.
I C has potentially good reduction over K if there isL/K (finite) such that C ×K L has a smooth modelover RL. Then:
I There is a minimal extension L/K with thisproperty ; it is Galois and called the monodromyextension.
I Gal(L/K ) is the monodromy group.I Its p-Sylow subgroup is the wild monodromy
group .
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Monodromy and automorphism groups.
I R is a strictly henselian DVR of inequalcharacteristic (0, p).K := FrR ; for example K/Qur
p finite.π a uniformizing parameter.k := RK /πRK .C/K smooth projective curve, g(C) ≥ 1.
I C has potentially good reduction over K if there isL/K (finite) such that C ×K L has a smooth modelover RL. Then:
I There is a minimal extension L/K with thisproperty ; it is Galois and called the monodromyextension.
I Gal(L/K ) is the monodromy group.I Its p-Sylow subgroup is the wild monodromy
group .
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Monodromy and automorphism groups.
I R is a strictly henselian DVR of inequalcharacteristic (0, p).K := FrR ; for example K/Qur
p finite.π a uniformizing parameter.k := RK /πRK .C/K smooth projective curve, g(C) ≥ 1.
I C has potentially good reduction over K if there isL/K (finite) such that C ×K L has a smooth modelover RL. Then:
I There is a minimal extension L/K with thisproperty ; it is Galois and called the monodromyextension.
I Gal(L/K ) is the monodromy group.I Its p-Sylow subgroup is the wild monodromy
group .
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Monodromy and automorphism groups.
I R is a strictly henselian DVR of inequalcharacteristic (0, p).K := FrR ; for example K/Qur
p finite.π a uniformizing parameter.k := RK /πRK .C/K smooth projective curve, g(C) ≥ 1.
I C has potentially good reduction over K if there isL/K (finite) such that C ×K L has a smooth modelover RL. Then:
I There is a minimal extension L/K with thisproperty ; it is Galois and called the monodromyextension.
I Gal(L/K ) is the monodromy group.I Its p-Sylow subgroup is the wild monodromy
group .
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I The base change C ×K K alg induces anhomomorphism ϕ : Gal(K alg/K ) → Autk Cs, whereCs is the special fiber of the smooth model overRL and L = (K alg)ker ϕ.
I Let ` be a prime number, then,n` := v`(|Gal(L/K )|) ≤ v`(|Autk Cs|).
I If ` /∈ {2, p}, then `n` is bounded by the maximalorder of an `-cyclic subgroup of GL2g(Z/`Z) i.e.`n` ≤ O(g).
I If p > 2, thennp ≤ inf 6̀=2,p vp(|GL2g(Z/`Z)|) = a + [a/p] + ...,where a = [ 2g
p−1 ].This gives an exponential type bound in g for|Autk Cs|. This justifies our interest in looking atStichtenoth ([St 73]) and Singh ([Si 73]).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I The base change C ×K K alg induces anhomomorphism ϕ : Gal(K alg/K ) → Autk Cs, whereCs is the special fiber of the smooth model overRL and L = (K alg)ker ϕ.
I Let ` be a prime number, then,n` := v`(|Gal(L/K )|) ≤ v`(|Autk Cs|).
I If ` /∈ {2, p}, then `n` is bounded by the maximalorder of an `-cyclic subgroup of GL2g(Z/`Z) i.e.`n` ≤ O(g).
I If p > 2, thennp ≤ inf 6̀=2,p vp(|GL2g(Z/`Z)|) = a + [a/p] + ...,where a = [ 2g
p−1 ].This gives an exponential type bound in g for|Autk Cs|. This justifies our interest in looking atStichtenoth ([St 73]) and Singh ([Si 73]).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I The base change C ×K K alg induces anhomomorphism ϕ : Gal(K alg/K ) → Autk Cs, whereCs is the special fiber of the smooth model overRL and L = (K alg)ker ϕ.
I Let ` be a prime number, then,n` := v`(|Gal(L/K )|) ≤ v`(|Autk Cs|).
I If ` /∈ {2, p}, then `n` is bounded by the maximalorder of an `-cyclic subgroup of GL2g(Z/`Z) i.e.`n` ≤ O(g).
I If p > 2, thennp ≤ inf 6̀=2,p vp(|GL2g(Z/`Z)|) = a + [a/p] + ...,where a = [ 2g
p−1 ].This gives an exponential type bound in g for|Autk Cs|. This justifies our interest in looking atStichtenoth ([St 73]) and Singh ([Si 73]).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I The base change C ×K K alg induces anhomomorphism ϕ : Gal(K alg/K ) → Autk Cs, whereCs is the special fiber of the smooth model overRL and L = (K alg)ker ϕ.
I Let ` be a prime number, then,n` := v`(|Gal(L/K )|) ≤ v`(|Autk Cs|).
I If ` /∈ {2, p}, then `n` is bounded by the maximalorder of an `-cyclic subgroup of GL2g(Z/`Z) i.e.`n` ≤ O(g).
I If p > 2, thennp ≤ inf 6̀=2,p vp(|GL2g(Z/`Z)|) = a + [a/p] + ...,where a = [ 2g
p−1 ].This gives an exponential type bound in g for|Autk Cs|. This justifies our interest in looking atStichtenoth ([St 73]) and Singh ([Si 73]).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Theorem([Ra 90]). Let YK → XK be a Galois cover with groupG. Let us assume that:
I G is nilpotent.I XK has a smooth model X.I The Zariski closure B of the branch locus BK in X
is étale over RK .
Then,
I the special fiber of the stable model YK istree-like, i.e. the Jacobian of YK has potentiallygood reduction.
I Raynaud’s proof is qualitative and it seems difficultto give a constructive one in the simplest cases.
I We have given in [Le-Ma3] such a proof in thecase of p-cyclic covers of the projective line.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Theorem([Ra 90]). Let YK → XK be a Galois cover with groupG. Let us assume that:
I G is nilpotent.I XK has a smooth model X.I The Zariski closure B of the branch locus BK in X
is étale over RK .
Then,
I the special fiber of the stable model YK istree-like, i.e. the Jacobian of YK has potentiallygood reduction.
I Raynaud’s proof is qualitative and it seems difficultto give a constructive one in the simplest cases.
I We have given in [Le-Ma3] such a proof in thecase of p-cyclic covers of the projective line.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Theorem([Ra 90]). Let YK → XK be a Galois cover with groupG. Let us assume that:
I G is nilpotent.I XK has a smooth model X.I The Zariski closure B of the branch locus BK in X
is étale over RK .
Then,
I the special fiber of the stable model YK istree-like, i.e. the Jacobian of YK has potentiallygood reduction.
I Raynaud’s proof is qualitative and it seems difficultto give a constructive one in the simplest cases.
I We have given in [Le-Ma3] such a proof in thecase of p-cyclic covers of the projective line.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Theorem([Ra 90]). Let YK → XK be a Galois cover with groupG. Let us assume that:
I G is nilpotent.I XK has a smooth model X.I The Zariski closure B of the branch locus BK in X
is étale over RK .
Then,
I the special fiber of the stable model YK istree-like, i.e. the Jacobian of YK has potentiallygood reduction.
I Raynaud’s proof is qualitative and it seems difficultto give a constructive one in the simplest cases.
I We have given in [Le-Ma3] such a proof in thecase of p-cyclic covers of the projective line.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Theorem([Ra 90]). Let YK → XK be a Galois cover with groupG. Let us assume that:
I G is nilpotent.I XK has a smooth model X.I The Zariski closure B of the branch locus BK in X
is étale over RK .
Then,
I the special fiber of the stable model YK istree-like, i.e. the Jacobian of YK has potentiallygood reduction.
I Raynaud’s proof is qualitative and it seems difficultto give a constructive one in the simplest cases.
I We have given in [Le-Ma3] such a proof in thecase of p-cyclic covers of the projective line.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Theorem([Ra 90]). Let YK → XK be a Galois cover with groupG. Let us assume that:
I G is nilpotent.I XK has a smooth model X.I The Zariski closure B of the branch locus BK in X
is étale over RK .
Then,
I the special fiber of the stable model YK istree-like, i.e. the Jacobian of YK has potentiallygood reduction.
I Raynaud’s proof is qualitative and it seems difficultto give a constructive one in the simplest cases.
I We have given in [Le-Ma3] such a proof in thecase of p-cyclic covers of the projective line.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 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 Cf : W p − W = f (X ). Let ∞ be the point of Cf
above X = ∞ and z a local parameter. Then,g := g(Cf ) = p−1
2 (m − 1) > 0.I G∞(f ) := {σ ∈ Autk Cf | σ(∞) = ∞}.I G∞,1(f ) := {σ ∈ Autk Cf | v∞(σ(z) − z) ≥ 2} , the
p-Sylow.I ([St 73]) Let g(Cf ) ≥ 2, then G∞,1(f ) is a p-Sylow
of Autk Cf .I It is normal except for f (X ) = X m where m|1 + p.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 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 Cf : W p − W = f (X ). Let ∞ be the point of Cf
above X = ∞ and z a local parameter. Then,g := g(Cf ) = p−1
2 (m − 1) > 0.I G∞(f ) := {σ ∈ Autk Cf | σ(∞) = ∞}.I G∞,1(f ) := {σ ∈ Autk Cf | v∞(σ(z) − z) ≥ 2} , the
p-Sylow.I ([St 73]) Let g(Cf ) ≥ 2, then G∞,1(f ) is a p-Sylow
of Autk Cf .I It is normal except for f (X ) = X m where m|1 + p.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 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 Cf : W p − W = f (X ). Let ∞ be the point of Cf
above X = ∞ and z a local parameter. Then,g := g(Cf ) = p−1
2 (m − 1) > 0.I G∞(f ) := {σ ∈ Autk Cf | σ(∞) = ∞}.I G∞,1(f ) := {σ ∈ Autk Cf | v∞(σ(z) − z) ≥ 2} , the
p-Sylow.I ([St 73]) Let g(Cf ) ≥ 2, then G∞,1(f ) is a p-Sylow
of Autk Cf .I It is normal except for f (X ) = X m where m|1 + p.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 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 Cf : W p − W = f (X ). Let ∞ be the point of Cf
above X = ∞ and z a local parameter. Then,g := g(Cf ) = p−1
2 (m − 1) > 0.I G∞(f ) := {σ ∈ Autk Cf | σ(∞) = ∞}.I G∞,1(f ) := {σ ∈ Autk Cf | v∞(σ(z) − z) ≥ 2} , the
p-Sylow.I ([St 73]) Let g(Cf ) ≥ 2, then G∞,1(f ) is a p-Sylow
of Autk Cf .I It is normal except for f (X ) = X m where m|1 + p.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 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 Cf : W p − W = f (X ). Let ∞ be the point of Cf
above X = ∞ and z a local parameter. Then,g := g(Cf ) = p−1
2 (m − 1) > 0.I G∞(f ) := {σ ∈ Autk Cf | σ(∞) = ∞}.I G∞,1(f ) := {σ ∈ Autk Cf | v∞(σ(z) − z) ≥ 2} , the
p-Sylow.I ([St 73]) Let g(Cf ) ≥ 2, then G∞,1(f ) is a p-Sylow
of Autk Cf .I It is normal except for f (X ) = X m where m|1 + p.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 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 Cf : W p − W = f (X ). Let ∞ be the point of Cf
above X = ∞ and z a local parameter. Then,g := g(Cf ) = p−1
2 (m − 1) > 0.I G∞(f ) := {σ ∈ Autk Cf | σ(∞) = ∞}.I G∞,1(f ) := {σ ∈ Autk Cf | v∞(σ(z) − z) ≥ 2} , the
p-Sylow.I ([St 73]) Let g(Cf ) ≥ 2, then G∞,1(f ) is a p-Sylow
of Autk Cf .I It is normal except for f (X ) = X m where m|1 + p.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Structure of G∞,1(f )
I Let ρ(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 Let τy(W ) := W + ay + P(X , y), ay ∈ Fp, then[τy , τz ] = ρε(y ,z), where ε : V × V → Fp is analternating form.
I ε is non degenerated iff < ρ >= Z (G∞,1).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Structure of G∞,1(f )
I Let ρ(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 Let τy(W ) := W + ay + P(X , y), ay ∈ Fp, then[τy , τz ] = ρε(y ,z), where ε : V × V → Fp is analternating form.
I ε is non degenerated iff < ρ >= Z (G∞,1).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Structure of G∞,1(f )
I Let ρ(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 Let τy(W ) := W + ay + P(X , y), ay ∈ Fp, then[τy , τz ] = ρε(y ,z), where ε : V × V → Fp is analternating form.
I ε is non degenerated iff < ρ >= Z (G∞,1).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Structure of G∞,1(f )
I Let ρ(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 Let τy(W ) := W + ay + P(X , y), ay ∈ Fp, then[τy , τz ] = ρε(y ,z), where ε : V × V → Fp is analternating form.
I ε is non degenerated iff < ρ >= Z (G∞,1).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Bounds for |G∞,1(f )|
LemmaIf f (X ) =
∑
1≤i≤m tiX i ∈ k [X ] is monic, then:I ∆(f )(X , Y ) := f (X + Y ) − f (X ) − f (Y ) =
R(X , Y ) + (F − Id)(Pf (X , Y )),where R ∈
⊕
bmp c≤ipn(i)<m, (i ,p)=1 k [Y ]X ipn(i)
and
Pf ∈ Xk [X , Y ].
I Pf = (Id + F + ... + F n−1)(∆(f )) mod X [ m−1p ]+1.
I Let us denote by Adf (Y ) the content ofR(X , Y ) ∈ k [Y ][X ].
I Adf (Y ) is an additive and separable polynomial.I Z (Adf (Y )) ' V.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Bounds for |G∞,1(f )|
LemmaIf f (X ) =
∑
1≤i≤m tiX i ∈ k [X ] is monic, then:I ∆(f )(X , Y ) := f (X + Y ) − f (X ) − f (Y ) =
R(X , Y ) + (F − Id)(Pf (X , Y )),where R ∈
⊕
bmp c≤ipn(i)<m, (i ,p)=1 k [Y ]X ipn(i)
and
Pf ∈ Xk [X , Y ].
I Pf = (Id + F + ... + F n−1)(∆(f )) mod X [ m−1p ]+1.
I Let us denote by Adf (Y ) the content ofR(X , Y ) ∈ k [Y ][X ].
I Adf (Y ) is an additive and separable polynomial.I Z (Adf (Y )) ' V.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Bounds for |G∞,1(f )|
LemmaIf f (X ) =
∑
1≤i≤m tiX i ∈ k [X ] is monic, then:I ∆(f )(X , Y ) := f (X + Y ) − f (X ) − f (Y ) =
R(X , Y ) + (F − Id)(Pf (X , Y )),where R ∈
⊕
bmp c≤ipn(i)<m, (i ,p)=1 k [Y ]X ipn(i)
and
Pf ∈ Xk [X , Y ].
I Pf = (Id + F + ... + F n−1)(∆(f )) mod X [ m−1p ]+1.
I Let us denote by Adf (Y ) the content ofR(X , Y ) ∈ k [Y ][X ].
I Adf (Y ) is an additive and separable polynomial.I Z (Adf (Y )) ' V.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Bounds for |G∞,1(f )|
LemmaIf f (X ) =
∑
1≤i≤m tiX i ∈ k [X ] is monic, then:I ∆(f )(X , Y ) := f (X + Y ) − f (X ) − f (Y ) =
R(X , Y ) + (F − Id)(Pf (X , Y )),where R ∈
⊕
bmp c≤ipn(i)<m, (i ,p)=1 k [Y ]X ipn(i)
and
Pf ∈ Xk [X , Y ].
I Pf = (Id + F + ... + F n−1)(∆(f )) mod X [ m−1p ]+1.
I Let us denote by Adf (Y ) the content ofR(X , Y ) ∈ k [Y ][X ].
I Adf (Y ) is an additive and separable polynomial.I Z (Adf (Y )) ' V.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Bounds for |G∞,1(f )|
LemmaIf f (X ) =
∑
1≤i≤m tiX i ∈ k [X ] is monic, then:I ∆(f )(X , Y ) := f (X + Y ) − f (X ) − f (Y ) =
R(X , Y ) + (F − Id)(Pf (X , Y )),where R ∈
⊕
bmp c≤ipn(i)<m, (i ,p)=1 k [Y ]X ipn(i)
and
Pf ∈ Xk [X , Y ].
I Pf = (Id + F + ... + F n−1)(∆(f )) mod X [ m−1p ]+1.
I Let us denote by Adf (Y ) the content ofR(X , Y ) ∈ k [Y ][X ].
I Adf (Y ) is an additive and separable polynomial.I Z (Adf (Y )) ' V.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Let m − 1 = `ps with (`, p) = 1.I ([St 73]) |G∞,1| = p deg Adf ≤ p(m − 1)2, i.e.
|G∞,1|
g2 ≤ 4p(p−1)2 .
I ([St 73]) s = 0 i.e. (m − 1, p) = 1, then |G∞,1| = p.I If s > 0,
I ` > 1, p = 2, then |G∞,1|g ≤ 2
3 .
I ` > 1, p > 2, then |G∞,1|g ≤ p
p−1 .
I ([St 73]) ` > 1, m = 1 + ps, then |G∞,1|g ≤ 2ps p
p−1
(with equality for f (X ) = X 1+ps).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Let m − 1 = `ps with (`, p) = 1.I ([St 73]) |G∞,1| = p deg Adf ≤ p(m − 1)2, i.e.
|G∞,1|
g2 ≤ 4p(p−1)2 .
I ([St 73]) s = 0 i.e. (m − 1, p) = 1, then |G∞,1| = p.I If s > 0,
I ` > 1, p = 2, then |G∞,1|g ≤ 2
3 .
I ` > 1, p > 2, then |G∞,1|g ≤ p
p−1 .
I ([St 73]) ` > 1, m = 1 + ps, then |G∞,1|g ≤ 2ps p
p−1
(with equality for f (X ) = X 1+ps).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Let m − 1 = `ps with (`, p) = 1.I ([St 73]) |G∞,1| = p deg Adf ≤ p(m − 1)2, i.e.
|G∞,1|
g2 ≤ 4p(p−1)2 .
I ([St 73]) s = 0 i.e. (m − 1, p) = 1, then |G∞,1| = p.I If s > 0,
I ` > 1, p = 2, then |G∞,1|g ≤ 2
3 .
I ` > 1, p > 2, then |G∞,1|g ≤ p
p−1 .
I ([St 73]) ` > 1, m = 1 + ps, then |G∞,1|g ≤ 2ps p
p−1
(with equality for f (X ) = X 1+ps).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Let m − 1 = `ps with (`, p) = 1.I ([St 73]) |G∞,1| = p deg Adf ≤ p(m − 1)2, i.e.
|G∞,1|
g2 ≤ 4p(p−1)2 .
I ([St 73]) s = 0 i.e. (m − 1, p) = 1, then |G∞,1| = p.I If s > 0,
I ` > 1, p = 2, then |G∞,1|g ≤ 2
3 .
I ` > 1, p > 2, then |G∞,1|g ≤ p
p−1 .
I ([St 73]) ` > 1, m = 1 + ps, then |G∞,1|g ≤ 2ps p
p−1
(with equality for f (X ) = X 1+ps).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Let m − 1 = `ps with (`, p) = 1.I ([St 73]) |G∞,1| = p deg Adf ≤ p(m − 1)2, i.e.
|G∞,1|
g2 ≤ 4p(p−1)2 .
I ([St 73]) s = 0 i.e. (m − 1, p) = 1, then |G∞,1| = p.I If s > 0,
I ` > 1, p = 2, then |G∞,1|g ≤ 2
3 .
I ` > 1, p > 2, then |G∞,1|g ≤ p
p−1 .
I ([St 73]) ` > 1, m = 1 + ps, then |G∞,1|g ≤ 2ps p
p−1
(with equality for f (X ) = X 1+ps).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Let m − 1 = `ps with (`, p) = 1.I ([St 73]) |G∞,1| = p deg Adf ≤ p(m − 1)2, i.e.
|G∞,1|
g2 ≤ 4p(p−1)2 .
I ([St 73]) s = 0 i.e. (m − 1, p) = 1, then |G∞,1| = p.I If s > 0,
I ` > 1, p = 2, then |G∞,1|g ≤ 2
3 .
I ` > 1, p > 2, then |G∞,1|g ≤ p
p−1 .
I ([St 73]) ` > 1, m = 1 + ps, then |G∞,1|g ≤ 2ps p
p−1
(with equality for f (X ) = X 1+ps).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Characterization of G∞,1(f )
I We consider the extensions of type0 → N ' Z/pZ → G → (Z/pZ)n → 0 (note thatG∞,1(f ) is an extension of this type). ThenG′ ⊂ N ⊂ Z (G).
I If G′ = Z (G), G is called extraspecial.I Then, |G| = p2s+1 and there are 2 isomorphism
classes for a given s.I If p > 2, we denote by E(p3) (resp. M(p3)) the
non abelian group of order p3 and exponent p(resp. p2). Then, G ' E(p3) ∗E(p3) ∗ ... ∗ E(p3) orM(p3) ∗ E(p3) ∗ ... ∗ E(p3), according as theexponent is p or p2.
I If p = 2, then G ' D8 ∗ D8 ∗ ... ∗ D8 orQ8 ∗ D8 ∗ ... ∗ D8 (in both cases, the exponent is22).
I If G′ ⊂ Z (G), G is a subgroup of an extraspecialgroup E with Z (E) = N ⊂ G.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Characterization of G∞,1(f )
I We consider the extensions of type0 → N ' Z/pZ → G → (Z/pZ)n → 0 (note thatG∞,1(f ) is an extension of this type). ThenG′ ⊂ N ⊂ Z (G).
I If G′ = Z (G), G is called extraspecial.I Then, |G| = p2s+1 and there are 2 isomorphism
classes for a given s.I If p > 2, we denote by E(p3) (resp. M(p3)) the
non abelian group of order p3 and exponent p(resp. p2). Then, G ' E(p3) ∗E(p3) ∗ ... ∗ E(p3) orM(p3) ∗ E(p3) ∗ ... ∗ E(p3), according as theexponent is p or p2.
I If p = 2, then G ' D8 ∗ D8 ∗ ... ∗ D8 orQ8 ∗ D8 ∗ ... ∗ D8 (in both cases, the exponent is22).
I If G′ ⊂ Z (G), G is a subgroup of an extraspecialgroup E with Z (E) = N ⊂ G.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Characterization of G∞,1(f )
I We consider the extensions of type0 → N ' Z/pZ → G → (Z/pZ)n → 0 (note thatG∞,1(f ) is an extension of this type). ThenG′ ⊂ N ⊂ Z (G).
I If G′ = Z (G), G is called extraspecial.I Then, |G| = p2s+1 and there are 2 isomorphism
classes for a given s.I If p > 2, we denote by E(p3) (resp. M(p3)) the
non abelian group of order p3 and exponent p(resp. p2). Then, G ' E(p3) ∗E(p3) ∗ ... ∗ E(p3) orM(p3) ∗ E(p3) ∗ ... ∗ E(p3), according as theexponent is p or p2.
I If p = 2, then G ' D8 ∗ D8 ∗ ... ∗ D8 orQ8 ∗ D8 ∗ ... ∗ D8 (in both cases, the exponent is22).
I If G′ ⊂ Z (G), G is a subgroup of an extraspecialgroup E with Z (E) = N ⊂ G.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Characterization of G∞,1(f )
I We consider the extensions of type0 → N ' Z/pZ → G → (Z/pZ)n → 0 (note thatG∞,1(f ) is an extension of this type). ThenG′ ⊂ N ⊂ Z (G).
I If G′ = Z (G), G is called extraspecial.I Then, |G| = p2s+1 and there are 2 isomorphism
classes for a given s.I If p > 2, we denote by E(p3) (resp. M(p3)) the
non abelian group of order p3 and exponent p(resp. p2). Then, G ' E(p3) ∗E(p3) ∗ ... ∗ E(p3) orM(p3) ∗ E(p3) ∗ ... ∗ E(p3), according as theexponent is p or p2.
I If p = 2, then G ' D8 ∗ D8 ∗ ... ∗ D8 orQ8 ∗ D8 ∗ ... ∗ D8 (in both cases, the exponent is22).
I If G′ ⊂ Z (G), G is a subgroup of an extraspecialgroup E with Z (E) = N ⊂ G.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Characterization of G∞,1(f )
I We consider the extensions of type0 → N ' Z/pZ → G → (Z/pZ)n → 0 (note thatG∞,1(f ) is an extension of this type). ThenG′ ⊂ N ⊂ Z (G).
I If G′ = Z (G), G is called extraspecial.I Then, |G| = p2s+1 and there are 2 isomorphism
classes for a given s.I If p > 2, we denote by E(p3) (resp. M(p3)) the
non abelian group of order p3 and exponent p(resp. p2). Then, G ' E(p3) ∗E(p3) ∗ ... ∗ E(p3) orM(p3) ∗ E(p3) ∗ ... ∗ E(p3), according as theexponent is p or p2.
I If p = 2, then G ' D8 ∗ D8 ∗ ... ∗ D8 orQ8 ∗ D8 ∗ ... ∗ D8 (in both cases, the exponent is22).
I If G′ ⊂ Z (G), G is a subgroup of an extraspecialgroup E with Z (E) = N ⊂ G.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Characterization of G∞,1(f )
I We consider the extensions of type0 → N ' Z/pZ → G → (Z/pZ)n → 0 (note thatG∞,1(f ) is an extension of this type). ThenG′ ⊂ N ⊂ Z (G).
I If G′ = Z (G), G is called extraspecial.I Then, |G| = p2s+1 and there are 2 isomorphism
classes for a given s.I If p > 2, we denote by E(p3) (resp. M(p3)) the
non abelian group of order p3 and exponent p(resp. p2). Then, G ' E(p3) ∗E(p3) ∗ ... ∗ E(p3) orM(p3) ∗ E(p3) ∗ ... ∗ E(p3), according as theexponent is p or p2.
I If p = 2, then G ' D8 ∗ D8 ∗ ... ∗ D8 orQ8 ∗ D8 ∗ ... ∗ D8 (in both cases, the exponent is22).
I If G′ ⊂ Z (G), G is a subgroup of an extraspecialgroup E with Z (E) = N ⊂ G.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Theorem([Le-Ma 1]). Let f (X ) = XΣ(F )(X ) ∈ Xk [X ],Σ(F ) =
∑
0≤i≤s aiF i ∈ k{F} an additivepolynomial with deg f = 1 + ps. Then,
I Adf (Y ) = F s(∑
0≤i≤s(ai F i + F−iai)(Y )), apalyndromic polynomial.
I G∞,1(f ) is an extraspecial group with cardinalp2s+1 and exponent p for p > 2, and of typeQ8 ∗ D8 ∗ ... ∗ D8 for p = 2.
I Theorem([Le-Ma 1]). If G is an extension of type0 → Z/pZ → G → (Z/pZ)n → 0, there isf ∈ Xk [X ] with G ' G∞,1(f ).
I Sketch proof: Extraspecial groups with exponentp2 are realized by a modification by a Witt cocycleof the polynomial f in the theorem above.
I We can see G as a subgroup of an extraspecialgroup E , then we realize E with fE and a suitablemodification of f E will limit G∞,1(fE ) to G.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Theorem([Le-Ma 1]). Let f (X ) = XΣ(F )(X ) ∈ Xk [X ],Σ(F ) =
∑
0≤i≤s aiF i ∈ k{F} an additivepolynomial with deg f = 1 + ps. Then,
I Adf (Y ) = F s(∑
0≤i≤s(ai F i + F−iai)(Y )), apalyndromic polynomial.
I G∞,1(f ) is an extraspecial group with cardinalp2s+1 and exponent p for p > 2, and of typeQ8 ∗ D8 ∗ ... ∗ D8 for p = 2.
I Theorem([Le-Ma 1]). If G is an extension of type0 → Z/pZ → G → (Z/pZ)n → 0, there isf ∈ Xk [X ] with G ' G∞,1(f ).
I Sketch proof: Extraspecial groups with exponentp2 are realized by a modification by a Witt cocycleof the polynomial f in the theorem above.
I We can see G as a subgroup of an extraspecialgroup E , then we realize E with fE and a suitablemodification of f E will limit G∞,1(fE ) to G.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Theorem([Le-Ma 1]). Let f (X ) = XΣ(F )(X ) ∈ Xk [X ],Σ(F ) =
∑
0≤i≤s aiF i ∈ k{F} an additivepolynomial with deg f = 1 + ps. Then,
I Adf (Y ) = F s(∑
0≤i≤s(ai F i + F−iai)(Y )), apalyndromic polynomial.
I G∞,1(f ) is an extraspecial group with cardinalp2s+1 and exponent p for p > 2, and of typeQ8 ∗ D8 ∗ ... ∗ D8 for p = 2.
I Theorem([Le-Ma 1]). If G is an extension of type0 → Z/pZ → G → (Z/pZ)n → 0, there isf ∈ Xk [X ] with G ' G∞,1(f ).
I Sketch proof: Extraspecial groups with exponentp2 are realized by a modification by a Witt cocycleof the polynomial f in the theorem above.
I We can see G as a subgroup of an extraspecialgroup E , then we realize E with fE and a suitablemodification of f E will limit G∞,1(fE ) to G.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Theorem([Le-Ma 1]). Let f (X ) = XΣ(F )(X ) ∈ Xk [X ],Σ(F ) =
∑
0≤i≤s aiF i ∈ k{F} an additivepolynomial with deg f = 1 + ps. Then,
I Adf (Y ) = F s(∑
0≤i≤s(ai F i + F−iai)(Y )), apalyndromic polynomial.
I G∞,1(f ) is an extraspecial group with cardinalp2s+1 and exponent p for p > 2, and of typeQ8 ∗ D8 ∗ ... ∗ D8 for p = 2.
I Theorem([Le-Ma 1]). If G is an extension of type0 → Z/pZ → G → (Z/pZ)n → 0, there isf ∈ Xk [X ] with G ' G∞,1(f ).
I Sketch proof: Extraspecial groups with exponentp2 are realized by a modification by a Witt cocycleof the polynomial f in the theorem above.
I We can see G as a subgroup of an extraspecialgroup E , then we realize E with fE and a suitablemodification of f E will limit G∞,1(fE ) to G.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Theorem([Le-Ma 1]). Let f (X ) = XΣ(F )(X ) ∈ Xk [X ],Σ(F ) =
∑
0≤i≤s aiF i ∈ k{F} an additivepolynomial with deg f = 1 + ps. Then,
I Adf (Y ) = F s(∑
0≤i≤s(ai F i + F−iai)(Y )), apalyndromic polynomial.
I G∞,1(f ) is an extraspecial group with cardinalp2s+1 and exponent p for p > 2, and of typeQ8 ∗ D8 ∗ ... ∗ D8 for p = 2.
I Theorem([Le-Ma 1]). If G is an extension of type0 → Z/pZ → G → (Z/pZ)n → 0, there isf ∈ Xk [X ] with G ' G∞,1(f ).
I Sketch proof: Extraspecial groups with exponentp2 are realized by a modification by a Witt cocycleof the polynomial f in the theorem above.
I We can see G as a subgroup of an extraspecialgroup E , then we realize E with fE and a suitablemodification of f E will limit G∞,1(fE ) to G.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Theorem([Le-Ma 1]). Let f (X ) = XΣ(F )(X ) ∈ Xk [X ],Σ(F ) =
∑
0≤i≤s aiF i ∈ k{F} an additivepolynomial with deg f = 1 + ps. Then,
I Adf (Y ) = F s(∑
0≤i≤s(ai F i + F−iai)(Y )), apalyndromic polynomial.
I G∞,1(f ) is an extraspecial group with cardinalp2s+1 and exponent p for p > 2, and of typeQ8 ∗ D8 ∗ ... ∗ D8 for p = 2.
I Theorem([Le-Ma 1]). If G is an extension of type0 → Z/pZ → G → (Z/pZ)n → 0, there isf ∈ Xk [X ] with G ' G∞,1(f ).
I Sketch proof: Extraspecial groups with exponentp2 are realized by a modification by a Witt cocycleof the polynomial f in the theorem above.
I We can see G as a subgroup of an extraspecialgroup E , then we realize E with fE and a suitablemodification of f E will limit G∞,1(fE ) to G.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Big actions (I)
I Theorem([Le-Ma 1]). Let f (X ) ∈ Xk [X ] with (deg f , p) = 1.
If |G∞,1|g > p
p−1 (23 for p = 2), then
f (X ) = cX + XΣ(F )(X ) ∈ k [X ].
I Sketch proof: One shows that monomials in f witha degree /∈ 1 + pN will limit the degree of Adf .
I Let (C, G) with G ⊂ Autk C. We say that (C, G) isa big action if G is a p-group and(N) gC > 0 and |G|
gC> 2p
p−1 .
It follows from ([Na 87]) that there is ∞ ∈ C, withI C → C/G ' P1
k −∞ is étale and G = G∞,1.I G∞,2 6= G∞,1 and C/G∞,2 ' P1
kI Then, G∞,1/G∞,2 acts as a group of translations
of the affine line C/G∞,2 − {∞}.I Transfert of condition (N) to quotients. Let
(C, G) a big action, if H C G and if g(C/H) > 0,then (C/H, G/H) is a big action.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Big actions (I)
I Theorem([Le-Ma 1]). Let f (X ) ∈ Xk [X ] with (deg f , p) = 1.
If |G∞,1|g > p
p−1 (23 for p = 2), then
f (X ) = cX + XΣ(F )(X ) ∈ k [X ].
I Sketch proof: One shows that monomials in f witha degree /∈ 1 + pN will limit the degree of Adf .
I Let (C, G) with G ⊂ Autk C. We say that (C, G) isa big action if G is a p-group and(N) gC > 0 and |G|
gC> 2p
p−1 .
It follows from ([Na 87]) that there is ∞ ∈ C, withI C → C/G ' P1
k −∞ is étale and G = G∞,1.I G∞,2 6= G∞,1 and C/G∞,2 ' P1
kI Then, G∞,1/G∞,2 acts as a group of translations
of the affine line C/G∞,2 − {∞}.I Transfert of condition (N) to quotients. Let
(C, G) a big action, if H C G and if g(C/H) > 0,then (C/H, G/H) is a big action.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Big actions (I)
I Theorem([Le-Ma 1]). Let f (X ) ∈ Xk [X ] with (deg f , p) = 1.
If |G∞,1|g > p
p−1 (23 for p = 2), then
f (X ) = cX + XΣ(F )(X ) ∈ k [X ].
I Sketch proof: One shows that monomials in f witha degree /∈ 1 + pN will limit the degree of Adf .
I Let (C, G) with G ⊂ Autk C. We say that (C, G) isa big action if G is a p-group and(N) gC > 0 and |G|
gC> 2p
p−1 .
It follows from ([Na 87]) that there is ∞ ∈ C, withI C → C/G ' P1
k −∞ is étale and G = G∞,1.I G∞,2 6= G∞,1 and C/G∞,2 ' P1
kI Then, G∞,1/G∞,2 acts as a group of translations
of the affine line C/G∞,2 − {∞}.I Transfert of condition (N) to quotients. Let
(C, G) a big action, if H C G and if g(C/H) > 0,then (C/H, G/H) is a big action.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Big actions (I)
I Theorem([Le-Ma 1]). Let f (X ) ∈ Xk [X ] with (deg f , p) = 1.
If |G∞,1|g > p
p−1 (23 for p = 2), then
f (X ) = cX + XΣ(F )(X ) ∈ k [X ].
I Sketch proof: One shows that monomials in f witha degree /∈ 1 + pN will limit the degree of Adf .
I Let (C, G) with G ⊂ Autk C. We say that (C, G) isa big action if G is a p-group and(N) gC > 0 and |G|
gC> 2p
p−1 .
It follows from ([Na 87]) that there is ∞ ∈ C, withI C → C/G ' P1
k −∞ is étale and G = G∞,1.I G∞,2 6= G∞,1 and C/G∞,2 ' P1
kI Then, G∞,1/G∞,2 acts as a group of translations
of the affine line C/G∞,2 − {∞}.I Transfert of condition (N) to quotients. Let
(C, G) a big action, if H C G and if g(C/H) > 0,then (C/H, G/H) is a big action.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Big actions (I)
I Theorem([Le-Ma 1]). Let f (X ) ∈ Xk [X ] with (deg f , p) = 1.
If |G∞,1|g > p
p−1 (23 for p = 2), then
f (X ) = cX + XΣ(F )(X ) ∈ k [X ].
I Sketch proof: One shows that monomials in f witha degree /∈ 1 + pN will limit the degree of Adf .
I Let (C, G) with G ⊂ Autk C. We say that (C, G) isa big action if G is a p-group and(N) gC > 0 and |G|
gC> 2p
p−1 .
It follows from ([Na 87]) that there is ∞ ∈ C, withI C → C/G ' P1
k −∞ is étale and G = G∞,1.I G∞,2 6= G∞,1 and C/G∞,2 ' P1
kI Then, G∞,1/G∞,2 acts as a group of translations
of the affine line C/G∞,2 − {∞}.I Transfert of condition (N) to quotients. Let
(C, G) a big action, if H C G and if g(C/H) > 0,then (C/H, G/H) is a big action.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Big actions (I)
I Theorem([Le-Ma 1]). Let f (X ) ∈ Xk [X ] with (deg f , p) = 1.
If |G∞,1|g > p
p−1 (23 for p = 2), then
f (X ) = cX + XΣ(F )(X ) ∈ k [X ].
I Sketch proof: One shows that monomials in f witha degree /∈ 1 + pN will limit the degree of Adf .
I Let (C, G) with G ⊂ Autk C. We say that (C, G) isa big action if G is a p-group and(N) gC > 0 and |G|
gC> 2p
p−1 .
It follows from ([Na 87]) that there is ∞ ∈ C, withI C → C/G ' P1
k −∞ is étale and G = G∞,1.I G∞,2 6= G∞,1 and C/G∞,2 ' P1
kI Then, G∞,1/G∞,2 acts as a group of translations
of the affine line C/G∞,2 − {∞}.I Transfert of condition (N) to quotients. Let
(C, G) a big action, if H C G and if g(C/H) > 0,then (C/H, G/H) is a big action.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Big actions (I)
I Theorem([Le-Ma 1]). Let f (X ) ∈ Xk [X ] with (deg f , p) = 1.
If |G∞,1|g > p
p−1 (23 for p = 2), then
f (X ) = cX + XΣ(F )(X ) ∈ k [X ].
I Sketch proof: One shows that monomials in f witha degree /∈ 1 + pN will limit the degree of Adf .
I Let (C, G) with G ⊂ Autk C. We say that (C, G) isa big action if G is a p-group and(N) gC > 0 and |G|
gC> 2p
p−1 .
It follows from ([Na 87]) that there is ∞ ∈ C, withI C → C/G ' P1
k −∞ is étale and G = G∞,1.I G∞,2 6= G∞,1 and C/G∞,2 ' P1
kI Then, G∞,1/G∞,2 acts as a group of translations
of the affine line C/G∞,2 − {∞}.I Transfert of condition (N) to quotients. Let
(C, G) a big action, if H C G and if g(C/H) > 0,then (C/H, G/H) is a big action.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Condition (N) and G2
In this section (C, G) is a big action. Let Gi be thelower ramification groups.
I Let H C G and H with index p in G2 (H exists!),then (C/H, G/H) is a big action.
I (G/H)2 = G2/H ' Z/pZ.I There is S(F ) ∈ k{F},
f1 = cX + XΣ(F )(X ) ∈ k [X ] with C/H ' Cf1 .I If G2 ' (Z/pZ)t , then k(C) = k(X , W1, ..., Wt ) and
℘(W1, ..., Wt ) = (f1(X ), f2(X ), ..., ft (X )) ∈ (k [X ])t
I f1(X ), .., ft (X ) are Fp-free mod ℘(k [X ]).I The group extension
0 → G2 → G1 → V = (Z/pZ)v → 0 induces arepresentation ρ : V → Glt (Fp)
I dual to the one given by V acting via translation:(v ∈ V )× (f1(X ), f2(X ), ..., ft (X )) mod ℘(k [X ])t →→ (f1(X + v), f2(X + v), ..., ft (X + v))mod ℘(k [X ])t
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Condition (N) and G2
In this section (C, G) is a big action. Let Gi be thelower ramification groups.
I Let H C G and H with index p in G2 (H exists!),then (C/H, G/H) is a big action.
I (G/H)2 = G2/H ' Z/pZ.I There is S(F ) ∈ k{F},
f1 = cX + XΣ(F )(X ) ∈ k [X ] with C/H ' Cf1 .I If G2 ' (Z/pZ)t , then k(C) = k(X , W1, ..., Wt ) and
℘(W1, ..., Wt ) = (f1(X ), f2(X ), ..., ft (X )) ∈ (k [X ])t
I f1(X ), .., ft (X ) are Fp-free mod ℘(k [X ]).I The group extension
0 → G2 → G1 → V = (Z/pZ)v → 0 induces arepresentation ρ : V → Glt (Fp)
I dual to the one given by V acting via translation:(v ∈ V )× (f1(X ), f2(X ), ..., ft (X )) mod ℘(k [X ])t →→ (f1(X + v), f2(X + v), ..., ft (X + v))mod ℘(k [X ])t
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Condition (N) and G2
In this section (C, G) is a big action. Let Gi be thelower ramification groups.
I Let H C G and H with index p in G2 (H exists!),then (C/H, G/H) is a big action.
I (G/H)2 = G2/H ' Z/pZ.I There is S(F ) ∈ k{F},
f1 = cX + XΣ(F )(X ) ∈ k [X ] with C/H ' Cf1 .I If G2 ' (Z/pZ)t , then k(C) = k(X , W1, ..., Wt ) and
℘(W1, ..., Wt ) = (f1(X ), f2(X ), ..., ft (X )) ∈ (k [X ])t
I f1(X ), .., ft (X ) are Fp-free mod ℘(k [X ]).I The group extension
0 → G2 → G1 → V = (Z/pZ)v → 0 induces arepresentation ρ : V → Glt (Fp)
I dual to the one given by V acting via translation:(v ∈ V )× (f1(X ), f2(X ), ..., ft (X )) mod ℘(k [X ])t →→ (f1(X + v), f2(X + v), ..., ft (X + v))mod ℘(k [X ])t
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Condition (N) and G2
In this section (C, G) is a big action. Let Gi be thelower ramification groups.
I Let H C G and H with index p in G2 (H exists!),then (C/H, G/H) is a big action.
I (G/H)2 = G2/H ' Z/pZ.I There is S(F ) ∈ k{F},
f1 = cX + XΣ(F )(X ) ∈ k [X ] with C/H ' Cf1 .I If G2 ' (Z/pZ)t , then k(C) = k(X , W1, ..., Wt ) and
℘(W1, ..., Wt ) = (f1(X ), f2(X ), ..., ft (X )) ∈ (k [X ])t
I f1(X ), .., ft (X ) are Fp-free mod ℘(k [X ]).I The group extension
0 → G2 → G1 → V = (Z/pZ)v → 0 induces arepresentation ρ : V → Glt (Fp)
I dual to the one given by V acting via translation:(v ∈ V )× (f1(X ), f2(X ), ..., ft (X )) mod ℘(k [X ])t →→ (f1(X + v), f2(X + v), ..., ft (X + v))mod ℘(k [X ])t
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Condition (N) and G2
In this section (C, G) is a big action. Let Gi be thelower ramification groups.
I Let H C G and H with index p in G2 (H exists!),then (C/H, G/H) is a big action.
I (G/H)2 = G2/H ' Z/pZ.I There is S(F ) ∈ k{F},
f1 = cX + XΣ(F )(X ) ∈ k [X ] with C/H ' Cf1 .I If G2 ' (Z/pZ)t , then k(C) = k(X , W1, ..., Wt ) and
℘(W1, ..., Wt ) = (f1(X ), f2(X ), ..., ft (X )) ∈ (k [X ])t
I f1(X ), .., ft (X ) are Fp-free mod ℘(k [X ]).I The group extension
0 → G2 → G1 → V = (Z/pZ)v → 0 induces arepresentation ρ : V → Glt (Fp)
I dual to the one given by V acting via translation:(v ∈ V )× (f1(X ), f2(X ), ..., ft (X )) mod ℘(k [X ])t →→ (f1(X + v), f2(X + v), ..., ft (X + v))mod ℘(k [X ])t
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Condition (N) and G2
In this section (C, G) is a big action. Let Gi be thelower ramification groups.
I Let H C G and H with index p in G2 (H exists!),then (C/H, G/H) is a big action.
I (G/H)2 = G2/H ' Z/pZ.I There is S(F ) ∈ k{F},
f1 = cX + XΣ(F )(X ) ∈ k [X ] with C/H ' Cf1 .I If G2 ' (Z/pZ)t , then k(C) = k(X , W1, ..., Wt ) and
℘(W1, ..., Wt ) = (f1(X ), f2(X ), ..., ft (X )) ∈ (k [X ])t
I f1(X ), .., ft (X ) are Fp-free mod ℘(k [X ]).I The group extension
0 → G2 → G1 → V = (Z/pZ)v → 0 induces arepresentation ρ : V → Glt (Fp)
I dual to the one given by V acting via translation:(v ∈ V )× (f1(X ), f2(X ), ..., ft (X )) mod ℘(k [X ])t →→ (f1(X + v), f2(X + v), ..., ft (X + v))mod ℘(k [X ])t
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Condition (N) and G2
In this section (C, G) is a big action. Let Gi be thelower ramification groups.
I Let H C G and H with index p in G2 (H exists!),then (C/H, G/H) is a big action.
I (G/H)2 = G2/H ' Z/pZ.I There is S(F ) ∈ k{F},
f1 = cX + XΣ(F )(X ) ∈ k [X ] with C/H ' Cf1 .I If G2 ' (Z/pZ)t , then k(C) = k(X , W1, ..., Wt ) and
℘(W1, ..., Wt ) = (f1(X ), f2(X ), ..., ft (X )) ∈ (k [X ])t
I f1(X ), .., ft (X ) are Fp-free mod ℘(k [X ]).I The group extension
0 → G2 → G1 → V = (Z/pZ)v → 0 induces arepresentation ρ : V → Glt (Fp)
I dual to the one given by V acting via translation:(v ∈ V )× (f1(X ), f2(X ), ..., ft (X )) mod ℘(k [X ])t →→ (f1(X + v), f2(X + v), ..., ft (X + v))mod ℘(k [X ])t
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Imρ is a unipotent subgroup of Glt(Fp) which isthe identity iff G2 ⊂ Z (G). In this casefi(X ) = ciX + XΣi(F )(X ) where Σi(F ) ∈ k{F}and v ∈ V is a commun zero to the palyndromicpolynomials Adfi ∈ k{F , F−1}.
I Let f1 := X (αF )(X ) = αX 1+p with αp + α = 0 ;then Adf1 = Y p2
− Y .I Let f2 := X 1+2p − X 2+p, thenI f2(X + Y ) − f2(X ) − f2(Y ) = 2(Y p − Y )X 1+p +
(Y − Y p2)X 2p + (Y 2p2
− Y 2 + 2Y 1+p − 2Y p+p2)X p
mod ℘(k [X , Y ])I If y ∈ Z (Adf1) = Fp2 one has
f2(X + y) = 2(yp−y)α f1(X ) + f2(X ) + ℘(P2).
I y → 2(yp−y)α is a non zero linear form over Fp2 with
value in Fp.I |G| = p2p2 and g = p−1
2 (p + p ∗ 2p).I
|G|g = 2p
p−1p2
1+2p .
I|G|g2 = 4p
(p−1)2p
(1+2p)2 .
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Imρ is a unipotent subgroup of Glt(Fp) which isthe identity iff G2 ⊂ Z (G). In this casefi(X ) = ciX + XΣi(F )(X ) where Σi(F ) ∈ k{F}and v ∈ V is a commun zero to the palyndromicpolynomials Adfi ∈ k{F , F−1}.
I Let f1 := X (αF )(X ) = αX 1+p with αp + α = 0 ;then Adf1 = Y p2
− Y .I Let f2 := X 1+2p − X 2+p, thenI f2(X + Y ) − f2(X ) − f2(Y ) = 2(Y p − Y )X 1+p +
(Y − Y p2)X 2p + (Y 2p2
− Y 2 + 2Y 1+p − 2Y p+p2)X p
mod ℘(k [X , Y ])I If y ∈ Z (Adf1) = Fp2 one has
f2(X + y) = 2(yp−y)α f1(X ) + f2(X ) + ℘(P2).
I y → 2(yp−y)α is a non zero linear form over Fp2 with
value in Fp.I |G| = p2p2 and g = p−1
2 (p + p ∗ 2p).I
|G|g = 2p
p−1p2
1+2p .
I|G|g2 = 4p
(p−1)2p
(1+2p)2 .
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Imρ is a unipotent subgroup of Glt(Fp) which isthe identity iff G2 ⊂ Z (G). In this casefi(X ) = ciX + XΣi(F )(X ) where Σi(F ) ∈ k{F}and v ∈ V is a commun zero to the palyndromicpolynomials Adfi ∈ k{F , F−1}.
I Let f1 := X (αF )(X ) = αX 1+p with αp + α = 0 ;then Adf1 = Y p2
− Y .I Let f2 := X 1+2p − X 2+p, thenI f2(X + Y ) − f2(X ) − f2(Y ) = 2(Y p − Y )X 1+p +
(Y − Y p2)X 2p + (Y 2p2
− Y 2 + 2Y 1+p − 2Y p+p2)X p
mod ℘(k [X , Y ])I If y ∈ Z (Adf1) = Fp2 one has
f2(X + y) = 2(yp−y)α f1(X ) + f2(X ) + ℘(P2).
I y → 2(yp−y)α is a non zero linear form over Fp2 with
value in Fp.I |G| = p2p2 and g = p−1
2 (p + p ∗ 2p).I
|G|g = 2p
p−1p2
1+2p .
I|G|g2 = 4p
(p−1)2p
(1+2p)2 .
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Imρ is a unipotent subgroup of Glt(Fp) which isthe identity iff G2 ⊂ Z (G). In this casefi(X ) = ciX + XΣi(F )(X ) where Σi(F ) ∈ k{F}and v ∈ V is a commun zero to the palyndromicpolynomials Adfi ∈ k{F , F−1}.
I Let f1 := X (αF )(X ) = αX 1+p with αp + α = 0 ;then Adf1 = Y p2
− Y .I Let f2 := X 1+2p − X 2+p, thenI f2(X + Y ) − f2(X ) − f2(Y ) = 2(Y p − Y )X 1+p +
(Y − Y p2)X 2p + (Y 2p2
− Y 2 + 2Y 1+p − 2Y p+p2)X p
mod ℘(k [X , Y ])I If y ∈ Z (Adf1) = Fp2 one has
f2(X + y) = 2(yp−y)α f1(X ) + f2(X ) + ℘(P2).
I y → 2(yp−y)α is a non zero linear form over Fp2 with
value in Fp.I |G| = p2p2 and g = p−1
2 (p + p ∗ 2p).I
|G|g = 2p
p−1p2
1+2p .
I|G|g2 = 4p
(p−1)2p
(1+2p)2 .
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Imρ is a unipotent subgroup of Glt(Fp) which isthe identity iff G2 ⊂ Z (G). In this casefi(X ) = ciX + XΣi(F )(X ) where Σi(F ) ∈ k{F}and v ∈ V is a commun zero to the palyndromicpolynomials Adfi ∈ k{F , F−1}.
I Let f1 := X (αF )(X ) = αX 1+p with αp + α = 0 ;then Adf1 = Y p2
− Y .I Let f2 := X 1+2p − X 2+p, thenI f2(X + Y ) − f2(X ) − f2(Y ) = 2(Y p − Y )X 1+p +
(Y − Y p2)X 2p + (Y 2p2
− Y 2 + 2Y 1+p − 2Y p+p2)X p
mod ℘(k [X , Y ])I If y ∈ Z (Adf1) = Fp2 one has
f2(X + y) = 2(yp−y)α f1(X ) + f2(X ) + ℘(P2).
I y → 2(yp−y)α is a non zero linear form over Fp2 with
value in Fp.I |G| = p2p2 and g = p−1
2 (p + p ∗ 2p).I
|G|g = 2p
p−1p2
1+2p .
I|G|g2 = 4p
(p−1)2p
(1+2p)2 .
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Imρ is a unipotent subgroup of Glt(Fp) which isthe identity iff G2 ⊂ Z (G). In this casefi(X ) = ciX + XΣi(F )(X ) where Σi(F ) ∈ k{F}and v ∈ V is a commun zero to the palyndromicpolynomials Adfi ∈ k{F , F−1}.
I Let f1 := X (αF )(X ) = αX 1+p with αp + α = 0 ;then Adf1 = Y p2
− Y .I Let f2 := X 1+2p − X 2+p, thenI f2(X + Y ) − f2(X ) − f2(Y ) = 2(Y p − Y )X 1+p +
(Y − Y p2)X 2p + (Y 2p2
− Y 2 + 2Y 1+p − 2Y p+p2)X p
mod ℘(k [X , Y ])I If y ∈ Z (Adf1) = Fp2 one has
f2(X + y) = 2(yp−y)α f1(X ) + f2(X ) + ℘(P2).
I y → 2(yp−y)α is a non zero linear form over Fp2 with
value in Fp.I |G| = p2p2 and g = p−1
2 (p + p ∗ 2p).I
|G|g = 2p
p−1p2
1+2p .
I|G|g2 = 4p
(p−1)2p
(1+2p)2 .
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Imρ is a unipotent subgroup of Glt(Fp) which isthe identity iff G2 ⊂ Z (G). In this casefi(X ) = ciX + XΣi(F )(X ) where Σi(F ) ∈ k{F}and v ∈ V is a commun zero to the palyndromicpolynomials Adfi ∈ k{F , F−1}.
I Let f1 := X (αF )(X ) = αX 1+p with αp + α = 0 ;then Adf1 = Y p2
− Y .I Let f2 := X 1+2p − X 2+p, thenI f2(X + Y ) − f2(X ) − f2(Y ) = 2(Y p − Y )X 1+p +
(Y − Y p2)X 2p + (Y 2p2
− Y 2 + 2Y 1+p − 2Y p+p2)X p
mod ℘(k [X , Y ])I If y ∈ Z (Adf1) = Fp2 one has
f2(X + y) = 2(yp−y)α f1(X ) + f2(X ) + ℘(P2).
I y → 2(yp−y)α is a non zero linear form over Fp2 with
value in Fp.I |G| = p2p2 and g = p−1
2 (p + p ∗ 2p).I
|G|g = 2p
p−1p2
1+2p .
I|G|g2 = 4p
(p−1)2p
(1+2p)2 .
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Imρ is a unipotent subgroup of Glt(Fp) which isthe identity iff G2 ⊂ Z (G). In this casefi(X ) = ciX + XΣi(F )(X ) where Σi(F ) ∈ k{F}and v ∈ V is a commun zero to the palyndromicpolynomials Adfi ∈ k{F , F−1}.
I Let f1 := X (αF )(X ) = αX 1+p with αp + α = 0 ;then Adf1 = Y p2
− Y .I Let f2 := X 1+2p − X 2+p, thenI f2(X + Y ) − f2(X ) − f2(Y ) = 2(Y p − Y )X 1+p +
(Y − Y p2)X 2p + (Y 2p2
− Y 2 + 2Y 1+p − 2Y p+p2)X p
mod ℘(k [X , Y ])I If y ∈ Z (Adf1) = Fp2 one has
f2(X + y) = 2(yp−y)α f1(X ) + f2(X ) + ℘(P2).
I y → 2(yp−y)α is a non zero linear form over Fp2 with
value in Fp.I |G| = p2p2 and g = p−1
2 (p + p ∗ 2p).I
|G|g = 2p
p−1p2
1+2p .
I|G|g2 = 4p
(p−1)2p
(1+2p)2 .
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Imρ is a unipotent subgroup of Glt(Fp) which isthe identity iff G2 ⊂ Z (G). In this casefi(X ) = ciX + XΣi(F )(X ) where Σi(F ) ∈ k{F}and v ∈ V is a commun zero to the palyndromicpolynomials Adfi ∈ k{F , F−1}.
I Let f1 := X (αF )(X ) = αX 1+p with αp + α = 0 ;then Adf1 = Y p2
− Y .I Let f2 := X 1+2p − X 2+p, thenI f2(X + Y ) − f2(X ) − f2(Y ) = 2(Y p − Y )X 1+p +
(Y − Y p2)X 2p + (Y 2p2
− Y 2 + 2Y 1+p − 2Y p+p2)X p
mod ℘(k [X , Y ])I If y ∈ Z (Adf1) = Fp2 one has
f2(X + y) = 2(yp−y)α f1(X ) + f2(X ) + ℘(P2).
I y → 2(yp−y)α is a non zero linear form over Fp2 with
value in Fp.I |G| = p2p2 and g = p−1
2 (p + p ∗ 2p).I
|G|g = 2p
p−1p2
1+2p .
I|G|g2 = 4p
(p−1)2p
(1+2p)2 .
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Theorem([Le-Ma 4]) Assume G2 is non abelian, thenG2 = G′.
I Sketch proof: If G′ 6= G2, there is H C G withG′ ⊂ H ⊂ G2 and [G2 : H] = p.(C/H, G/H) is abig action ;
I C/H : W p − W = f := XΣ(F )(X ),deg(f ) = 1 + ps.
I (AutC/H)∞,1 := E is extraspecial with orderp2s+1.
I G/H is abelian and normal in E .I ([Hu 67] Satz 13.7 p. 353) |G/H| ≤ ps+1 and so
|G/H|/g(C/H) ≤ 2ps+1
(p−1)ps = 2pp−1 , a contradiction.
We deduce the following corollary from ([Su 86]4.21 p.75).
I CorollaryIf |G2| = p3, then G2 is abelian.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Theorem([Le-Ma 4]) Assume G2 is non abelian, thenG2 = G′.
I Sketch proof: If G′ 6= G2, there is H C G withG′ ⊂ H ⊂ G2 and [G2 : H] = p.(C/H, G/H) is abig action ;
I C/H : W p − W = f := XΣ(F )(X ),deg(f ) = 1 + ps.
I (AutC/H)∞,1 := E is extraspecial with orderp2s+1.
I G/H is abelian and normal in E .I ([Hu 67] Satz 13.7 p. 353) |G/H| ≤ ps+1 and so
|G/H|/g(C/H) ≤ 2ps+1
(p−1)ps = 2pp−1 , a contradiction.
We deduce the following corollary from ([Su 86]4.21 p.75).
I CorollaryIf |G2| = p3, then G2 is abelian.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Theorem([Le-Ma 4]) Assume G2 is non abelian, thenG2 = G′.
I Sketch proof: If G′ 6= G2, there is H C G withG′ ⊂ H ⊂ G2 and [G2 : H] = p.(C/H, G/H) is abig action ;
I C/H : W p − W = f := XΣ(F )(X ),deg(f ) = 1 + ps.
I (AutC/H)∞,1 := E is extraspecial with orderp2s+1.
I G/H is abelian and normal in E .I ([Hu 67] Satz 13.7 p. 353) |G/H| ≤ ps+1 and so
|G/H|/g(C/H) ≤ 2ps+1
(p−1)ps = 2pp−1 , a contradiction.
We deduce the following corollary from ([Su 86]4.21 p.75).
I CorollaryIf |G2| = p3, then G2 is abelian.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Theorem([Le-Ma 4]) Assume G2 is non abelian, thenG2 = G′.
I Sketch proof: If G′ 6= G2, there is H C G withG′ ⊂ H ⊂ G2 and [G2 : H] = p.(C/H, G/H) is abig action ;
I C/H : W p − W = f := XΣ(F )(X ),deg(f ) = 1 + ps.
I (AutC/H)∞,1 := E is extraspecial with orderp2s+1.
I G/H is abelian and normal in E .I ([Hu 67] Satz 13.7 p. 353) |G/H| ≤ ps+1 and so
|G/H|/g(C/H) ≤ 2ps+1
(p−1)ps = 2pp−1 , a contradiction.
We deduce the following corollary from ([Su 86]4.21 p.75).
I CorollaryIf |G2| = p3, then G2 is abelian.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Theorem([Le-Ma 4]) Assume G2 is non abelian, thenG2 = G′.
I Sketch proof: If G′ 6= G2, there is H C G withG′ ⊂ H ⊂ G2 and [G2 : H] = p.(C/H, G/H) is abig action ;
I C/H : W p − W = f := XΣ(F )(X ),deg(f ) = 1 + ps.
I (AutC/H)∞,1 := E is extraspecial with orderp2s+1.
I G/H is abelian and normal in E .I ([Hu 67] Satz 13.7 p. 353) |G/H| ≤ ps+1 and so
|G/H|/g(C/H) ≤ 2ps+1
(p−1)ps = 2pp−1 , a contradiction.
We deduce the following corollary from ([Su 86]4.21 p.75).
I CorollaryIf |G2| = p3, then G2 is abelian.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Theorem([Le-Ma 4]) Assume G2 is non abelian, thenG2 = G′.
I Sketch proof: If G′ 6= G2, there is H C G withG′ ⊂ H ⊂ G2 and [G2 : H] = p.(C/H, G/H) is abig action ;
I C/H : W p − W = f := XΣ(F )(X ),deg(f ) = 1 + ps.
I (AutC/H)∞,1 := E is extraspecial with orderp2s+1.
I G/H is abelian and normal in E .I ([Hu 67] Satz 13.7 p. 353) |G/H| ≤ ps+1 and so
|G/H|/g(C/H) ≤ 2ps+1
(p−1)ps = 2pp−1 , a contradiction.
We deduce the following corollary from ([Su 86]4.21 p.75).
I CorollaryIf |G2| = p3, then G2 is abelian.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Theorem([Le-Ma 4]) Assume G2 is non abelian, thenG2 = G′.
I Sketch proof: If G′ 6= G2, there is H C G withG′ ⊂ H ⊂ G2 and [G2 : H] = p.(C/H, G/H) is abig action ;
I C/H : W p − W = f := XΣ(F )(X ),deg(f ) = 1 + ps.
I (AutC/H)∞,1 := E is extraspecial with orderp2s+1.
I G/H is abelian and normal in E .I ([Hu 67] Satz 13.7 p. 353) |G/H| ≤ ps+1 and so
|G/H|/g(C/H) ≤ 2ps+1
(p−1)ps = 2pp−1 , a contradiction.
We deduce the following corollary from ([Su 86]4.21 p.75).
I CorollaryIf |G2| = p3, then G2 is abelian.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Riemann surfacesI In characteristic 0, an analogue of big actions is
given by the actions of a finite group G on acompact Riemann surface C with gC ≥ 2 suchthat |G| = 84(gC − 1) (we say that C is anHurwitz curve) ([Co 90]).
I Let us mention Klein’s quartic (G ' PSL2(F7)) ([El99]).
I The Fricke-Macbeath curves with genus 7(G ' PSL2(F8)) ([Mc] 65).
I Let C be an Hurwitz curve with genus gC . Letn > 1 and Cn the maximal unramified Galoiscover whose group is abelian with exponent n.The Galois group of Cn/C is (Z/nZ)2gC . It followsfrom the unicity of Cn that the k -automorphisms ofC have n2g prolongations to Cn. ThereforegCn − 1 = n2g(gC − 1) and n2g |Autk C| ≤ |Autk Cn|,where |Autk Cn| ≥ 84(gCn − 1) ; Cn is an Hurwitzcurve ([Mc] 61).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Riemann surfacesI In characteristic 0, an analogue of big actions is
given by the actions of a finite group G on acompact Riemann surface C with gC ≥ 2 suchthat |G| = 84(gC − 1) (we say that C is anHurwitz curve) ([Co 90]).
I Let us mention Klein’s quartic (G ' PSL2(F7)) ([El99]).
I The Fricke-Macbeath curves with genus 7(G ' PSL2(F8)) ([Mc] 65).
I Let C be an Hurwitz curve with genus gC . Letn > 1 and Cn the maximal unramified Galoiscover whose group is abelian with exponent n.The Galois group of Cn/C is (Z/nZ)2gC . It followsfrom the unicity of Cn that the k -automorphisms ofC have n2g prolongations to Cn. ThereforegCn − 1 = n2g(gC − 1) and n2g |Autk C| ≤ |Autk Cn|,where |Autk Cn| ≥ 84(gCn − 1) ; Cn is an Hurwitzcurve ([Mc] 61).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Riemann surfacesI In characteristic 0, an analogue of big actions is
given by the actions of a finite group G on acompact Riemann surface C with gC ≥ 2 suchthat |G| = 84(gC − 1) (we say that C is anHurwitz curve) ([Co 90]).
I Let us mention Klein’s quartic (G ' PSL2(F7)) ([El99]).
I The Fricke-Macbeath curves with genus 7(G ' PSL2(F8)) ([Mc] 65).
I Let C be an Hurwitz curve with genus gC . Letn > 1 and Cn the maximal unramified Galoiscover whose group is abelian with exponent n.The Galois group of Cn/C is (Z/nZ)2gC . It followsfrom the unicity of Cn that the k -automorphisms ofC have n2g prolongations to Cn. ThereforegCn − 1 = n2g(gC − 1) and n2g |Autk C| ≤ |Autk Cn|,where |Autk Cn| ≥ 84(gCn − 1) ; Cn is an Hurwitzcurve ([Mc] 61).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Riemann surfacesI In characteristic 0, an analogue of big actions is
given by the actions of a finite group G on acompact Riemann surface C with gC ≥ 2 suchthat |G| = 84(gC − 1) (we say that C is anHurwitz curve) ([Co 90]).
I Let us mention Klein’s quartic (G ' PSL2(F7)) ([El99]).
I The Fricke-Macbeath curves with genus 7(G ' PSL2(F8)) ([Mc] 65).
I Let C be an Hurwitz curve with genus gC . Letn > 1 and Cn the maximal unramified Galoiscover whose group is abelian with exponent n.The Galois group of Cn/C is (Z/nZ)2gC . It followsfrom the unicity of Cn that the k -automorphisms ofC have n2g prolongations to Cn. ThereforegCn − 1 = n2g(gC − 1) and n2g |Autk C| ≤ |Autk Cn|,where |Autk Cn| ≥ 84(gCn − 1) ; Cn is an Hurwitzcurve ([Mc] 61).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Ray class fields
I If (C, G) is a big action then C → C/G is an étalecover of the affine line whose group is a p-group ;it follows that the Hasse-Witt invariant of C iszero ; therefore, in order to adapt the previousproof to char. p > 0, one needs to acceptramification. This is done with the so called rayclass fields of function fields over finite fields.
I Let K := Fq(X ) where q = pe, S the set of finiterational places (X − v), v ∈ Fq and m ∈ N. LetK alg be an algebraic closure. Let K m
S ⊂ K alg, thebiggest abelian extension L of K with conductor≤ m∞ and such that the places in S arecompletely decomposed.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Ray class fields
I If (C, G) is a big action then C → C/G is an étalecover of the affine line whose group is a p-group ;it follows that the Hasse-Witt invariant of C iszero ; therefore, in order to adapt the previousproof to char. p > 0, one needs to acceptramification. This is done with the so called rayclass fields of function fields over finite fields.
I Let K := Fq(X ) where q = pe, S the set of finiterational places (X − v), v ∈ Fq and m ∈ N. LetK alg be an algebraic closure. Let K m
S ⊂ K alg, thebiggest abelian extension L of K with conductor≤ m∞ and such that the places in S arecompletely decomposed.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I ([La 99], [Au 00]) The constant field of KmS is Fq
and GS(m) := Gal(K mS /K ) ' (1 + TFq[[T ]])/ <
1 + T mFq[[T ]], 1 − vT , v ∈ Fq >, is a p-group.I ([Le-Ma 4]) Let Cm/Fq be the smooth projective
curve with function field KmS . The translations
X → X + v , v ∈ Fq stabilize S and ∞ ; they canbe extended to Fq-automorphisms of K m
S . In thisway, we get an action of a p-group G(m) on Cm
with 0 → GS(m) → G(m) → Fq → 0I ([Au 00]) If nm := |GS(m)|, then
gCm = 1 + nm(−1 + m/2) − (1/2)∑
0≤j≤m−1 nj ≤nm(−1 + m/2)
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I ([La 99], [Au 00]) The constant field of KmS is Fq
and GS(m) := Gal(K mS /K ) ' (1 + TFq[[T ]])/ <
1 + T mFq[[T ]], 1 − vT , v ∈ Fq >, is a p-group.I ([Le-Ma 4]) Let Cm/Fq be the smooth projective
curve with function field KmS . The translations
X → X + v , v ∈ Fq stabilize S and ∞ ; they canbe extended to Fq-automorphisms of K m
S . In thisway, we get an action of a p-group G(m) on Cm
with 0 → GS(m) → G(m) → Fq → 0I ([Au 00]) If nm := |GS(m)|, then
gCm = 1 + nm(−1 + m/2) − (1/2)∑
0≤j≤m−1 nj ≤nm(−1 + m/2)
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I ([La 99], [Au 00]) The constant field of KmS is Fq
and GS(m) := Gal(K mS /K ) ' (1 + TFq[[T ]])/ <
1 + T mFq[[T ]], 1 − vT , v ∈ Fq >, is a p-group.I ([Le-Ma 4]) Let Cm/Fq be the smooth projective
curve with function field KmS . The translations
X → X + v , v ∈ Fq stabilize S and ∞ ; they canbe extended to Fq-automorphisms of K m
S . In thisway, we get an action of a p-group G(m) on Cm
with 0 → GS(m) → G(m) → Fq → 0I ([Au 00]) If nm := |GS(m)|, then
gCm = 1 + nm(−1 + m/2) − (1/2)∑
0≤j≤m−1 nj ≤nm(−1 + m/2)
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I|G(m)|
gCm≥ nmq
nm(−1+m/2) = q−1+m/2 . This is a “big
action” as soon as q−1+m/2 > 2p
p−1 (we haveG2 = GS(m))
I Let Nq := |Cm(Fq)|. Then, Nq = 1 + |G(m)|, andthe quotient |G(m)|
gCm∼ Nq
gCm.
I ([La 99]) If q = pe,m2 := pde/2e+1 + p + 1 is thesmallest conductor m such that the exponent ofGm
S is > p.I If e > 2, (Cm2 , G(m2)) is a big action and G2 is
abelian with exponent p2.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I|G(m)|
gCm≥ nmq
nm(−1+m/2) = q−1+m/2 . This is a “big
action” as soon as q−1+m/2 > 2p
p−1 (we haveG2 = GS(m))
I Let Nq := |Cm(Fq)|. Then, Nq = 1 + |G(m)|, andthe quotient |G(m)|
gCm∼ Nq
gCm.
I ([La 99]) If q = pe,m2 := pde/2e+1 + p + 1 is thesmallest conductor m such that the exponent ofGm
S is > p.I If e > 2, (Cm2 , G(m2)) is a big action and G2 is
abelian with exponent p2.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I|G(m)|
gCm≥ nmq
nm(−1+m/2) = q−1+m/2 . This is a “big
action” as soon as q−1+m/2 > 2p
p−1 (we haveG2 = GS(m))
I Let Nq := |Cm(Fq)|. Then, Nq = 1 + |G(m)|, andthe quotient |G(m)|
gCm∼ Nq
gCm.
I ([La 99]) If q = pe,m2 := pde/2e+1 + p + 1 is thesmallest conductor m such that the exponent ofGm
S is > p.I If e > 2, (Cm2 , G(m2)) is a big action and G2 is
abelian with exponent p2.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I|G(m)|
gCm≥ nmq
nm(−1+m/2) = q−1+m/2 . This is a “big
action” as soon as q−1+m/2 > 2p
p−1 (we haveG2 = GS(m))
I Let Nq := |Cm(Fq)|. Then, Nq = 1 + |G(m)|, andthe quotient |G(m)|
gCm∼ Nq
gCm.
I ([La 99]) If q = pe,m2 := pde/2e+1 + p + 1 is thesmallest conductor m such that the exponent ofGm
S is > p.I If e > 2, (Cm2 , G(m2)) is a big action and G2 is
abelian with exponent p2.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Big actions (II)From now on, k is any algebraically closed fieldand (C, G) is a big action.
I If G2 ' Z/pnZ, then n = 1 ([Le-Ma 4]).I Sketch proof: Let H = Gpn−2
2 then (C/H, G/H) is abig action, so one can assume that n = 2. ThenC → C/G2 is given by ℘(W0, W1) = (f0, f1) withf0 = XΣ(F )(X ), deg f0 = 1 + ps.
I Let v ∈ V := Z (Adf0) and P ∈ k [X ] withf0(X + v) = f0(X ) + ℘(P) alors f1(X + v)− f1(X ) =
`(v)f0(X ) + 1p (f0(X )p + P(X )p −P(X )p2
− (f0(X ) +
P(X ))p − f0(X + v)p + (f0(X + v) + P(X )p)p)
I = `(v)f0(X ) +∑
1≤i≤p−1(−1)i−1
i v iX p−i+ps+1
mod X ps+1where ` : V → Fp is a linear form.
I For G2 abelian with exponent pe, e ≥ 2, one canexpect for its p-rank a lower bound in O(log(gC)).This is the case in the preceding situation i.e.(C, G) = (Cm2 , G(m2)) ([M. Rocher, thesis inpreparation]).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Big actions (II)From now on, k is any algebraically closed fieldand (C, G) is a big action.
I If G2 ' Z/pnZ, then n = 1 ([Le-Ma 4]).I Sketch proof: Let H = Gpn−2
2 then (C/H, G/H) is abig action, so one can assume that n = 2. ThenC → C/G2 is given by ℘(W0, W1) = (f0, f1) withf0 = XΣ(F )(X ), deg f0 = 1 + ps.
I Let v ∈ V := Z (Adf0) and P ∈ k [X ] withf0(X + v) = f0(X ) + ℘(P) alors f1(X + v)− f1(X ) =
`(v)f0(X ) + 1p (f0(X )p + P(X )p −P(X )p2
− (f0(X ) +
P(X ))p − f0(X + v)p + (f0(X + v) + P(X )p)p)
I = `(v)f0(X ) +∑
1≤i≤p−1(−1)i−1
i v iX p−i+ps+1
mod X ps+1where ` : V → Fp is a linear form.
I For G2 abelian with exponent pe, e ≥ 2, one canexpect for its p-rank a lower bound in O(log(gC)).This is the case in the preceding situation i.e.(C, G) = (Cm2 , G(m2)) ([M. Rocher, thesis inpreparation]).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Big actions (II)From now on, k is any algebraically closed fieldand (C, G) is a big action.
I If G2 ' Z/pnZ, then n = 1 ([Le-Ma 4]).I Sketch proof: Let H = Gpn−2
2 then (C/H, G/H) is abig action, so one can assume that n = 2. ThenC → C/G2 is given by ℘(W0, W1) = (f0, f1) withf0 = XΣ(F )(X ), deg f0 = 1 + ps.
I Let v ∈ V := Z (Adf0) and P ∈ k [X ] withf0(X + v) = f0(X ) + ℘(P) alors f1(X + v)− f1(X ) =
`(v)f0(X ) + 1p (f0(X )p + P(X )p −P(X )p2
− (f0(X ) +
P(X ))p − f0(X + v)p + (f0(X + v) + P(X )p)p)
I = `(v)f0(X ) +∑
1≤i≤p−1(−1)i−1
i v iX p−i+ps+1
mod X ps+1where ` : V → Fp is a linear form.
I For G2 abelian with exponent pe, e ≥ 2, one canexpect for its p-rank a lower bound in O(log(gC)).This is the case in the preceding situation i.e.(C, G) = (Cm2 , G(m2)) ([M. Rocher, thesis inpreparation]).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Big actions (II)From now on, k is any algebraically closed fieldand (C, G) is a big action.
I If G2 ' Z/pnZ, then n = 1 ([Le-Ma 4]).I Sketch proof: Let H = Gpn−2
2 then (C/H, G/H) is abig action, so one can assume that n = 2. ThenC → C/G2 is given by ℘(W0, W1) = (f0, f1) withf0 = XΣ(F )(X ), deg f0 = 1 + ps.
I Let v ∈ V := Z (Adf0) and P ∈ k [X ] withf0(X + v) = f0(X ) + ℘(P) alors f1(X + v)− f1(X ) =
`(v)f0(X ) + 1p (f0(X )p + P(X )p −P(X )p2
− (f0(X ) +
P(X ))p − f0(X + v)p + (f0(X + v) + P(X )p)p)
I = `(v)f0(X ) +∑
1≤i≤p−1(−1)i−1
i v iX p−i+ps+1
mod X ps+1where ` : V → Fp is a linear form.
I For G2 abelian with exponent pe, e ≥ 2, one canexpect for its p-rank a lower bound in O(log(gC)).This is the case in the preceding situation i.e.(C, G) = (Cm2 , G(m2)) ([M. Rocher, thesis inpreparation]).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Big actions (II)From now on, k is any algebraically closed fieldand (C, G) is a big action.
I If G2 ' Z/pnZ, then n = 1 ([Le-Ma 4]).I Sketch proof: Let H = Gpn−2
2 then (C/H, G/H) is abig action, so one can assume that n = 2. ThenC → C/G2 is given by ℘(W0, W1) = (f0, f1) withf0 = XΣ(F )(X ), deg f0 = 1 + ps.
I Let v ∈ V := Z (Adf0) and P ∈ k [X ] withf0(X + v) = f0(X ) + ℘(P) alors f1(X + v)− f1(X ) =
`(v)f0(X ) + 1p (f0(X )p + P(X )p −P(X )p2
− (f0(X ) +
P(X ))p − f0(X + v)p + (f0(X + v) + P(X )p)p)
I = `(v)f0(X ) +∑
1≤i≤p−1(−1)i−1
i v iX p−i+ps+1
mod X ps+1where ` : V → Fp is a linear form.
I For G2 abelian with exponent pe, e ≥ 2, one canexpect for its p-rank a lower bound in O(log(gC)).This is the case in the preceding situation i.e.(C, G) = (Cm2 , G(m2)) ([M. Rocher, thesis inpreparation]).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Maximal curves (I)Let us assume that (C, G) is a big action.
I Let i0 with G2 = G3 = .... = Gi0 % Gi0+1. Theng(C/Gi0+1) = 1
2(|G2/Gi0+1| − 1)(i0 − 1).
I If 0 < M ≤ |G|
g2C
, then
I |Gi0+1| ≤1M
|G/Gi0+1|
g2C/Gi0+1
≤ 1M
4|G2/Gi0+1|
(|G2/Gi0+1|−1)2 .
I Theorem([Le-Ma 1]) If |G|
g2C≥ 4
(p−1)2 there is Σ(F ) ∈ k{F}
and f = cX + XΣ(F )(X ) ∈ k [X ] with C ' Cf .Moreover there are two possibilities for G:
I|G|
g2C
= 4p(p−1)2 and G = G∞,1(f ) or
I|G|
g2C
= 4(p−1)2 and G ⊂ G∞,1(f ) has index p.
I Note that the sequence pn
(pn−1)2 is decreasing and
that |Gi0+1| ∈ pN . We deduce bounds for|G2/Gi0+1|, |Gi0+1| and so for |G2|.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Maximal curves (I)Let us assume that (C, G) is a big action.
I Let i0 with G2 = G3 = .... = Gi0 % Gi0+1. Theng(C/Gi0+1) = 1
2(|G2/Gi0+1| − 1)(i0 − 1).
I If 0 < M ≤ |G|
g2C
, then
I |Gi0+1| ≤1M
|G/Gi0+1|
g2C/Gi0+1
≤ 1M
4|G2/Gi0+1|
(|G2/Gi0+1|−1)2 .
I Theorem([Le-Ma 1]) If |G|
g2C≥ 4
(p−1)2 there is Σ(F ) ∈ k{F}
and f = cX + XΣ(F )(X ) ∈ k [X ] with C ' Cf .Moreover there are two possibilities for G:
I|G|
g2C
= 4p(p−1)2 and G = G∞,1(f ) or
I|G|
g2C
= 4(p−1)2 and G ⊂ G∞,1(f ) has index p.
I Note that the sequence pn
(pn−1)2 is decreasing and
that |Gi0+1| ∈ pN . We deduce bounds for|G2/Gi0+1|, |Gi0+1| and so for |G2|.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Maximal curves (I)Let us assume that (C, G) is a big action.
I Let i0 with G2 = G3 = .... = Gi0 % Gi0+1. Theng(C/Gi0+1) = 1
2(|G2/Gi0+1| − 1)(i0 − 1).
I If 0 < M ≤ |G|
g2C
, then
I |Gi0+1| ≤1M
|G/Gi0+1|
g2C/Gi0+1
≤ 1M
4|G2/Gi0+1|
(|G2/Gi0+1|−1)2 .
I Theorem([Le-Ma 1]) If |G|
g2C≥ 4
(p−1)2 there is Σ(F ) ∈ k{F}
and f = cX + XΣ(F )(X ) ∈ k [X ] with C ' Cf .Moreover there are two possibilities for G:
I|G|
g2C
= 4p(p−1)2 and G = G∞,1(f ) or
I|G|
g2C
= 4(p−1)2 and G ⊂ G∞,1(f ) has index p.
I Note that the sequence pn
(pn−1)2 is decreasing and
that |Gi0+1| ∈ pN . We deduce bounds for|G2/Gi0+1|, |Gi0+1| and so for |G2|.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Maximal curves (I)Let us assume that (C, G) is a big action.
I Let i0 with G2 = G3 = .... = Gi0 % Gi0+1. Theng(C/Gi0+1) = 1
2(|G2/Gi0+1| − 1)(i0 − 1).
I If 0 < M ≤ |G|
g2C
, then
I |Gi0+1| ≤1M
|G/Gi0+1|
g2C/Gi0+1
≤ 1M
4|G2/Gi0+1|
(|G2/Gi0+1|−1)2 .
I Theorem([Le-Ma 1]) If |G|
g2C≥ 4
(p−1)2 there is Σ(F ) ∈ k{F}
and f = cX + XΣ(F )(X ) ∈ k [X ] with C ' Cf .Moreover there are two possibilities for G:
I|G|
g2C
= 4p(p−1)2 and G = G∞,1(f ) or
I|G|
g2C
= 4(p−1)2 and G ⊂ G∞,1(f ) has index p.
I Note that the sequence pn
(pn−1)2 is decreasing and
that |Gi0+1| ∈ pN . We deduce bounds for|G2/Gi0+1|, |Gi0+1| and so for |G2|.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Maximal curves (I)Let us assume that (C, G) is a big action.
I Let i0 with G2 = G3 = .... = Gi0 % Gi0+1. Theng(C/Gi0+1) = 1
2(|G2/Gi0+1| − 1)(i0 − 1).
I If 0 < M ≤ |G|
g2C
, then
I |Gi0+1| ≤1M
|G/Gi0+1|
g2C/Gi0+1
≤ 1M
4|G2/Gi0+1|
(|G2/Gi0+1|−1)2 .
I Theorem([Le-Ma 1]) If |G|
g2C≥ 4
(p−1)2 there is Σ(F ) ∈ k{F}
and f = cX + XΣ(F )(X ) ∈ k [X ] with C ' Cf .Moreover there are two possibilities for G:
I|G|
g2C
= 4p(p−1)2 and G = G∞,1(f ) or
I|G|
g2C
= 4(p−1)2 and G ⊂ G∞,1(f ) has index p.
I Note that the sequence pn
(pn−1)2 is decreasing and
that |Gi0+1| ∈ pN . We deduce bounds for|G2/Gi0+1|, |Gi0+1| and so for |G2|.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Maximal curves (I)Let us assume that (C, G) is a big action.
I Let i0 with G2 = G3 = .... = Gi0 % Gi0+1. Theng(C/Gi0+1) = 1
2(|G2/Gi0+1| − 1)(i0 − 1).
I If 0 < M ≤ |G|
g2C
, then
I |Gi0+1| ≤1M
|G/Gi0+1|
g2C/Gi0+1
≤ 1M
4|G2/Gi0+1|
(|G2/Gi0+1|−1)2 .
I Theorem([Le-Ma 1]) If |G|
g2C≥ 4
(p−1)2 there is Σ(F ) ∈ k{F}
and f = cX + XΣ(F )(X ) ∈ k [X ] with C ' Cf .Moreover there are two possibilities for G:
I|G|
g2C
= 4p(p−1)2 and G = G∞,1(f ) or
I|G|
g2C
= 4(p−1)2 and G ⊂ G∞,1(f ) has index p.
I Note that the sequence pn
(pn−1)2 is decreasing and
that |Gi0+1| ∈ pN . We deduce bounds for|G2/Gi0+1|, |Gi0+1| and so for |G2|.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Maximal curves (I)Let us assume that (C, G) is a big action.
I Let i0 with G2 = G3 = .... = Gi0 % Gi0+1. Theng(C/Gi0+1) = 1
2(|G2/Gi0+1| − 1)(i0 − 1).
I If 0 < M ≤ |G|
g2C
, then
I |Gi0+1| ≤1M
|G/Gi0+1|
g2C/Gi0+1
≤ 1M
4|G2/Gi0+1|
(|G2/Gi0+1|−1)2 .
I Theorem([Le-Ma 1]) If |G|
g2C≥ 4
(p−1)2 there is Σ(F ) ∈ k{F}
and f = cX + XΣ(F )(X ) ∈ k [X ] with C ' Cf .Moreover there are two possibilities for G:
I|G|
g2C
= 4p(p−1)2 and G = G∞,1(f ) or
I|G|
g2C
= 4(p−1)2 and G ⊂ G∞,1(f ) has index p.
I Note that the sequence pn
(pn−1)2 is decreasing and
that |Gi0+1| ∈ pN . We deduce bounds for|G2/Gi0+1|, |Gi0+1| and so for |G2|.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Maximal curves (II)
We still assume that (C, G) is a big action.I One can push the “classification ” of big actions
up to the condition |G|
g2C≥ 4
(p2−1)2 . Namely
I One first show that |G2| divides p3.I The condition G2 = G′
1 implies that G2 is abelian.I Applying ([Mr 71]) to the case of abelian
extensions with group Z/pZ × Z/p2Z one showsthat G2 has exponent p (we have seen above thatG2 is cyclic iff G2 = Z/pZ).
I Theorem([Le-Ma 4]) For all M > 0, the set |G|
g2C
> M, for
(C, G) a big action with G2 abelian with exponentp, is finite.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Maximal curves (II)
We still assume that (C, G) is a big action.I One can push the “classification ” of big actions
up to the condition |G|
g2C≥ 4
(p2−1)2 . Namely
I One first show that |G2| divides p3.I The condition G2 = G′
1 implies that G2 is abelian.I Applying ([Mr 71]) to the case of abelian
extensions with group Z/pZ × Z/p2Z one showsthat G2 has exponent p (we have seen above thatG2 is cyclic iff G2 = Z/pZ).
I Theorem([Le-Ma 4]) For all M > 0, the set |G|
g2C
> M, for
(C, G) a big action with G2 abelian with exponentp, is finite.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Maximal curves (II)
We still assume that (C, G) is a big action.I One can push the “classification ” of big actions
up to the condition |G|
g2C≥ 4
(p2−1)2 . Namely
I One first show that |G2| divides p3.I The condition G2 = G′
1 implies that G2 is abelian.I Applying ([Mr 71]) to the case of abelian
extensions with group Z/pZ × Z/p2Z one showsthat G2 has exponent p (we have seen above thatG2 is cyclic iff G2 = Z/pZ).
I Theorem([Le-Ma 4]) For all M > 0, the set |G|
g2C
> M, for
(C, G) a big action with G2 abelian with exponentp, is finite.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Maximal curves (II)
We still assume that (C, G) is a big action.I One can push the “classification ” of big actions
up to the condition |G|
g2C≥ 4
(p2−1)2 . Namely
I One first show that |G2| divides p3.I The condition G2 = G′
1 implies that G2 is abelian.I Applying ([Mr 71]) to the case of abelian
extensions with group Z/pZ × Z/p2Z one showsthat G2 has exponent p (we have seen above thatG2 is cyclic iff G2 = Z/pZ).
I Theorem([Le-Ma 4]) For all M > 0, the set |G|
g2C
> M, for
(C, G) a big action with G2 abelian with exponentp, is finite.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Maximal curves (II)
We still assume that (C, G) is a big action.I One can push the “classification ” of big actions
up to the condition |G|
g2C≥ 4
(p2−1)2 . Namely
I One first show that |G2| divides p3.I The condition G2 = G′
1 implies that G2 is abelian.I Applying ([Mr 71]) to the case of abelian
extensions with group Z/pZ × Z/p2Z one showsthat G2 has exponent p (we have seen above thatG2 is cyclic iff G2 = Z/pZ).
I Theorem([Le-Ma 4]) For all M > 0, the set |G|
g2C
> M, for
(C, G) a big action with G2 abelian with exponentp, is finite.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Sketch proof: We saw that |G2| and so t arebounded above. We can assume that the G2
cover is given by the fi , 1 ≤ i ≤ t withm1 ≤ m2 ≤ ... ≤ mt where mi := deg fi and insuch a way thatdeg(
∑
1≤i≤t λi fi) ∈ {mi , 1 ≤ i ≤ t} for[λi ] ∈ Pt−1(Fp).
I If Imρ is trivial.I Then mi − 1 = pνi and ν1 ≤ ... ≤ νtI |G| = pt |V | ≤ pt+2ν1 .I gC = (p−1)
2 (∑
1≤i≤t pi−1pνi )
I M ≤ pt |V |g2 ≤ 4pt
(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2
I νi − ν1 is bounded above.I
p2ν1
|V | ≤ 4pt
M(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2 and so { p2ν1
|V | } is
finite.I { |G|
g2C
= 4pt |V |p−2ν1
(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2 } is finite.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Sketch proof: We saw that |G2| and so t arebounded above. We can assume that the G2
cover is given by the fi , 1 ≤ i ≤ t withm1 ≤ m2 ≤ ... ≤ mt where mi := deg fi and insuch a way thatdeg(
∑
1≤i≤t λi fi) ∈ {mi , 1 ≤ i ≤ t} for[λi ] ∈ Pt−1(Fp).
I If Imρ is trivial.I Then mi − 1 = pνi and ν1 ≤ ... ≤ νtI |G| = pt |V | ≤ pt+2ν1 .I gC = (p−1)
2 (∑
1≤i≤t pi−1pνi )
I M ≤ pt |V |g2 ≤ 4pt
(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2
I νi − ν1 is bounded above.I
p2ν1
|V | ≤ 4pt
M(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2 and so { p2ν1
|V | } is
finite.I { |G|
g2C
= 4pt |V |p−2ν1
(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2 } is finite.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Sketch proof: We saw that |G2| and so t arebounded above. We can assume that the G2
cover is given by the fi , 1 ≤ i ≤ t withm1 ≤ m2 ≤ ... ≤ mt where mi := deg fi and insuch a way thatdeg(
∑
1≤i≤t λi fi) ∈ {mi , 1 ≤ i ≤ t} for[λi ] ∈ Pt−1(Fp).
I If Imρ is trivial.I Then mi − 1 = pνi and ν1 ≤ ... ≤ νtI |G| = pt |V | ≤ pt+2ν1 .I gC = (p−1)
2 (∑
1≤i≤t pi−1pνi )
I M ≤ pt |V |g2 ≤ 4pt
(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2
I νi − ν1 is bounded above.I
p2ν1
|V | ≤ 4pt
M(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2 and so { p2ν1
|V | } is
finite.I { |G|
g2C
= 4pt |V |p−2ν1
(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2 } is finite.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Sketch proof: We saw that |G2| and so t arebounded above. We can assume that the G2
cover is given by the fi , 1 ≤ i ≤ t withm1 ≤ m2 ≤ ... ≤ mt where mi := deg fi and insuch a way thatdeg(
∑
1≤i≤t λi fi) ∈ {mi , 1 ≤ i ≤ t} for[λi ] ∈ Pt−1(Fp).
I If Imρ is trivial.I Then mi − 1 = pνi and ν1 ≤ ... ≤ νtI |G| = pt |V | ≤ pt+2ν1 .I gC = (p−1)
2 (∑
1≤i≤t pi−1pνi )
I M ≤ pt |V |g2 ≤ 4pt
(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2
I νi − ν1 is bounded above.I
p2ν1
|V | ≤ 4pt
M(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2 and so { p2ν1
|V | } is
finite.I { |G|
g2C
= 4pt |V |p−2ν1
(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2 } is finite.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Sketch proof: We saw that |G2| and so t arebounded above. We can assume that the G2
cover is given by the fi , 1 ≤ i ≤ t withm1 ≤ m2 ≤ ... ≤ mt where mi := deg fi and insuch a way thatdeg(
∑
1≤i≤t λi fi) ∈ {mi , 1 ≤ i ≤ t} for[λi ] ∈ Pt−1(Fp).
I If Imρ is trivial.I Then mi − 1 = pνi and ν1 ≤ ... ≤ νtI |G| = pt |V | ≤ pt+2ν1 .I gC = (p−1)
2 (∑
1≤i≤t pi−1pνi )
I M ≤ pt |V |g2 ≤ 4pt
(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2
I νi − ν1 is bounded above.I
p2ν1
|V | ≤ 4pt
M(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2 and so { p2ν1
|V | } is
finite.I { |G|
g2C
= 4pt |V |p−2ν1
(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2 } is finite.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Sketch proof: We saw that |G2| and so t arebounded above. We can assume that the G2
cover is given by the fi , 1 ≤ i ≤ t withm1 ≤ m2 ≤ ... ≤ mt where mi := deg fi and insuch a way thatdeg(
∑
1≤i≤t λi fi) ∈ {mi , 1 ≤ i ≤ t} for[λi ] ∈ Pt−1(Fp).
I If Imρ is trivial.I Then mi − 1 = pνi and ν1 ≤ ... ≤ νtI |G| = pt |V | ≤ pt+2ν1 .I gC = (p−1)
2 (∑
1≤i≤t pi−1pνi )
I M ≤ pt |V |g2 ≤ 4pt
(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2
I νi − ν1 is bounded above.I
p2ν1
|V | ≤ 4pt
M(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2 and so { p2ν1
|V | } is
finite.I { |G|
g2C
= 4pt |V |p−2ν1
(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2 } is finite.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Sketch proof: We saw that |G2| and so t arebounded above. We can assume that the G2
cover is given by the fi , 1 ≤ i ≤ t withm1 ≤ m2 ≤ ... ≤ mt where mi := deg fi and insuch a way thatdeg(
∑
1≤i≤t λi fi) ∈ {mi , 1 ≤ i ≤ t} for[λi ] ∈ Pt−1(Fp).
I If Imρ is trivial.I Then mi − 1 = pνi and ν1 ≤ ... ≤ νtI |G| = pt |V | ≤ pt+2ν1 .I gC = (p−1)
2 (∑
1≤i≤t pi−1pνi )
I M ≤ pt |V |g2 ≤ 4pt
(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2
I νi − ν1 is bounded above.I
p2ν1
|V | ≤ 4pt
M(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2 and so { p2ν1
|V | } is
finite.I { |G|
g2C
= 4pt |V |p−2ν1
(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2 } is finite.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Sketch proof: We saw that |G2| and so t arebounded above. We can assume that the G2
cover is given by the fi , 1 ≤ i ≤ t withm1 ≤ m2 ≤ ... ≤ mt where mi := deg fi and insuch a way thatdeg(
∑
1≤i≤t λi fi) ∈ {mi , 1 ≤ i ≤ t} for[λi ] ∈ Pt−1(Fp).
I If Imρ is trivial.I Then mi − 1 = pνi and ν1 ≤ ... ≤ νtI |G| = pt |V | ≤ pt+2ν1 .I gC = (p−1)
2 (∑
1≤i≤t pi−1pνi )
I M ≤ pt |V |g2 ≤ 4pt
(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2
I νi − ν1 is bounded above.I
p2ν1
|V | ≤ 4pt
M(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2 and so { p2ν1
|V | } is
finite.I { |G|
g2C
= 4pt |V |p−2ν1
(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2 } is finite.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Sketch proof: We saw that |G2| and so t arebounded above. We can assume that the G2
cover is given by the fi , 1 ≤ i ≤ t withm1 ≤ m2 ≤ ... ≤ mt where mi := deg fi and insuch a way thatdeg(
∑
1≤i≤t λi fi) ∈ {mi , 1 ≤ i ≤ t} for[λi ] ∈ Pt−1(Fp).
I If Imρ is trivial.I Then mi − 1 = pνi and ν1 ≤ ... ≤ νtI |G| = pt |V | ≤ pt+2ν1 .I gC = (p−1)
2 (∑
1≤i≤t pi−1pνi )
I M ≤ pt |V |g2 ≤ 4pt
(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2
I νi − ν1 is bounded above.I
p2ν1
|V | ≤ 4pt
M(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2 and so { p2ν1
|V | } is
finite.I { |G|
g2C
= 4pt |V |p−2ν1
(p−1)2(P
1≤i≤t pi−1pνi−ν1 )2 } is finite.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I If Imρ isn’t trivial.I There is a smallest i0 such that
fi0+1(X ) 6= cX + XΣ(F )(X ).I For v ∈ V
fi0+1(X + v) = fi0+1(X ) +∑
1≤i≤i0`i(v)fi(X )
mod ℘(k [X ])I `i is a non zero linear form on the Fp-space V .I Let W := ∩1≤i≤i0 ker `i ; |W | ≥ |V |
pi0.
I gC = (p−1)2 (
∑
1≤i≤t pi−1(mi − 1)) ≥(p−1)
2 (pi0(mi0+1 − 1)).I
2p|W |(p−1)(mi0+1−1) ≤ 2p
p−1
I gC ≥ p−12 pi0(mi0+1 − 1) ≥ p−1
2 |V |
I M ≤ pt |V |g2 ≤ 4pt |V |
(p−1)2|V |2
I |V | is bounded above and g2C ≤ pt |V |
M is alsobounded above .
I { |G|
g2C
= |G2||V |
g2C
} is finite. ///
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I If Imρ isn’t trivial.I There is a smallest i0 such that
fi0+1(X ) 6= cX + XΣ(F )(X ).I For v ∈ V
fi0+1(X + v) = fi0+1(X ) +∑
1≤i≤i0`i(v)fi(X )
mod ℘(k [X ])I `i is a non zero linear form on the Fp-space V .I Let W := ∩1≤i≤i0 ker `i ; |W | ≥ |V |
pi0.
I gC = (p−1)2 (
∑
1≤i≤t pi−1(mi − 1)) ≥(p−1)
2 (pi0(mi0+1 − 1)).I
2p|W |(p−1)(mi0+1−1) ≤ 2p
p−1
I gC ≥ p−12 pi0(mi0+1 − 1) ≥ p−1
2 |V |
I M ≤ pt |V |g2 ≤ 4pt |V |
(p−1)2|V |2
I |V | is bounded above and g2C ≤ pt |V |
M is alsobounded above .
I { |G|
g2C
= |G2||V |
g2C
} is finite. ///
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I If Imρ isn’t trivial.I There is a smallest i0 such that
fi0+1(X ) 6= cX + XΣ(F )(X ).I For v ∈ V
fi0+1(X + v) = fi0+1(X ) +∑
1≤i≤i0`i(v)fi(X )
mod ℘(k [X ])I `i is a non zero linear form on the Fp-space V .I Let W := ∩1≤i≤i0 ker `i ; |W | ≥ |V |
pi0.
I gC = (p−1)2 (
∑
1≤i≤t pi−1(mi − 1)) ≥(p−1)
2 (pi0(mi0+1 − 1)).I
2p|W |(p−1)(mi0+1−1) ≤ 2p
p−1
I gC ≥ p−12 pi0(mi0+1 − 1) ≥ p−1
2 |V |
I M ≤ pt |V |g2 ≤ 4pt |V |
(p−1)2|V |2
I |V | is bounded above and g2C ≤ pt |V |
M is alsobounded above .
I { |G|
g2C
= |G2||V |
g2C
} is finite. ///
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I If Imρ isn’t trivial.I There is a smallest i0 such that
fi0+1(X ) 6= cX + XΣ(F )(X ).I For v ∈ V
fi0+1(X + v) = fi0+1(X ) +∑
1≤i≤i0`i(v)fi(X )
mod ℘(k [X ])I `i is a non zero linear form on the Fp-space V .I Let W := ∩1≤i≤i0 ker `i ; |W | ≥ |V |
pi0.
I gC = (p−1)2 (
∑
1≤i≤t pi−1(mi − 1)) ≥(p−1)
2 (pi0(mi0+1 − 1)).I
2p|W |(p−1)(mi0+1−1) ≤ 2p
p−1
I gC ≥ p−12 pi0(mi0+1 − 1) ≥ p−1
2 |V |
I M ≤ pt |V |g2 ≤ 4pt |V |
(p−1)2|V |2
I |V | is bounded above and g2C ≤ pt |V |
M is alsobounded above .
I { |G|
g2C
= |G2||V |
g2C
} is finite. ///
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I If Imρ isn’t trivial.I There is a smallest i0 such that
fi0+1(X ) 6= cX + XΣ(F )(X ).I For v ∈ V
fi0+1(X + v) = fi0+1(X ) +∑
1≤i≤i0`i(v)fi(X )
mod ℘(k [X ])I `i is a non zero linear form on the Fp-space V .I Let W := ∩1≤i≤i0 ker `i ; |W | ≥ |V |
pi0.
I gC = (p−1)2 (
∑
1≤i≤t pi−1(mi − 1)) ≥(p−1)
2 (pi0(mi0+1 − 1)).I
2p|W |(p−1)(mi0+1−1) ≤ 2p
p−1
I gC ≥ p−12 pi0(mi0+1 − 1) ≥ p−1
2 |V |
I M ≤ pt |V |g2 ≤ 4pt |V |
(p−1)2|V |2
I |V | is bounded above and g2C ≤ pt |V |
M is alsobounded above .
I { |G|
g2C
= |G2||V |
g2C
} is finite. ///
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I If Imρ isn’t trivial.I There is a smallest i0 such that
fi0+1(X ) 6= cX + XΣ(F )(X ).I For v ∈ V
fi0+1(X + v) = fi0+1(X ) +∑
1≤i≤i0`i(v)fi(X )
mod ℘(k [X ])I `i is a non zero linear form on the Fp-space V .I Let W := ∩1≤i≤i0 ker `i ; |W | ≥ |V |
pi0.
I gC = (p−1)2 (
∑
1≤i≤t pi−1(mi − 1)) ≥(p−1)
2 (pi0(mi0+1 − 1)).I
2p|W |(p−1)(mi0+1−1) ≤ 2p
p−1
I gC ≥ p−12 pi0(mi0+1 − 1) ≥ p−1
2 |V |
I M ≤ pt |V |g2 ≤ 4pt |V |
(p−1)2|V |2
I |V | is bounded above and g2C ≤ pt |V |
M is alsobounded above .
I { |G|
g2C
= |G2||V |
g2C
} is finite. ///
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I If Imρ isn’t trivial.I There is a smallest i0 such that
fi0+1(X ) 6= cX + XΣ(F )(X ).I For v ∈ V
fi0+1(X + v) = fi0+1(X ) +∑
1≤i≤i0`i(v)fi(X )
mod ℘(k [X ])I `i is a non zero linear form on the Fp-space V .I Let W := ∩1≤i≤i0 ker `i ; |W | ≥ |V |
pi0.
I gC = (p−1)2 (
∑
1≤i≤t pi−1(mi − 1)) ≥(p−1)
2 (pi0(mi0+1 − 1)).I
2p|W |(p−1)(mi0+1−1) ≤ 2p
p−1
I gC ≥ p−12 pi0(mi0+1 − 1) ≥ p−1
2 |V |
I M ≤ pt |V |g2 ≤ 4pt |V |
(p−1)2|V |2
I |V | is bounded above and g2C ≤ pt |V |
M is alsobounded above .
I { |G|
g2C
= |G2||V |
g2C
} is finite. ///
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I If Imρ isn’t trivial.I There is a smallest i0 such that
fi0+1(X ) 6= cX + XΣ(F )(X ).I For v ∈ V
fi0+1(X + v) = fi0+1(X ) +∑
1≤i≤i0`i(v)fi(X )
mod ℘(k [X ])I `i is a non zero linear form on the Fp-space V .I Let W := ∩1≤i≤i0 ker `i ; |W | ≥ |V |
pi0.
I gC = (p−1)2 (
∑
1≤i≤t pi−1(mi − 1)) ≥(p−1)
2 (pi0(mi0+1 − 1)).I
2p|W |(p−1)(mi0+1−1) ≤ 2p
p−1
I gC ≥ p−12 pi0(mi0+1 − 1) ≥ p−1
2 |V |
I M ≤ pt |V |g2 ≤ 4pt |V |
(p−1)2|V |2
I |V | is bounded above and g2C ≤ pt |V |
M is alsobounded above .
I { |G|
g2C
= |G2||V |
g2C
} is finite. ///
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I If Imρ isn’t trivial.I There is a smallest i0 such that
fi0+1(X ) 6= cX + XΣ(F )(X ).I For v ∈ V
fi0+1(X + v) = fi0+1(X ) +∑
1≤i≤i0`i(v)fi(X )
mod ℘(k [X ])I `i is a non zero linear form on the Fp-space V .I Let W := ∩1≤i≤i0 ker `i ; |W | ≥ |V |
pi0.
I gC = (p−1)2 (
∑
1≤i≤t pi−1(mi − 1)) ≥(p−1)
2 (pi0(mi0+1 − 1)).I
2p|W |(p−1)(mi0+1−1) ≤ 2p
p−1
I gC ≥ p−12 pi0(mi0+1 − 1) ≥ p−1
2 |V |
I M ≤ pt |V |g2 ≤ 4pt |V |
(p−1)2|V |2
I |V | is bounded above and g2C ≤ pt |V |
M is alsobounded above .
I { |G|
g2C
= |G2||V |
g2C
} is finite. ///
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I If Imρ isn’t trivial.I There is a smallest i0 such that
fi0+1(X ) 6= cX + XΣ(F )(X ).I For v ∈ V
fi0+1(X + v) = fi0+1(X ) +∑
1≤i≤i0`i(v)fi(X )
mod ℘(k [X ])I `i is a non zero linear form on the Fp-space V .I Let W := ∩1≤i≤i0 ker `i ; |W | ≥ |V |
pi0.
I gC = (p−1)2 (
∑
1≤i≤t pi−1(mi − 1)) ≥(p−1)
2 (pi0(mi0+1 − 1)).I
2p|W |(p−1)(mi0+1−1) ≤ 2p
p−1
I gC ≥ p−12 pi0(mi0+1 − 1) ≥ p−1
2 |V |
I M ≤ pt |V |g2 ≤ 4pt |V |
(p−1)2|V |2
I |V | is bounded above and g2C ≤ pt |V |
M is alsobounded above .
I { |G|
g2C
= |G2||V |
g2C
} is finite. ///
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I If Imρ isn’t trivial.I There is a smallest i0 such that
fi0+1(X ) 6= cX + XΣ(F )(X ).I For v ∈ V
fi0+1(X + v) = fi0+1(X ) +∑
1≤i≤i0`i(v)fi(X )
mod ℘(k [X ])I `i is a non zero linear form on the Fp-space V .I Let W := ∩1≤i≤i0 ker `i ; |W | ≥ |V |
pi0.
I gC = (p−1)2 (
∑
1≤i≤t pi−1(mi − 1)) ≥(p−1)
2 (pi0(mi0+1 − 1)).I
2p|W |(p−1)(mi0+1−1) ≤ 2p
p−1
I gC ≥ p−12 pi0(mi0+1 − 1) ≥ p−1
2 |V |
I M ≤ pt |V |g2 ≤ 4pt |V |
(p−1)2|V |2
I |V | is bounded above and g2C ≤ pt |V |
M is alsobounded above .
I { |G|
g2C
= |G2||V |
g2C
} is finite. ///
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Monodromy polynomial ([Le-Ma 3])
I Let C −→ P1K birationally given by the equation:
Z p0 = f (X0) =
∏
1≤i≤m(X0 − xi)ni ∈ R[X0],
(ni , p) = 1 and (deg f , p) = 1,v(xi − xj) = v(xi ) = 0 for i 6= j .
I f ′(Y )/f (Y ) = S1(Y )/S0(Y ), (S0(Y ), S1(Y )) = 1 ;then deg(S1(Y )) = m − 1 and deg(S0(Y )) = m .
I f (X + Y ) = f (Y )((1 + a1(Y )X + ... + ar (Y )X r )p −∑
r+1≤i≤n Ai(Y )X i), où r + 1 = [n/p],ai(Y ), Ai(Y ) ∈ K (Y ).
I There is a unique α such that r < pα < n < pα+1
I There is T (Y ) ∈ R[Y ] with
Apα(Y ) = −(
1p
pα−1
)
p.S1(Y )pα
+pT (Y )S0(Y )pα .
I L(Y ) := S1(Y )pα
+ pT (Y ). This is a polynomial ofdegree pα(m − 1) which is called the monodromypolynomial of f (Y ).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Monodromy polynomial ([Le-Ma 3])
I Let C −→ P1K birationally given by the equation:
Z p0 = f (X0) =
∏
1≤i≤m(X0 − xi)ni ∈ R[X0],
(ni , p) = 1 and (deg f , p) = 1,v(xi − xj) = v(xi ) = 0 for i 6= j .
I f ′(Y )/f (Y ) = S1(Y )/S0(Y ), (S0(Y ), S1(Y )) = 1 ;then deg(S1(Y )) = m − 1 and deg(S0(Y )) = m .
I f (X + Y ) = f (Y )((1 + a1(Y )X + ... + ar (Y )X r )p −∑
r+1≤i≤n Ai(Y )X i), où r + 1 = [n/p],ai(Y ), Ai(Y ) ∈ K (Y ).
I There is a unique α such that r < pα < n < pα+1
I There is T (Y ) ∈ R[Y ] with
Apα(Y ) = −(
1p
pα−1
)
p.S1(Y )pα
+pT (Y )S0(Y )pα .
I L(Y ) := S1(Y )pα
+ pT (Y ). This is a polynomial ofdegree pα(m − 1) which is called the monodromypolynomial of f (Y ).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Monodromy polynomial ([Le-Ma 3])
I Let C −→ P1K birationally given by the equation:
Z p0 = f (X0) =
∏
1≤i≤m(X0 − xi)ni ∈ R[X0],
(ni , p) = 1 and (deg f , p) = 1,v(xi − xj) = v(xi ) = 0 for i 6= j .
I f ′(Y )/f (Y ) = S1(Y )/S0(Y ), (S0(Y ), S1(Y )) = 1 ;then deg(S1(Y )) = m − 1 and deg(S0(Y )) = m .
I f (X + Y ) = f (Y )((1 + a1(Y )X + ... + ar (Y )X r )p −∑
r+1≤i≤n Ai(Y )X i), où r + 1 = [n/p],ai(Y ), Ai(Y ) ∈ K (Y ).
I There is a unique α such that r < pα < n < pα+1
I There is T (Y ) ∈ R[Y ] with
Apα(Y ) = −(
1p
pα−1
)
p.S1(Y )pα
+pT (Y )S0(Y )pα .
I L(Y ) := S1(Y )pα
+ pT (Y ). This is a polynomial ofdegree pα(m − 1) which is called the monodromypolynomial of f (Y ).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Monodromy polynomial ([Le-Ma 3])
I Let C −→ P1K birationally given by the equation:
Z p0 = f (X0) =
∏
1≤i≤m(X0 − xi)ni ∈ R[X0],
(ni , p) = 1 and (deg f , p) = 1,v(xi − xj) = v(xi ) = 0 for i 6= j .
I f ′(Y )/f (Y ) = S1(Y )/S0(Y ), (S0(Y ), S1(Y )) = 1 ;then deg(S1(Y )) = m − 1 and deg(S0(Y )) = m .
I f (X + Y ) = f (Y )((1 + a1(Y )X + ... + ar (Y )X r )p −∑
r+1≤i≤n Ai(Y )X i), où r + 1 = [n/p],ai(Y ), Ai(Y ) ∈ K (Y ).
I There is a unique α such that r < pα < n < pα+1
I There is T (Y ) ∈ R[Y ] with
Apα(Y ) = −(
1p
pα−1
)
p.S1(Y )pα
+pT (Y )S0(Y )pα .
I L(Y ) := S1(Y )pα
+ pT (Y ). This is a polynomial ofdegree pα(m − 1) which is called the monodromypolynomial of f (Y ).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Monodromy polynomial ([Le-Ma 3])
I Let C −→ P1K birationally given by the equation:
Z p0 = f (X0) =
∏
1≤i≤m(X0 − xi)ni ∈ R[X0],
(ni , p) = 1 and (deg f , p) = 1,v(xi − xj) = v(xi ) = 0 for i 6= j .
I f ′(Y )/f (Y ) = S1(Y )/S0(Y ), (S0(Y ), S1(Y )) = 1 ;then deg(S1(Y )) = m − 1 and deg(S0(Y )) = m .
I f (X + Y ) = f (Y )((1 + a1(Y )X + ... + ar (Y )X r )p −∑
r+1≤i≤n Ai(Y )X i), où r + 1 = [n/p],ai(Y ), Ai(Y ) ∈ K (Y ).
I There is a unique α such that r < pα < n < pα+1
I There is T (Y ) ∈ R[Y ] with
Apα(Y ) = −(
1p
pα−1
)
p.S1(Y )pα
+pT (Y )S0(Y )pα .
I L(Y ) := S1(Y )pα
+ pT (Y ). This is a polynomial ofdegree pα(m − 1) which is called the monodromypolynomial of f (Y ).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Monodromy polynomial ([Le-Ma 3])
I Let C −→ P1K birationally given by the equation:
Z p0 = f (X0) =
∏
1≤i≤m(X0 − xi)ni ∈ R[X0],
(ni , p) = 1 and (deg f , p) = 1,v(xi − xj) = v(xi ) = 0 for i 6= j .
I f ′(Y )/f (Y ) = S1(Y )/S0(Y ), (S0(Y ), S1(Y )) = 1 ;then deg(S1(Y )) = m − 1 and deg(S0(Y )) = m .
I f (X + Y ) = f (Y )((1 + a1(Y )X + ... + ar (Y )X r )p −∑
r+1≤i≤n Ai(Y )X i), où r + 1 = [n/p],ai(Y ), Ai(Y ) ∈ K (Y ).
I There is a unique α such that r < pα < n < pα+1
I There is T (Y ) ∈ R[Y ] with
Apα(Y ) = −(
1p
pα−1
)
p.S1(Y )pα
+pT (Y )S0(Y )pα .
I L(Y ) := S1(Y )pα
+ pT (Y ). This is a polynomial ofdegree pα(m − 1) which is called the monodromypolynomial of f (Y ).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Special fiber of the easy model
We mean the R-model CR defined byZ p
0 = f (X0) =∏
1≤i≤m(X0 − xi)ni ∈ R[X0] (cf. fig 1).
FIG.: CR ⊗R k −→ P1k with singularities and branch locus
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Theorem([Le-Ma 3]) The components with genus > 0 ofthe marked stable model of C correspondbijectively to the Gauss valuations vXj
withρjXj = X0 − yj , where yj is a zero of themonodromy polynomial L(Y )
I ρj ∈ Ralg satisfies
v(ρj ) = max{1i v
(
λp
Ai (yj )
)
for r + 1 ≤ i ≤ n}.
I The dual graph of the special fiber of the markedstable model of C is an oriented tree whose endsare in bijection with the components of genus > 0.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Theorem([Le-Ma 3]) The components with genus > 0 ofthe marked stable model of C correspondbijectively to the Gauss valuations vXj
withρjXj = X0 − yj , where yj is a zero of themonodromy polynomial L(Y )
I ρj ∈ Ralg satisfies
v(ρj ) = max{1i v
(
λp
Ai (yj )
)
for r + 1 ≤ i ≤ n}.
I The dual graph of the special fiber of the markedstable model of C is an oriented tree whose endsare in bijection with the components of genus > 0.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I Theorem([Le-Ma 3]) The components with genus > 0 ofthe marked stable model of C correspondbijectively to the Gauss valuations vXj
withρjXj = X0 − yj , where yj is a zero of themonodromy polynomial L(Y )
I ρj ∈ Ralg satisfies
v(ρj ) = max{1i v
(
λp
Ai (yj )
)
for r + 1 ≤ i ≤ n}.
I The dual graph of the special fiber of the markedstable model of C is an oriented tree whose endsare in bijection with the components of genus > 0.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Potentially good reduction with m = 1 + ps
Theorem([Le-Ma 3])
I p > 2, q = pn, n ≥ 1, K = Qurp (pp/(q+1)) and
C −→ P1K is birationally defined by the equation
Z p0 = f (X0) = 1 + pp/(q+1)X q
0 + X q+10 .
I Then, C has potentially good reduction and L(Y )is irreducible over K .
I The monodromy L/K is the extension of thedecomposition field of L(Y ) obtained by adjoiningthe p-roots f (y)1/p , for y describing the zeroes ofL(Y ).
I The monodromy group is the extraspecial groupwith exponent p2 and order pq2 (which is maximalfor this conductor).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Potentially good reduction with m = 1 + ps
Theorem([Le-Ma 3])
I p > 2, q = pn, n ≥ 1, K = Qurp (pp/(q+1)) and
C −→ P1K is birationally defined by the equation
Z p0 = f (X0) = 1 + pp/(q+1)X q
0 + X q+10 .
I Then, C has potentially good reduction and L(Y )is irreducible over K .
I The monodromy L/K is the extension of thedecomposition field of L(Y ) obtained by adjoiningthe p-roots f (y)1/p , for y describing the zeroes ofL(Y ).
I The monodromy group is the extraspecial groupwith exponent p2 and order pq2 (which is maximalfor this conductor).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Potentially good reduction with m = 1 + ps
Theorem([Le-Ma 3])
I p > 2, q = pn, n ≥ 1, K = Qurp (pp/(q+1)) and
C −→ P1K is birationally defined by the equation
Z p0 = f (X0) = 1 + pp/(q+1)X q
0 + X q+10 .
I Then, C has potentially good reduction and L(Y )is irreducible over K .
I The monodromy L/K is the extension of thedecomposition field of L(Y ) obtained by adjoiningthe p-roots f (y)1/p , for y describing the zeroes ofL(Y ).
I The monodromy group is the extraspecial groupwith exponent p2 and order pq2 (which is maximalfor this conductor).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Potentially good reduction with m = 1 + ps
Theorem([Le-Ma 3])
I p > 2, q = pn, n ≥ 1, K = Qurp (pp/(q+1)) and
C −→ P1K is birationally defined by the equation
Z p0 = f (X0) = 1 + pp/(q+1)X q
0 + X q+10 .
I Then, C has potentially good reduction and L(Y )is irreducible over K .
I The monodromy L/K is the extension of thedecomposition field of L(Y ) obtained by adjoiningthe p-roots f (y)1/p , for y describing the zeroes ofL(Y ).
I The monodromy group is the extraspecial groupwith exponent p2 and order pq2 (which is maximalfor this conductor).
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Curves of genus 2 ([Le-Ma 3])
I Case p = 2 and m = 5 ( i.e. curves with genus 2over a 2-adic field ⊂ Qtame
2 ).I There are 3 types of degeneration for the marked
stable model.I
genus 1curves
genus 1curves
genus 2curveP1
k
P1k
Type 1Gal(K ′/K )w ↪→ Q8 × Q8 Gal(K ′/K )w ↪→ (Q8 × Q8) o Z/2Z Gal(K ′/K )w ↪→ Q8 ∗ D8
original component
Type 2
original componentP1
k original componentP1
k
Type 3
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Curves of genus 2 ([Le-Ma 3])
I Case p = 2 and m = 5 ( i.e. curves with genus 2over a 2-adic field ⊂ Qtame
2 ).I There are 3 types of degeneration for the marked
stable model.I
genus 1curves
genus 1curves
genus 2curveP1
k
P1k
Type 1Gal(K ′/K )w ↪→ Q8 × Q8 Gal(K ′/K )w ↪→ (Q8 × Q8) o Z/2Z Gal(K ′/K )w ↪→ Q8 ∗ D8
original component
Type 2
original componentP1
k original componentP1
k
Type 3
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
Curves of genus 2 ([Le-Ma 3])
I Case p = 2 and m = 5 ( i.e. curves with genus 2over a 2-adic field ⊂ Qtame
2 ).I There are 3 types of degeneration for the marked
stable model.I
genus 1curves
genus 1curves
genus 2curveP1
k
P1k
Type 1Gal(K ′/K )w ↪→ Q8 × Q8 Gal(K ′/K )w ↪→ (Q8 × Q8) o Z/2Z Gal(K ′/K )w ↪→ Q8 ∗ D8
original component
Type 2
original componentP1
k original componentP1
k
Type 3
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I C −→ P1K is birationally defined by the equation
Z p0 = f (X0) with
f (X0) = 1 + b2X 20 + b3X 3
0 + b4X 40 + X 5
0 ∈ R[X0].I Now, we see that the monodromy can be maximal
for the 3 types of degeneration.
I a) f (X0) = 1 + 23/5X 20 + X 3
0 + 22/5X 40 + X 5
0 andK = Qur
2 (21/15) ;I C has a marked stable model of type 1.I The maximal monodromy group is ' Q8 × Q8.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I C −→ P1K is birationally defined by the equation
Z p0 = f (X0) with
f (X0) = 1 + b2X 20 + b3X 3
0 + b4X 40 + X 5
0 ∈ R[X0].I Now, we see that the monodromy can be maximal
for the 3 types of degeneration.
I a) f (X0) = 1 + 23/5X 20 + X 3
0 + 22/5X 40 + X 5
0 andK = Qur
2 (21/15) ;I C has a marked stable model of type 1.I The maximal monodromy group is ' Q8 × Q8.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I C −→ P1K is birationally defined by the equation
Z p0 = f (X0) with
f (X0) = 1 + b2X 20 + b3X 3
0 + b4X 40 + X 5
0 ∈ R[X0].I Now, we see that the monodromy can be maximal
for the 3 types of degeneration.
I a) f (X0) = 1 + 23/5X 20 + X 3
0 + 22/5X 40 + X 5
0 andK = Qur
2 (21/15) ;I C has a marked stable model of type 1.I The maximal monodromy group is ' Q8 × Q8.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I C −→ P1K is birationally defined by the equation
Z p0 = f (X0) with
f (X0) = 1 + b2X 20 + b3X 3
0 + b4X 40 + X 5
0 ∈ R[X0].I Now, we see that the monodromy can be maximal
for the 3 types of degeneration.
I a) f (X0) = 1 + 23/5X 20 + X 3
0 + 22/5X 40 + X 5
0 andK = Qur
2 (21/15) ;I C has a marked stable model of type 1.I The maximal monodromy group is ' Q8 × Q8.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I C −→ P1K is birationally defined by the equation
Z p0 = f (X0) with
f (X0) = 1 + b2X 20 + b3X 3
0 + b4X 40 + X 5
0 ∈ R[X0].I Now, we see that the monodromy can be maximal
for the 3 types of degeneration.
I a) f (X0) = 1 + 23/5X 20 + X 3
0 + 22/5X 40 + X 5
0 andK = Qur
2 (21/15) ;I C has a marked stable model of type 1.I The maximal monodromy group is ' Q8 × Q8.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I b) Let K = Qur2 (a) with a9 = 2 and
f (X0) = 1 + a3X 20 + a6X 3
0 + X 50 .
I C has a marked stable model of type 2.I The maximal monodromy group is
' (Q8 × Q8) o Z/2Z, where Z/2Z exchanges the2 factors.
I c) K = Qur2 and . f (X0) = 1 + X 4
0 + X 50 .
I C has potentially good reduction (i.e. is of type 3)I The maximal monodromy group is ' Q8 ∗ D8.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I b) Let K = Qur2 (a) with a9 = 2 and
f (X0) = 1 + a3X 20 + a6X 3
0 + X 50 .
I C has a marked stable model of type 2.I The maximal monodromy group is
' (Q8 × Q8) o Z/2Z, where Z/2Z exchanges the2 factors.
I c) K = Qur2 and . f (X0) = 1 + X 4
0 + X 50 .
I C has potentially good reduction (i.e. is of type 3)I The maximal monodromy group is ' Q8 ∗ D8.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I b) Let K = Qur2 (a) with a9 = 2 and
f (X0) = 1 + a3X 20 + a6X 3
0 + X 50 .
I C has a marked stable model of type 2.I The maximal monodromy group is
' (Q8 × Q8) o Z/2Z, where Z/2Z exchanges the2 factors.
I c) K = Qur2 and . f (X0) = 1 + X 4
0 + X 50 .
I C has potentially good reduction (i.e. is of type 3)I The maximal monodromy group is ' Q8 ∗ D8.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I b) Let K = Qur2 (a) with a9 = 2 and
f (X0) = 1 + a3X 20 + a6X 3
0 + X 50 .
I C has a marked stable model of type 2.I The maximal monodromy group is
' (Q8 × Q8) o Z/2Z, where Z/2Z exchanges the2 factors.
I c) K = Qur2 and . f (X0) = 1 + X 4
0 + X 50 .
I C has potentially good reduction (i.e. is of type 3)I The maximal monodromy group is ' Q8 ∗ D8.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I b) Let K = Qur2 (a) with a9 = 2 and
f (X0) = 1 + a3X 20 + a6X 3
0 + X 50 .
I C has a marked stable model of type 2.I The maximal monodromy group is
' (Q8 × Q8) o Z/2Z, where Z/2Z exchanges the2 factors.
I c) K = Qur2 and . f (X0) = 1 + X 4
0 + X 50 .
I C has potentially good reduction (i.e. is of type 3)I The maximal monodromy group is ' Q8 ∗ D8.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
I b) Let K = Qur2 (a) with a9 = 2 and
f (X0) = 1 + a3X 20 + a6X 3
0 + X 50 .
I C has a marked stable model of type 2.I The maximal monodromy group is
' (Q8 × Q8) o Z/2Z, where Z/2Z exchanges the2 factors.
I c) K = Qur2 and . f (X0) = 1 + X 4
0 + X 50 .
I C has potentially good reduction (i.e. is of type 3)I The maximal monodromy group is ' Q8 ∗ D8.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
R. Auer, Ray class fields of global function fieldswith many rational places, Acta Arith. 95 (2000),no. 2, 97–122.
M. Conder, Hurwitz groups: a brief survey, Bull.Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 359–370.
N. Elkies, The Klein quartic in number theory, Theeightfold way, 51–101, Math. Sci. Res. Inst. Publ.,35, Cambridge Univ. Press, Cambridge, 1999.
K. Lauter, A formula for constructing curves overfinite fields with many rational points, J. NumberTheory 74 (1999), no. 1, 56–72.
C. Lehr, M. Matignon, Automorphism groups forp-cyclic covers of the affine line, Compos. Math.141 (2005), no. 5, 1213–1237.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
C. Lehr, M. Matignon,Automorphisms of curvesand stable reduction, in Problems from theworkshop on "Automorphisms of Curves" (Leiden,August, 2004), edited by G. Cornelissen andF.Oort, Rend. Sem. Math. Univ. Padova. Vol. 113(2005), 151-158.
C. Lehr, M. Matignon, Wild monodromy andautomorphisms of curves , Duke math. J. àparaître.
C. Lehr, M. Matignon, Curves with a big p-groupaction, En préparation.
A. M. Macbeath, On a theorem of Hurwitz, Proc.Glasgow Math. Assoc. 5 1961 90–96 (1961).
A. M. Macbeath, On a curve of genus 7, Proc.London Math. Soc. (3) 15 1965 527–542.
Automorphisms
C. LEHR , M. MATIGNON
Plan
IntroductionMonodromy and automorphism groups
Automorphism groupsCovers of the affine lineStructure of G∞,1(f )Bounds for |G∞,1(f )|Characterization of G∞,1(f )
Actions of p-groupsNakajima conditionMore about G2Riemann surfacesRay class fieldsBig actionsMaximal curves
MonodromyMarked stable modelPotentially good reductionGenus 2
Références
M. Marshall, Ramification groups of abelian localfield extensions, Canad. J. Math. 23 (1971)271–281.
S. Nakajima, p-ranks and automorphism groups ofalgebraic curves, Trans. Amer. Math. Soc. 303,595-607 (1987).
M. Raynaud, p-groupes et réduction semi-stabledes courbes, The Grothendieck Festschrift, Vol.3,Basel-Boston-Berlin: Birkhäuser (1990).
B. Singh, On the group of automorphisms offunction field of genus at least two, J. Pure Appl.Algebra 4 (1974), 205–229.
H. Stichtenoth, Über die Automorphismengruppeeines algebraischen Funktionenkörpers vonPrimzahlcharakteristik. I, II. Arch. Math. 24 (1973)527–544 , 615–631.
M. Suzuki, Group theory II, Grundlehren derMathematischen Wissenschaften 248.Springer-Verlag, New York, 1986.