Automorphisms and monodromy - u-bordeaux.frmmatigno/chuoslides.pdfAutomorphisms C. LEHR, M. MATIGNON...

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C. LEHR , M. MATIGNON

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

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Automorphisms and monodromy

C. LEHR 1 M. MATIGNON2

1MPI Bonn 2Laboratoire A2X Bordeaux

Workshop Chuo University April 2006

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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

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PlanIntroduction

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PlanIntroduction

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PlanIntroduction

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PlanIntroduction

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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 .

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group .

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group .

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group .

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group .

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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]).

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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 .

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.

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is étale over RK .

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is étale over RK .

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is étale over RK .

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is étale over RK .

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is étale over RK .

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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.

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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).

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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)

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.

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LemmaIf f (X ) =

Pf ∈ Xk [X , Y ].

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LemmaIf f (X ) =

Pf ∈ Xk [X , Y ].

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LemmaIf f (X ) =

Pf ∈ Xk [X , Y ].

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LemmaIf f (X ) =

Pf ∈ Xk [X , Y ].

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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

I ` > 1, p > 2, then |G∞,1|g ≤ p

p−1 .

I ([St 73]) ` > 1, m = 1 + ps, then |G∞,1|g ≤ 2ps p

(with equality for f (X ) = X 1+ps).

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|G∞,1|

g2 ≤ 4p(p−1)2 .

I ` > 1, p = 2, then |G∞,1|g ≤ 2

I ` > 1, p > 2, then |G∞,1|g ≤ p

p−1 .

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|G∞,1|

g2 ≤ 4p(p−1)2 .

I ` > 1, p = 2, then |G∞,1|g ≤ 2

I ` > 1, p > 2, then |G∞,1|g ≤ p

p−1 .

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|G∞,1|

g2 ≤ 4p(p−1)2 .

I ` > 1, p = 2, then |G∞,1|g ≤ 2

I ` > 1, p > 2, then |G∞,1|g ≤ p

p−1 .

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|G∞,1|

g2 ≤ 4p(p−1)2 .

I ` > 1, p = 2, then |G∞,1|g ≤ 2

I ` > 1, p > 2, then |G∞,1|g ≤ p

p−1 .

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|G∞,1|

g2 ≤ 4p(p−1)2 .

I ` > 1, p = 2, then |G∞,1|g ≤ 2

I ` > 1, p > 2, then |G∞,1|g ≤ p

p−1 .

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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.

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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.

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I Adf (Y ) = F s(∑

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I Adf (Y ) = F s(∑

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I Adf (Y ) = F s(∑

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I Adf (Y ) = F s(∑

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I Adf (Y ) = F s(∑

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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.

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Big actions (I)

If |G∞,1|g > p

p−1 (23 for p = 2), then

f (X ) = cX + XΣ(F )(X ) ∈ k [X ].

gC> 2p

p−1 .

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Big actions (I)

If |G∞,1|g > p

p−1 (23 for p = 2), then

f (X ) = cX + XΣ(F )(X ) ∈ k [X ].

gC> 2p

p−1 .

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Big actions (I)

If |G∞,1|g > p

p−1 (23 for p = 2), then

f (X ) = cX + XΣ(F )(X ) ∈ k [X ].

gC> 2p

p−1 .

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Big actions (I)

If |G∞,1|g > p

p−1 (23 for p = 2), then

f (X ) = cX + XΣ(F )(X ) ∈ k [X ].

gC> 2p

p−1 .

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Big actions (I)

If |G∞,1|g > p

p−1 (23 for p = 2), then

f (X ) = cX + XΣ(F )(X ) ∈ k [X ].

gC> 2p

p−1 .

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Big actions (I)

If |G∞,1|g > p

p−1 (23 for p = 2), then

f (X ) = cX + XΣ(F )(X ) ∈ k [X ].

gC> 2p

p−1 .

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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

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℘(W1, ..., Wt ) = (f1(X ), f2(X ), ..., ft (X )) ∈ (k [X ])t

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℘(W1, ..., Wt ) = (f1(X ), f2(X ), ..., ft (X )) ∈ (k [X ])t

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℘(W1, ..., Wt ) = (f1(X ), f2(X ), ..., ft (X )) ∈ (k [X ])t

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℘(W1, ..., Wt ) = (f1(X ), f2(X ), ..., ft (X )) ∈ (k [X ])t

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℘(W1, ..., Wt ) = (f1(X ), f2(X ), ..., ft (X )) ∈ (k [X ])t

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℘(W1, ..., Wt ) = (f1(X ), f2(X ), ..., ft (X )) ∈ (k [X ])t

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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 .

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(Y − Y p2)X 2p + (Y 2p2

− Y 2 + 2Y 1+p − 2Y p+p2)X p

f2(X + y) = 2(yp−y)α f1(X ) + f2(X ) + ℘(P2).

2 (p + p ∗ 2p).I

|G|g = 2p

p−1p2

1+2p .

I|G|g2 = 4p

(p−1)2p

(1+2p)2 .

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(Y − Y p2)X 2p + (Y 2p2

− Y 2 + 2Y 1+p − 2Y p+p2)X p

f2(X + y) = 2(yp−y)α f1(X ) + f2(X ) + ℘(P2).

2 (p + p ∗ 2p).I

|G|g = 2p

p−1p2

1+2p .

I|G|g2 = 4p

(p−1)2p

(1+2p)2 .

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(Y − Y p2)X 2p + (Y 2p2

− Y 2 + 2Y 1+p − 2Y p+p2)X p

f2(X + y) = 2(yp−y)α f1(X ) + f2(X ) + ℘(P2).

2 (p + p ∗ 2p).I

|G|g = 2p

p−1p2

1+2p .

I|G|g2 = 4p

(p−1)2p

(1+2p)2 .

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(Y − Y p2)X 2p + (Y 2p2

− Y 2 + 2Y 1+p − 2Y p+p2)X p

f2(X + y) = 2(yp−y)α f1(X ) + f2(X ) + ℘(P2).

2 (p + p ∗ 2p).I

|G|g = 2p

p−1p2

1+2p .

I|G|g2 = 4p

(p−1)2p

(1+2p)2 .

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(Y − Y p2)X 2p + (Y 2p2

− Y 2 + 2Y 1+p − 2Y p+p2)X p

f2(X + y) = 2(yp−y)α f1(X ) + f2(X ) + ℘(P2).

2 (p + p ∗ 2p).I

|G|g = 2p

p−1p2

1+2p .

I|G|g2 = 4p

(p−1)2p

(1+2p)2 .

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(Y − Y p2)X 2p + (Y 2p2

− Y 2 + 2Y 1+p − 2Y p+p2)X p

f2(X + y) = 2(yp−y)α f1(X ) + f2(X ) + ℘(P2).

2 (p + p ∗ 2p).I

|G|g = 2p

p−1p2

1+2p .

I|G|g2 = 4p

(p−1)2p

(1+2p)2 .

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(Y − Y p2)X 2p + (Y 2p2

− Y 2 + 2Y 1+p − 2Y p+p2)X p

f2(X + y) = 2(yp−y)α f1(X ) + f2(X ) + ℘(P2).

2 (p + p ∗ 2p).I

|G|g = 2p

p−1p2

1+2p .

I|G|g2 = 4p

(p−1)2p

(1+2p)2 .

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(Y − Y p2)X 2p + (Y 2p2

− Y 2 + 2Y 1+p − 2Y p+p2)X p

f2(X + y) = 2(yp−y)α f1(X ) + f2(X ) + ℘(P2).

2 (p + p ∗ 2p).I

|G|g = 2p

p−1p2

1+2p .

I|G|g2 = 4p

(p−1)2p

(1+2p)2 .

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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.

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|G/H|/g(C/H) ≤ 2ps+1

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|G/H|/g(C/H) ≤ 2ps+1

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|G/H|/g(C/H) ≤ 2ps+1

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|G/H|/g(C/H) ≤ 2ps+1

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|G/H|/g(C/H) ≤ 2ps+1

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|G/H|/g(C/H) ≤ 2ps+1

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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).

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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.

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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.

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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)

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Références

gCm = 1 + nm(−1 + m/2) − (1/2)∑

0≤j≤m−1 nj ≤nm(−1 + m/2)

Automorphisms

Références

gCm = 1 + nm(−1 + m/2) − (1/2)∑

0≤j≤m−1 nj ≤nm(−1 + m/2)

Automorphisms

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))

gCm∼ Nq

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

Références

I|G(m)|

gCm≥ nmq

gCm∼ Nq

Automorphisms

Références

I|G(m)|

gCm≥ nmq

gCm∼ Nq

Automorphisms

Références

I|G(m)|

gCm≥ nmq

gCm∼ Nq

Automorphisms

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

Références

`(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

Automorphisms

Références

`(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

Automorphisms

Références

`(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

Automorphisms

Références

`(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

Automorphisms

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|

, 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:

= 4p(p−1)2 and G = G∞,1(f ) or

= 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

Références

2(|G2/Gi0+1| − 1)(i0 − 1).

I If 0 < M ≤ |G|

, then

I |Gi0+1| ≤1M

|G/Gi0+1|

g2C/Gi0+1

≤ 1M

4|G2/Gi0+1|

(|G2/Gi0+1|−1)2 .

g2C≥ 4

= 4p(p−1)2 and G = G∞,1(f ) or

Automorphisms

Références

2(|G2/Gi0+1| − 1)(i0 − 1).

I If 0 < M ≤ |G|

, then

I |Gi0+1| ≤1M

|G/Gi0+1|

g2C/Gi0+1

≤ 1M

4|G2/Gi0+1|

(|G2/Gi0+1|−1)2 .

g2C≥ 4

= 4p(p−1)2 and G = G∞,1(f ) or

Automorphisms

Références

2(|G2/Gi0+1| − 1)(i0 − 1).

I If 0 < M ≤ |G|

, then

I |Gi0+1| ≤1M

|G/Gi0+1|

g2C/Gi0+1

≤ 1M

4|G2/Gi0+1|

(|G2/Gi0+1|−1)2 .

g2C≥ 4

= 4p(p−1)2 and G = G∞,1(f ) or

Automorphisms

Références

2(|G2/Gi0+1| − 1)(i0 − 1).

I If 0 < M ≤ |G|

, then

I |Gi0+1| ≤1M

|G/Gi0+1|

g2C/Gi0+1

≤ 1M

4|G2/Gi0+1|

(|G2/Gi0+1|−1)2 .

g2C≥ 4

= 4p(p−1)2 and G = G∞,1(f ) or

Automorphisms

Références

2(|G2/Gi0+1| − 1)(i0 − 1).

I If 0 < M ≤ |G|

, then

I |Gi0+1| ≤1M

|G/Gi0+1|

g2C/Gi0+1

≤ 1M

4|G2/Gi0+1|

(|G2/Gi0+1|−1)2 .

g2C≥ 4

= 4p(p−1)2 and G = G∞,1(f ) or

Automorphisms

Références

2(|G2/Gi0+1| − 1)(i0 − 1).

I If 0 < M ≤ |G|

, then

I |Gi0+1| ≤1M

|G/Gi0+1|

g2C/Gi0+1

≤ 1M

4|G2/Gi0+1|

(|G2/Gi0+1|−1)2 .

g2C≥ 4

= 4p(p−1)2 and G = G∞,1(f ) or

Automorphisms

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|

> M, for

(C, G) a big action with G2 abelian with exponentp, is finite.

Automorphisms

Références

Maximal curves (II)

g2C≥ 4

(p2−1)2 . Namely

> M, for

Automorphisms

Références

Maximal curves (II)

g2C≥ 4

(p2−1)2 . Namely

> M, for

Automorphisms

Références

Maximal curves (II)

g2C≥ 4

(p2−1)2 . Namely

> M, for

Automorphisms

Références

Maximal curves (II)

g2C≥ 4

(p2−1)2 . Namely

> M, for

Automorphisms

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

|V | ≤ 4pt

M(p−1)2(P

1≤i≤t pi−1pνi−ν1 )2 and so { p2ν1

|V | } is

finite.I { |G|

= 4pt |V |p−2ν1

(p−1)2(P

1≤i≤t pi−1pνi−ν1 )2 } is finite.

Automorphisms

Références

2 (∑

I M ≤ pt |V |g2 ≤ 4pt

(p−1)2(P

|V | ≤ 4pt

M(p−1)2(P

|V | } is

finite.I { |G|

= 4pt |V |p−2ν1

(p−1)2(P

Automorphisms

Références

2 (∑

I M ≤ pt |V |g2 ≤ 4pt

(p−1)2(P

|V | ≤ 4pt

M(p−1)2(P

|V | } is

finite.I { |G|

= 4pt |V |p−2ν1

(p−1)2(P

Automorphisms

Références

2 (∑

I M ≤ pt |V |g2 ≤ 4pt

(p−1)2(P

|V | ≤ 4pt

M(p−1)2(P

|V | } is

finite.I { |G|

= 4pt |V |p−2ν1

(p−1)2(P

Automorphisms

Références

2 (∑

I M ≤ pt |V |g2 ≤ 4pt

(p−1)2(P

|V | ≤ 4pt

M(p−1)2(P

|V | } is

finite.I { |G|

= 4pt |V |p−2ν1

(p−1)2(P

Automorphisms

Références

2 (∑

I M ≤ pt |V |g2 ≤ 4pt

(p−1)2(P

|V | ≤ 4pt

M(p−1)2(P

|V | } is

finite.I { |G|

= 4pt |V |p−2ν1

(p−1)2(P

Automorphisms

Références

2 (∑

I M ≤ pt |V |g2 ≤ 4pt

(p−1)2(P

|V | ≤ 4pt

M(p−1)2(P

|V | } is

finite.I { |G|

= 4pt |V |p−2ν1

(p−1)2(P

Automorphisms

Références

2 (∑

I M ≤ pt |V |g2 ≤ 4pt

(p−1)2(P

|V | ≤ 4pt

M(p−1)2(P

|V | } is

finite.I { |G|

= 4pt |V |p−2ν1

(p−1)2(P

Automorphisms

Références

2 (∑

I M ≤ pt |V |g2 ≤ 4pt

(p−1)2(P

|V | ≤ 4pt

M(p−1)2(P

|V | } is

finite.I { |G|

= 4pt |V |p−2ν1

(p−1)2(P

Automorphisms

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 |

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

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|

= |G2||V |

} is finite. ///

Automorphisms

Références

fi0+1(X + v) = fi0+1(X ) +∑

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

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 { |G|

= |G2||V |

} is finite. ///

Automorphisms

Références

fi0+1(X + v) = fi0+1(X ) +∑

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

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 { |G|

= |G2||V |

} is finite. ///

Automorphisms

Références

fi0+1(X + v) = fi0+1(X ) +∑

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

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 { |G|

= |G2||V |

} is finite. ///

Automorphisms

Références

fi0+1(X + v) = fi0+1(X ) +∑

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

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 { |G|

= |G2||V |

} is finite. ///

Automorphisms

Références

fi0+1(X + v) = fi0+1(X ) +∑

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

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 { |G|

= |G2||V |

} is finite. ///

Automorphisms

Références

fi0+1(X + v) = fi0+1(X ) +∑

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

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 { |G|

= |G2||V |

} is finite. ///

Automorphisms

Références

fi0+1(X + v) = fi0+1(X ) +∑

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

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 { |G|

= |G2||V |

} is finite. ///

Automorphisms

Références

fi0+1(X + v) = fi0+1(X ) +∑

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

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 { |G|

= |G2||V |

} is finite. ///

Automorphisms

Références

fi0+1(X + v) = fi0+1(X ) +∑

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

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 { |G|

= |G2||V |

} is finite. ///

Automorphisms

Références

fi0+1(X + v) = fi0+1(X ) +∑

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

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 { |G|

= |G2||V |

} is finite. ///

Automorphisms

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 ) = −(

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

Références

Z p0 = f (X0) =

1≤i≤m(X0 − xi)ni ∈ R[X0],

Apα(Y ) = −(

pα−1

p.S1(Y )pα

+pT (Y )S0(Y )pα .

I L(Y ) := S1(Y )pα

Automorphisms

Références

Z p0 = f (X0) =

1≤i≤m(X0 − xi)ni ∈ R[X0],

Apα(Y ) = −(

pα−1

p.S1(Y )pα

+pT (Y )S0(Y )pα .

I L(Y ) := S1(Y )pα

Automorphisms

Références

Z p0 = f (X0) =

1≤i≤m(X0 − xi)ni ∈ R[X0],

Apα(Y ) = −(

pα−1

p.S1(Y )pα

+pT (Y )S0(Y )pα .

I L(Y ) := S1(Y )pα

Automorphisms

Références

Z p0 = f (X0) =

1≤i≤m(X0 − xi)ni ∈ R[X0],

Apα(Y ) = −(

pα−1

p.S1(Y )pα

+pT (Y )S0(Y )pα .

I L(Y ) := S1(Y )pα

Automorphisms

Références

Z p0 = f (X0) =

1≤i≤m(X0 − xi)ni ∈ R[X0],

Apα(Y ) = −(

pα−1

p.S1(Y )pα

+pT (Y )S0(Y )pα .

I L(Y ) := S1(Y )pα

Automorphisms

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

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

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

Références

v(ρj ) = max{1i v

Ai (yj )

for r + 1 ≤ i ≤ n}.

Automorphisms

Références

v(ρj ) = max{1i v

Ai (yj )

for r + 1 ≤ i ≤ n}.

Automorphisms

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

Références

Theorem([Le-Ma 3])

Z p0 = f (X0) = 1 + pp/(q+1)X q

0 + X q+10 .

Automorphisms

Références

Theorem([Le-Ma 3])

Z p0 = f (X0) = 1 + pp/(q+1)X q

0 + X q+10 .

Automorphisms

Références

Theorem([Le-Ma 3])

Z p0 = f (X0) = 1 + pp/(q+1)X q

0 + X q+10 .

Automorphisms

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 2curveP1

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

Type 3

Automorphisms

Références

stable model.I

genus 1curves

genus 2curveP1

original component

Type 2

Type 3

Automorphisms

Références

stable model.I

genus 1curves

genus 2curveP1

original component

Type 2

Type 3

Automorphisms

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

Références

Z p0 = f (X0) with

f (X0) = 1 + b2X 20 + b3X 3

0 + b4X 40 + X 5

I a) f (X0) = 1 + 23/5X 20 + X 3

0 + 22/5X 40 + X 5

0 andK = Qur

Automorphisms

Références

Z p0 = f (X0) with

f (X0) = 1 + b2X 20 + b3X 3

0 + b4X 40 + X 5

I a) f (X0) = 1 + 23/5X 20 + X 3

0 + 22/5X 40 + X 5

0 andK = Qur

Automorphisms

Références

Z p0 = f (X0) with

f (X0) = 1 + b2X 20 + b3X 3

0 + b4X 40 + X 5

I a) f (X0) = 1 + 23/5X 20 + X 3

0 + 22/5X 40 + X 5

0 andK = Qur

Automorphisms

Références

Z p0 = f (X0) with

f (X0) = 1 + b2X 20 + b3X 3

0 + b4X 40 + X 5

I a) f (X0) = 1 + 23/5X 20 + X 3

0 + 22/5X 40 + X 5

0 andK = Qur

Automorphisms

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

Références

f (X0) = 1 + a3X 20 + a6X 3

0 + X 50 .

I c) K = Qur2 and . f (X0) = 1 + X 4

0 + X 50 .

Automorphisms

Références

f (X0) = 1 + a3X 20 + a6X 3

0 + X 50 .

I c) K = Qur2 and . f (X0) = 1 + X 4

0 + X 50 .

Automorphisms

Références

f (X0) = 1 + a3X 20 + a6X 3

0 + X 50 .

I c) K = Qur2 and . f (X0) = 1 + X 4

0 + X 50 .

Automorphisms

Références

f (X0) = 1 + a3X 20 + a6X 3

0 + X 50 .

I c) K = Qur2 and . f (X0) = 1 + X 4

0 + X 50 .

Automorphisms

Références

f (X0) = 1 + a3X 20 + a6X 3

0 + X 50 .

I c) K = Qur2 and . f (X0) = 1 + X 4

0 + X 50 .

Automorphisms

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

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

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.

Automorphisms and monodromy - u-bordeaux.frmmatigno/chuoslides.pdfAutomorphisms C. LEHR, M. MATIGNON...

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