p-Groups and Automorphism groups of curves in char. p > 0

(1)

p-Groups and Automorphism groups of curves in char. p>0

M. MATIGNON

Plan Introduction

Monodromy and automorphism groups

Automorphism groups

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

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

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

Monodromy

Marked stable model Potentially good reduction Genus 2

Références

p-Groups and Automorphism groups of curves in char. p > 0

M. M ATIGNON ¹

1

Laboratoire A2X Bordeaux

GTEM Kick-Off Seminar October 2006

(2)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

Plan

Introduction

Monodromy and automorphism groups Automorphism groups of curves in char.p > 0

p -cyclic covers of the affine line Structure of G _∞,1 (f )

Bounds for |G _∞,1 (f )|

Characterization of G _∞,1 (f )

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

Condition (N) and G ₂ Riemann surfaces Ray class fields Maximal curves Monodromy polynomial

Marked stable model Potentially good reduction Genus 2

References

(3)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

Plan

Introduction

Monodromy and automorphism groups Automorphism groups of curves in char.p > 0

p -cyclic covers of the affine line Structure of G _∞,1 (f )

Bounds for |G _∞,1 (f )|

Characterization of G _∞,1 (f )

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

Condition (N) and G ₂ Riemann surfaces Ray class fields Maximal curves Monodromy polynomial

Marked stable model Potentially good reduction Genus 2

References

(4)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

Plan

Introduction

Monodromy and automorphism groups Automorphism groups of curves in char.p > 0

p -cyclic covers of the affine line Structure of G _∞,1 (f )

Bounds for |G _∞,1 (f )|

Characterization of G _∞,1 (f )

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

Condition (N) and G ₂ Riemann surfaces Ray class fields Maximal curves Monodromy polynomial

Marked stable model Potentially good reduction Genus 2

References

(5)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

Plan

Introduction

Monodromy and automorphism groups Automorphism groups of curves in char.p > 0

p -cyclic covers of the affine line Structure of G _∞,1 (f )

Bounds for |G _∞,1 (f )|

Characterization of G _∞,1 (f )

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

Condition (N) and G ₂ Riemann surfaces Ray class fields Maximal curves Monodromy polynomial

Marked stable model Potentially good reduction Genus 2

References

(6)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

Plan

Introduction

Monodromy and automorphism groups Automorphism groups of curves in char.p > 0

p -cyclic covers of the affine line Structure of G _∞,1 (f )

Bounds for |G _∞,1 (f )|

Characterization of G _∞,1 (f )

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

Condition (N) and G ₂ Riemann surfaces Ray class fields Maximal curves Monodromy polynomial

Marked stable model Potentially good reduction Genus 2

References

(7)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

Monodromy and automorphism groups.

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

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

π a uniformizing parameter.

k := R _K /πR K .

C/K smooth projective curve, g(C) ≥ 1.

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

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

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

I Its p-Sylow subgroup is the wild monodromy

group .

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M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

Monodromy and automorphism groups.

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

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

π a uniformizing parameter.

k := R _K /πR K .

C/K smooth projective curve, g(C) ≥ 1.

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

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

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

I Its p-Sylow subgroup is the wild monodromy

group .

(9)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

Monodromy and automorphism groups.

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

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

π a uniformizing parameter.

k := R _K /πR K .

C/K smooth projective curve, g(C) ≥ 1.

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

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

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

I Its p-Sylow subgroup is the wild monodromy

group .

(10)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

Monodromy and automorphism groups.

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

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

π a uniformizing parameter.

k := R _K /πR K .

C/K smooth projective curve, g(C) ≥ 1.

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

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

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

I Its p-Sylow subgroup is the wild monodromy

group .

(11)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

Monodromy and automorphism groups.

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

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

π a uniformizing parameter.

k := R _K /πR K .

C/K smooth projective curve, g(C) ≥ 1.

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

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

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

I Its p-Sylow subgroup is the wild monodromy

group .

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M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

I The base change C × _K K ^alg induces an

homomorphism ϕ : Gal(K ^alg /K ) → Aut k C _s , where C _s is the special fiber of the smooth model over R _L and L = (K ^alg ) ^ker ^ϕ .

I Let ` be a prime number, then, n _` := v _` (|Gal(L/K )|) ≤ v _` (|Aut _k C s |).

I If ` / ∈ {2, p}, then n _` ≤ 2g.

I If p > 2, then

n p ≤ inf _`6=2,p v p (|GL _2g ( Z /` Z )|) = a + [a/p] + ..., where a = [ _p−1 ^2g ].

I This gives an exponential type bound in g for

|Aut k C _s |. This justifies our interest in looking at

polynomial bounds, Stichtenoth ([St 73]) and

Singh ([Si 73]).

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M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

I The base change C × _K K ^alg induces an

homomorphism ϕ : Gal(K ^alg /K ) → Aut k C _s , where C _s is the special fiber of the smooth model over R _L and L = (K ^alg ) ^ker ^ϕ .

I Let ` be a prime number, then, n _` := v _` (|Gal(L/K )|) ≤ v _` (|Aut _k C s |).

I If ` / ∈ {2, p}, then n _` ≤ 2g.

I If p > 2, then

n p ≤ inf _`6=2,p v p (|GL _2g ( Z /` Z )|) = a + [a/p] + ..., where a = [ _p−1 ^2g ].

I This gives an exponential type bound in g for

|Aut k C _s |. This justifies our interest in looking at

polynomial bounds, Stichtenoth ([St 73]) and

Singh ([Si 73]).

(14)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

I The base change C × _K K ^alg induces an

homomorphism ϕ : Gal(K ^alg /K ) → Aut k C _s , where C _s is the special fiber of the smooth model over R _L and L = (K ^alg ) ^ker ^ϕ .

I Let ` be a prime number, then, n _` := v _` (|Gal(L/K )|) ≤ v _` (|Aut _k C s |).

I If ` / ∈ {2, p}, then n _` ≤ 2g.

I If p > 2, then

n p ≤ inf _`6=2,p v p (|GL _2g ( Z /` Z )|) = a + [a/p] + ..., where a = [ _p−1 ^2g ].

I This gives an exponential type bound in g for

|Aut k C _s |. This justifies our interest in looking at

polynomial bounds, Stichtenoth ([St 73]) and

Singh ([Si 73]).

(15)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

I The base change C × _K K ^alg induces an

homomorphism ϕ : Gal(K ^alg /K ) → Aut k C _s , where C _s is the special fiber of the smooth model over R _L and L = (K ^alg ) ^ker ^ϕ .

I Let ` be a prime number, then, n _` := v _` (|Gal(L/K )|) ≤ v _` (|Aut _k C s |).

I If ` / ∈ {2, p}, then n _` ≤ 2g.

I If p > 2, then

n p ≤ inf _`6=2,p v p (|GL _2g ( Z /` Z )|) = a + [a/p] + ..., where a = [ _p−1 ^2g ].

I This gives an exponential type bound in g for

|Aut k C _s |. This justifies our interest in looking at

polynomial bounds, Stichtenoth ([St 73]) and

Singh ([Si 73]).

(16)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

I The base change C × _K K ^alg induces an

homomorphism ϕ : Gal(K ^alg /K ) → Aut k C _s , where C _s is the special fiber of the smooth model over R _L and L = (K ^alg ) ^ker ^ϕ .

I Let ` be a prime number, then, n _` := v _` (|Gal(L/K )|) ≤ v _` (|Aut _k C s |).

I If ` / ∈ {2, p}, then n _` ≤ 2g.

I If p > 2, then

n p ≤ inf _`6=2,p v p (|GL _2g ( Z /` Z )|) = a + [a/p] + ..., where a = [ _p−1 ^2g ].

I This gives an exponential type bound in g for

|Aut k C _s |. This justifies our interest in looking at

polynomial bounds, Stichtenoth ([St 73]) and

Singh ([Si 73]).

(17)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

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.

I C _f : W ^p − W = f (X ). Let ∞ be the point of C _f above X = ∞ and z a local parameter. Then, g := g(C _f ) = ^p−1 ₂ (m − 1) > 0.

I G _∞ (f ) := {σ ∈ Aut k C _f | σ(∞) = ∞}.

I G _∞,1 (f ) := {σ ∈ Aut k C _f | v _∞ (σ(z) − z) ≥ 2} , the p-Sylow.

I ([St 73]) Let g(C _f ) ≥ 2, then G _∞,1 (f ) is a p-Sylow of Aut _k C _f .

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

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M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

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.

I C _f : W ^p − W = f (X ). Let ∞ be the point of C _f above X = ∞ and z a local parameter. Then, g := g(C _f ) = ^p−1 ₂ (m − 1) > 0.

I G _∞ (f ) := {σ ∈ Aut k C _f | σ(∞) = ∞}.

I G _∞,1 (f ) := {σ ∈ Aut k C _f | v _∞ (σ(z) − z) ≥ 2} , the p-Sylow.

I ([St 73]) Let g(C _f ) ≥ 2, then G _∞,1 (f ) is a p-Sylow of Aut _k C _f .

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

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M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

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.

I C _f : W ^p − W = f (X ). Let ∞ be the point of C _f above X = ∞ and z a local parameter. Then, g := g(C _f ) = ^p−1 ₂ (m − 1) > 0.

I G _∞ (f ) := {σ ∈ Aut k C _f | σ(∞) = ∞}.

I G _∞,1 (f ) := {σ ∈ Aut k C _f | v _∞ (σ(z) − z) ≥ 2} , the p-Sylow.

I ([St 73]) Let g(C _f ) ≥ 2, then G _∞,1 (f ) is a p-Sylow of Aut _k C _f .

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

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M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

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.

I C _f : W ^p − W = f (X ). Let ∞ be the point of C _f above X = ∞ and z a local parameter. Then, g := g(C _f ) = ^p−1 ₂ (m − 1) > 0.

I G _∞ (f ) := {σ ∈ Aut k C _f | σ(∞) = ∞}.

I G _∞,1 (f ) := {σ ∈ Aut k C _f | v _∞ (σ(z) − z) ≥ 2} , the p-Sylow.

I ([St 73]) Let g(C _f ) ≥ 2, then G _∞,1 (f ) is a p-Sylow of Aut _k C _f .

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

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M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

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.

I C _f : W ^p − W = f (X ). Let ∞ be the point of C _f above X = ∞ and z a local parameter. Then, g := g(C _f ) = ^p−1 ₂ (m − 1) > 0.

I G _∞ (f ) := {σ ∈ Aut k C _f | σ(∞) = ∞}.

I G _∞,1 (f ) := {σ ∈ Aut k C _f | v _∞ (σ(z) − z) ≥ 2} , the p-Sylow.

I ([St 73]) Let g(C _f ) ≥ 2, then G _∞,1 (f ) is a p-Sylow of Aut _k C _f .

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

(22)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

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.

I C _f : W ^p − W = f (X ). Let ∞ be the point of C _f above X = ∞ and z a local parameter. Then, g := g(C _f ) = ^p−1 ₂ (m − 1) > 0.

I G _∞ (f ) := {σ ∈ Aut k C _f | σ(∞) = ∞}.

I G _∞,1 (f ) := {σ ∈ Aut k C _f | v _∞ (σ(z) − z) ≥ 2} , the p-Sylow.

I ([St 73]) Let g(C _f ) ≥ 2, then G _∞,1 (f ) is a p-Sylow of Aut _k C _f .

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

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M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

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 = {τ y | τ y (X) = X + y, y ∈ k }.

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

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

I Let τ _y (W ) := W + a _y + P(X , y ), a _y ∈ F p , then [τ y , τ z ] = ρ ^(y,z) , where : V × V → F p is an alternating form.

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

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M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

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 = {τ y | τ y (X) = X + y, y ∈ k }.

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

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

I Let τ _y (W ) := W + a _y + P(X , y ), a _y ∈ F p , then [τ y , τ z ] = ρ ^(y,z) , where : V × V → F p is an alternating form.

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

(25)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

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 = {τ y | τ y (X) = X + y, y ∈ k }.

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

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

I Let τ _y (W ) := W + a _y + P(X , y ), a _y ∈ F p , then [τ y , τ z ] = ρ ^(y,z) , where : V × V → F p is an alternating form.

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

(26)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

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 = {τ y | τ y (X) = X + y, y ∈ k }.

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

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

I Let τ _y (W ) := W + a _y + P(X , y ), a _y ∈ F p , then [τ y , τ z ] = ρ ^(y,z) , where : V × V → F p is an alternating form.

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

(27)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

Bounds for |G _∞,1 (f )|

Lemma If f (X ) = P

1≤i≤m t _i X ⁱ ∈ k[X ] is monic, then :

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

where R ∈ L

b

^m_p

c≤ip

ⁿ⁽ⁱ⁾

<m, (i,p)=1 k [Y ]X ^ip

ⁿ⁽ⁱ⁾

and P _f ∈ Xk[X , Y ].

I P _f = (Id + F + ... + F ⁿ⁻¹ )(∆(f )) mod X ^[

^m−1^p

^]+1 .

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

I Ad _f (Y ) is an additive and separable polynomial.

I Z (Ad f (Y )) ' V .

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M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

Bounds for |G _∞,1 (f )|

Lemma If f (X ) = P

1≤i≤m t _i X ⁱ ∈ k[X ] is monic, then :

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

where R ∈ L

b

^m_p

c≤ip

ⁿ⁽ⁱ⁾

<m, (i,p)=1 k [Y ]X ^ip

ⁿ⁽ⁱ⁾

and P _f ∈ Xk[X , Y ].

I P _f = (Id + F + ... + F ⁿ⁻¹ )(∆(f )) mod X ^[

^m−1^p

^]+1 .

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

I Ad _f (Y ) is an additive and separable polynomial.

I Z (Ad f (Y )) ' V .

(29)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

Bounds for |G _∞,1 (f )|

Lemma If f (X ) = P

1≤i≤m t _i X ⁱ ∈ k[X ] is monic, then :

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

where R ∈ L

b

^m_p

c≤ip

ⁿ⁽ⁱ⁾

<m, (i,p)=1 k [Y ]X ^ip

ⁿ⁽ⁱ⁾

and P _f ∈ Xk[X , Y ].

I P _f = (Id + F + ... + F ⁿ⁻¹ )(∆(f )) mod X ^[

^m−1^p

^]+1 .

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

I Ad _f (Y ) is an additive and separable polynomial.

I Z (Ad f (Y )) ' V .

(30)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

Bounds for |G _∞,1 (f )|

Lemma If f (X ) = P

1≤i≤m t _i X ⁱ ∈ k[X ] is monic, then :

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

where R ∈ L

b

^m_p

c≤ip

ⁿ⁽ⁱ⁾

<m, (i,p)=1 k [Y ]X ^ip

ⁿ⁽ⁱ⁾

and P _f ∈ Xk[X , Y ].

I P _f = (Id + F + ... + F ⁿ⁻¹ )(∆(f )) mod X ^[

^m−1^p

^]+1 .

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

I Ad _f (Y ) is an additive and separable polynomial.

I Z (Ad f (Y )) ' V .

(31)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

Bounds for |G _∞,1 (f )|

Lemma If f (X ) = P

1≤i≤m t _i X ⁱ ∈ k[X ] is monic, then :

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

where R ∈ L

b

^m_p

c≤ip

ⁿ⁽ⁱ⁾

<m, (i,p)=1 k [Y ]X ^ip

ⁿ⁽ⁱ⁾

and P _f ∈ Xk[X , Y ].

I P _f = (Id + F + ... + F ⁿ⁻¹ )(∆(f )) mod X ^[

^m−1^p

^]+1 .

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

I Ad _f (Y ) is an additive and separable polynomial.

I Z (Ad f (Y )) ' V .

(32)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

Let m − 1 = `p ^s with (`, p) = 1.

I ([St 73]) |G _∞,1 | = p deg Ad f ≤ p(m − 1) ² , i.e.

|G

∞,1

|

g

²

≤ _(p−1) ^4p

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 ^|

≤

²₃

.

I

` > 1, p > 2, then

^|G^∞,1_g ^|

≤

_p−1^p

.

I

([St 73]) ` = 1, m = 1 + p

^s

, then

^|G^∞,1_g ^|

≤ 2p

^s_p−1^p

(with equality for f(X ) = X

^1+p^s

).

(33)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

Let m − 1 = `p ^s with (`, p) = 1.

I ([St 73]) |G _∞,1 | = p deg Ad f ≤ p(m − 1) ² , i.e.

|G

∞,1

|

g

²

≤ _(p−1) ^4p

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 ^|

≤

²₃

.

I

` > 1, p > 2, then

^|G^∞,1_g ^|

≤

_p−1^p

.

I

([St 73]) ` = 1, m = 1 + p

^s

, then

^|G^∞,1_g ^|

≤ 2p

^s_p−1^p

(with equality for f(X ) = X

^1+p^s

).

(34)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

Let m − 1 = `p ^s with (`, p) = 1.

I ([St 73]) |G _∞,1 | = p deg Ad f ≤ p(m − 1) ² , i.e.

|G

∞,1

|

g

²

≤ _(p−1) ^4p

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 ^|

≤

²₃

.

I

` > 1, p > 2, then

^|G^∞,1_g ^|

≤

_p−1^p

.

I

([St 73]) ` = 1, m = 1 + p

^s

, then

^|G^∞,1_g ^|

≤ 2p

^s_p−1^p

(with equality for f(X ) = X

^1+p^s

).

(35)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

Let m − 1 = `p ^s with (`, p) = 1.

I ([St 73]) |G _∞,1 | = p deg Ad f ≤ p(m − 1) ² , i.e.

|G

∞,1

|

g

²

≤ _(p−1) ^4p

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 ^|

≤

²₃

.

I

` > 1, p > 2, then

^|G^∞,1_g ^|

≤

_p−1^p

.

I

([St 73]) ` = 1, m = 1 + p

^s

, then

^|G^∞,1_g ^|

≤ 2p

^s_p−1^p

(with equality for f(X ) = X

^1+p^s

).

(36)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

Let m − 1 = `p ^s with (`, p) = 1.

I ([St 73]) |G _∞,1 | = p deg Ad f ≤ p(m − 1) ² , i.e.

|G

∞,1

|

g

²

≤ _(p−1) ^4p

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 ^|

≤

²₃

.

I

` > 1, p > 2, then

^|G^∞,1_g ^|

≤

_p−1^p

.

I

([St 73]) ` = 1, m = 1 + p

^s

, then

^|G^∞,1_g ^|

≤ 2p

^s_p−1^p

(with equality for f(X ) = X

^1+p^s

).

(37)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

Let m − 1 = `p ^s with (`, p) = 1.

I ([St 73]) |G _∞,1 | = p deg Ad f ≤ p(m − 1) ² , i.e.

|G

∞,1

|

g

²

≤ _(p−1) ^4p

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 ^|

≤

²₃

.

I

` > 1, p > 2, then

^|G^∞,1_g ^|

≤

_p−1^p

.

I

([St 73]) ` = 1, m = 1 + p

^s

, then

^|G^∞,1_g ^|

≤ 2p

^s_p−1^p

(with equality for f(X ) = X

^1+p^s

).

(38)

M. MATIGNON

Plan Introduction

Automorphism groups

Monodromy

Références

Characterization of G _∞,1 (f )

I We consider the extensions of type

0 → N ' Z /p Z → G → ( Z /p Z ) ⁿ → 0 (note that G _∞,1 (f ) is an extension of this type). Then G ⁰ ⊂ N ⊂ Z (G).

I If G ⁰ = Z (G), G is called extraspecial.

I

Then, |G| = p

^2s+1

and there are 2 isomorphism classes for a given s.

I

If p > 2, we denote by E(p

³

) (resp. M(p

³

)) the non abelian group of order p

³

and exponent p (resp. p

²

). Then, G ' E(p

³

) ∗ E(p

³

) ∗ ... ∗ E(p

³

) or M(p

³

) ∗ E (p

³

) ∗ ... ∗ E (p

³

), according as the exponent is p or p

²

.

I

If p = 2, then G ' D

₈

∗ D

₈

∗ ... ∗ D

₈

or

Q

₈

∗ D

₈

∗ ... ∗ D

₈

(in both cases, the exponent is 2

²

p-Groups and Automorphism groups of curves in char. p > 0

p-Groups and Automorphism groups of curves in char. p > 0

M. M ATIGNON 1

Laboratoire A2X Bordeaux

GTEM Kick-Off Seminar October 2006

Plan

Introduction

Monodromy and automorphism groups Automorphism groups of curves in char.p > 0

p -cyclic covers of the affine line Structure of G ∞,1 (f )

Bounds for |G ∞,1 (f )|

Characterization of G ∞,1 (f )

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

Condition (N) and G 2 Riemann surfaces Ray class fields Maximal curves Monodromy polynomial

Marked stable model Potentially good reduction Genus 2

References

Plan

Introduction

Monodromy and automorphism groups Automorphism groups of curves in char.p > 0

p -cyclic covers of the affine line Structure of G ∞,1 (f )

Bounds for |G ∞,1 (f )|

Characterization of G ∞,1 (f )

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

Condition (N) and G 2 Riemann surfaces Ray class fields Maximal curves Monodromy polynomial

Marked stable model Potentially good reduction Genus 2

References

Plan

Introduction

Monodromy and automorphism groups Automorphism groups of curves in char.p > 0

p -cyclic covers of the affine line Structure of G ∞,1 (f )

Bounds for |G ∞,1 (f )|

Characterization of G ∞,1 (f )

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

Condition (N) and G 2 Riemann surfaces Ray class fields Maximal curves Monodromy polynomial

Marked stable model Potentially good reduction Genus 2

References

Plan

Introduction

Monodromy and automorphism groups Automorphism groups of curves in char.p > 0

p -cyclic covers of the affine line Structure of G ∞,1 (f )

Bounds for |G ∞,1 (f )|

Characterization of G ∞,1 (f )

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

Condition (N) and G 2 Riemann surfaces Ray class fields Maximal curves Monodromy polynomial

Marked stable model Potentially good reduction Genus 2

References

Plan

Introduction

Monodromy and automorphism groups Automorphism groups of curves in char.p > 0

p -cyclic covers of the affine line Structure of G ∞,1 (f )

Bounds for |G ∞,1 (f )|

Characterization of G ∞,1 (f )

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

Condition (N) and G 2 Riemann surfaces Ray class fields Maximal curves Monodromy polynomial

Marked stable model Potentially good reduction Genus 2

References

Monodromy and automorphism groups.

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

K := FrR ; for example K / Q ur p finite.

π a uniformizing parameter.

k := R K /πR K .

C/K smooth projective curve, g(C) ≥ 1.

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

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

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

I Its p-Sylow subgroup is the wild monodromy

group .

Monodromy and automorphism groups.

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

K := FrR ; for example K / Q ur p finite.

π a uniformizing parameter.

k := R K /πR K .

C/K smooth projective curve, g(C) ≥ 1.

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

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

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

I Its p-Sylow subgroup is the wild monodromy

group .

Monodromy and automorphism groups.

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

K := FrR ; for example K / Q ur p finite.

M. M ATIGNON ¹

p -cyclic covers of the affine line Structure of G _∞,1 (f )

Bounds for |G _∞,1 (f )|

Characterization of G _∞,1 (f )

Condition (N) and G ₂ Riemann surfaces Ray class fields Maximal curves Monodromy polynomial

p -cyclic covers of the affine line Structure of G _∞,1 (f )

Bounds for |G _∞,1 (f )|

Characterization of G _∞,1 (f )

Condition (N) and G ₂ Riemann surfaces Ray class fields Maximal curves Monodromy polynomial

p -cyclic covers of the affine line Structure of G _∞,1 (f )

Bounds for |G _∞,1 (f )|

Characterization of G _∞,1 (f )

Condition (N) and G ₂ Riemann surfaces Ray class fields Maximal curves Monodromy polynomial

p -cyclic covers of the affine line Structure of G _∞,1 (f )

Bounds for |G _∞,1 (f )|

Characterization of G _∞,1 (f )

Condition (N) and G ₂ Riemann surfaces Ray class fields Maximal curves Monodromy polynomial

p -cyclic covers of the affine line Structure of G _∞,1 (f )

Bounds for |G _∞,1 (f )|

Characterization of G _∞,1 (f )

Condition (N) and G ₂ Riemann surfaces Ray class fields Maximal curves Monodromy polynomial

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

k := R _K /πR K .

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

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

k := R _K /πR K .

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

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

k := R _K /πR K .

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

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

k := R _K /πR K .

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

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

k := R _K /πR K .

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

I The base change C × _K K ^alg induces an

homomorphism ϕ : Gal(K ^alg /K ) → Aut k C _s , where C _s is the special fiber of the smooth model over R _L and L = (K ^alg ) ^ker ^ϕ .

I Let ` be a prime number, then, n _` := v _` (|Gal(L/K )|) ≤ v _` (|Aut _k C s |).

I If ` / ∈ {2, p}, then n _` ≤ 2g.

n p ≤ inf _`6=2,p v p (|GL _2g ( Z /` Z )|) = a + [a/p] + ..., where a = [ _p−1 ^2g ].

|Aut k C _s |. This justifies our interest in looking at

I The base change C × _K K ^alg induces an

homomorphism ϕ : Gal(K ^alg /K ) → Aut k C _s , where C _s is the special fiber of the smooth model over R _L and L = (K ^alg ) ^ker ^ϕ .

I Let ` be a prime number, then, n _` := v _` (|Gal(L/K )|) ≤ v _` (|Aut _k C s |).

I If ` / ∈ {2, p}, then n _` ≤ 2g.

n p ≤ inf _`6=2,p v p (|GL _2g ( Z /` Z )|) = a + [a/p] + ..., where a = [ _p−1 ^2g ].

|Aut k C _s |. This justifies our interest in looking at

I The base change C × _K K ^alg induces an

homomorphism ϕ : Gal(K ^alg /K ) → Aut k C _s , where C _s is the special fiber of the smooth model over R _L and L = (K ^alg ) ^ker ^ϕ .

I Let ` be a prime number, then, n _` := v _` (|Gal(L/K )|) ≤ v _` (|Aut _k C s |).

I If ` / ∈ {2, p}, then n _` ≤ 2g.

n p ≤ inf _`6=2,p v p (|GL _2g ( Z /` Z )|) = a + [a/p] + ..., where a = [ _p−1 ^2g ].

|Aut k C _s |. This justifies our interest in looking at

I The base change C × _K K ^alg induces an

homomorphism ϕ : Gal(K ^alg /K ) → Aut k C _s , where C _s is the special fiber of the smooth model over R _L and L = (K ^alg ) ^ker ^ϕ .

I Let ` be a prime number, then, n _` := v _` (|Gal(L/K )|) ≤ v _` (|Aut _k C s |).

I If ` / ∈ {2, p}, then n _` ≤ 2g.

n p ≤ inf _`6=2,p v p (|GL _2g ( Z /` Z )|) = a + [a/p] + ..., where a = [ _p−1 ^2g ].

|Aut k C _s |. This justifies our interest in looking at