Magali Rocher

Large p-groups actions with |G| g 2 ≥ 4 (p 2 -1) 2

Let k be an algebraically closed field of characteristic p > 0 and C a connected nonsingular projective curve over k with genus g ≥ 2. Let (C, G) be a "big action" , i.e. a pair (C, G) where G is a p-subgroup of the k-automorphism group of C such that |G| g > 2 p p-1 . We first study finiteness results on the values taken by the quotient |G| g 2 when (C, G) runs over the big actions satisfying |G| g 2 ≥ M , for a given positive real M > 0. Then, we exhibit a classification and a parametrization of such big actions when M = 4 (p 2 -1) 2 .

1 Introduction.

Setting. Let k be an algebraically closed field of positive characteristic p > 0 and C a connected nonsingular projective curve over k, with genus g ≥ 2. As in characteristic zero, the k-automorphism group of the curve C, Aut k (C), is a finite group whose order is bounded from above by a polynomial in g (cf.

). The difference is due to the appearance of wild ramification. More precisely, let G be a subgroup of Aut k (C). If the order of G is prime to p, then the Hurwitz bound still holds, i.e. |G| ≤ 84 (g -1). Now, if G is a p-Sylow subgroup of Aut k (C), Nakajima (cf. [Na87]) proves that |G| can be larger according to the value of the p-rank γ of the curve C. Indeed, if γ > 0, then |G| ≤ 2 p p-1 g, whereas for γ = 0, |G| ≤ max{g, 4 p (p-1) 2 g 2 } , knowing that the quadratic upper bound 4 p (p-1) 2 g 2 can really be attained. Following Nakajima's work, Lehr and Matignon explore the "big actions", that is to say the pairs (C, G) where G is a p-subgroup of Aut k (C) such that |G| g > 2 p p-1

). In this case, the ramification locus of the cover π : C → C/G is located at one point of C, say ∞. In [MR08], we display necessary conditions on G 2 , the second ramification group of G at ∞ in lower notation, for (C, G) to be a big action. In particular, we show that G 2 coincides with the derived subgroup G ′ of G.

Motivation and purpose. The aim of this paper is to pursue the classification of big actions as initiated in [START_REF] Lehr | Automorphism groups for p-cyclic covers of the affine line[END_REF]. Indeed, when searching for a classification of big actions, it naturally occurs that the quotient |G| g 2 has a "sieve" effect. Lehr and Matignon first prove that the big actions such that |G| g 2 ≥ 4 (p-1) 2 correspond to the p-cyclic étale covers of the affine line parametrized by an Artin-Schreier equation:

W p -W = f (X) := X S(X) + c X ∈ k[X],
where S(X) runs over the additive polynomials of k[X]. In [START_REF] Matignon | On smooth curves endowed with a large automorphism p-group in characteristic p[END_REF], we show that the big actions satisfying |G| g 2 ≥ 4 (p 2 -1) 2 correspond to the étale covers of the affine line with Galois group G ′ ≃ (Z/pZ) n , with n ≤ 3. This motivated the study of big actions with a p-elementary abelian G ′ , say G ′ ≃ (Z/pZ) n , which is the main topic of [START_REF] Rocher | Large p-groups actions with a p-elementary abelian second ramification group[END_REF] where we generalize the structure theorem obtained in the p-cyclic case. Namely, we prove that when G ′ ≃ (Z/pZ) n with n ≥ 1, then the function field of the curve is parametrized by n Artin Schreier equations:

W p i -W i = f i (X) ∈ k[X]
where each function f i can be written as a linear combination over k of products of at most i + 1 additive polynomials. In this paper, we display the parametrization of the functions f i 's in the case of big actions satisfying |G| g 2 ≥ 4 (p 2 -1) 2 . In what follows, this condition is called condition ( * ).

Outline ot the paper. The paper falls into two main parts. The first one is focused on finiteness results for big actions (C, G) satisfying |G| g 2 ≥ M for a given positive real M > 0, called big actions satisfying G M , whereas the second part is dedicated to the classification of such big actions when M = 4 (p 2 -1) 2 . More precisely, we prove in section 4 that, for a given M > 0, the order of G ′ only takes a finite number of values for (C, G) a big action satisfying G M . When exploring similar finiteness results for g and |G|, we are lead to a purely group-theoretic discussion around the inclusion

F ratt(G ′ ) ⊂ [G ′ , G],
where F ratt(G ′ ) means the Frattini subgroup of G ′ and [G ′ , G] denotes the commutator subgroup of G ′ and G (cf. section 4). When the inclusion is strict, |G| and g also take a finite number of values for (C, G) satisfying G M . This is no more true when F ratt(G ′ ) = [G ′ , G]. In this case, we can only conclude that, for p > 2, the quotient |G| g 2 takes a finite number of values for (C, G) satisfying G M with an abelian G ′ . Note that we do not know yet examples of big actions with a non-abelian G ′ . Another central question to is the link between the subgroups G of Aut k (C) such that (C, G) is a big action and a p-Sylow subgroup of Aut k (C) containing G (section 3). Among other things, we prove that they have the same derived subgroup. This, together with the fact that the order of G ′ takes a finite number of values for big actions satisfying G M , implies, on the one hand, that the order of G ′ is a key criterion to classify big actions and, on the other hand, that we can concentrate on p-Sylow subgroups of A. In section 5, we eventually display the classification and the parametrization of big actions (C, G) under condition ( * ) according to the order of G ′ . Pursuing the preceding discussion, we have to distinguish the cases [G ′ , G] = F ratt(G ′ )(= {e}) and [G ′ , G] F ratt(G ′ )(= {e}).

Notation and preliminary remarks. Let k be an algebraically closed field of characteristic p > 0. We denote by F the Frobenius endomorphism for a k-algebra. Then, ℘ means the Frobenius operator minus identity. We denote by k{F } the k-subspace of k[X] generated by the polynomials F i (X), with i ∈ N. It is a ring under the composition. Furthermore, for all α in k, F α = α p F . The elements of k{F } are the additive polynomials, i.e. the polynomials P (X) of k[X] such that for all α and β in k, P (α + β) = P (α) + P (β). Moreover, a separable polynomial is additive if and only if the set of its roots is a subgroup of k (see [START_REF] Goss | Basic structures of function field arithmetic[END_REF] chap. 1).

Let f (X) be a polynomial of k[X]. Then, there is a unique polynomial red(f )(X) in k[X], called the reduced representative of f , which is p-power free, i.e. red(f )(X) ∈ (i,p)=1 k X i , and such that red(f )(X) = f (X) mod ℘(k[X]). We say that the polynomial f is reduced mod ℘(k[X]) if and only if it coincides with its reduced representative red(f ). The equation W p -W = f (X) defines a p-cyclic étale cover of the affine line that we denote by C f . Conversely, any p-cyclic étale cover of the affine line Spec k[X] corresponds to a curve C f where f is a polynomial of k[X] (see [START_REF] Milne | Etale cohomology[END_REF] III.4.12, p. 127). By Artin-Schreier theory, the covers C f and C red(f ) define the same p-cyclic covers of the affine line. The curve C f is irreducible if and only if red(f ) = 0.

Throughout the text, C denotes a connected nonsingular projective curve over k, with genus g ≥ 2. We denote by A := Aut k C the k-automorphism group of the curve C and by S(A) p any p-Sylow subgroup of A. For any point P ∈ C and any i ≥ -1, we denote by A P,i the i-th ramification group of A at P in lower notation, namely A P,i := {σ ∈ A, v P (σ(t P ) -t P ) ≥ i + 1} where t P denotes a uniformizing parameter at P and v P means the order function at P .

2 The setting: generalities about big actions.

Definition 2.1. Let C be a connected nonsingular projective curve over k, with genus g ≥ 2. Let G be a subgroup of A. We say that the pair (C, G) is a big action if G is a finite p-group such that

|G| g > 2 p p -1
To precise the background of this work, we first recall basic properties of big actions established in [START_REF] Lehr | Automorphism groups for p-cyclic covers of the affine line[END_REF] and [START_REF] Matignon | On smooth curves endowed with a large automorphism p-group in characteristic p[END_REF].

Recall 2.2. Assume that (C, G) is a big action. Then, there is a point of C (say ∞) such that G is the wild inertia subgroup of G at ∞: G 1 . Moreover, the quotient C/G is isomorphic to the projective line P 1 k and the ramification locus (respectively branch locus) of the cover π : C → C/G is the point ∞ (respectively π(∞)). For all i ≥ 0, we denote by G i the i-th lower ramification group of G at ∞:

G i := {σ ∈ G, v ∞ (σ(t ∞ ) -t ∞ ) ≥ i + 1}
where t ∞ denotes a uniformizing parameter at ∞ and v ∞ means the order function at ∞. 1. Then, G 2 is non trivial and it is strictly included in G 1 .

2. The quotient curve C/G 2 is isomorphic to the projective line P 1 k .

3. The quotient group G/G 2 acts as a group of translations of the affine line C/G 2 -{∞} = Spec k[X], through X → X + y, where y runs over a subgroup V of k. Then, V is an F psubvector space of k. We denote by v its dimension. This gives the following exact sequence: [START_REF] Matignon | On smooth curves endowed with a large automorphism p-group in characteristic p[END_REF] Thm. 2.6.4). Let (C, G) be a big action. Then,

0 -→ G 2 -→ G = G 1 π -→ V ≃ (Z/ p Z) v -→ 0 where π : G → V g → g(X) -X Recall 2.3. ([
G 2 = G ′ = F ratt(G)
where G ′ means the commutator subgroup of G and F ratt(G) = G ′ G p the Frattini subgroup of G.

To conclude this first section, we introduce new definitions used in our future classification.

Definition 2.4. Let C be a connected nonsingular projective curve over k, with genus g ≥ 2. Let G be a subgroup of A. Let M > 0 be a positive real. We say that:

1. G satisfies G(C) (or (C, G) satisfies G) if (C, G) is a big action. 2. G satisfies G M (C) (or (C, G) satisfies G M ) if (C, G) is a big action with |G| g 2 ≥ M. 3. If (C, G) satisfies G M with M = 4 (p 2 -1) 2 , we say that (C, G) satisfies condition ( * ).
Remark 2.5. There exists big actions (C, G) satisfying G M if and only if M ≤ 4 p (p-1) 2 (see [START_REF] Stichtenoth | Über die Automorphismengruppe eines algebraischen Funktionkorpers von Primzahlcharakteristik I, II[END_REF]).

3 A study on p-Sylow subgroups of Aut k (C) inducing big actions.

In this section, we more specifically concentrate on the p-Sylow subgroup(s) of A satisfying G(C) (resp. G M (C)).

Remark 3.1. Let C be a connected nonsingular projective curve over k, with genus g ≥ 2. Assume that there exists a subgroup G ⊂ A satisfying G(C).

1. Then, every p-Sylow subgroup of A satisfies G(C).

2. Moreover, A has a unique p-Sylow subgroup except in the three following cases (cf. [START_REF] Hansen | Deligne-Lusztig varieties and group codes. Coding theory and algebraic geometry[END_REF] and [START_REF] Giulietti | On large automorphism groups of algebraic curves in positive characteristic[END_REF]):

(a) The Hermitian curve

C H : W q + W = X 1+q
with p ≥ 2, q = p s , s ≥ 1. Then, g = 1 2 (q 2 -q) and A ≃ P SU (3, q) or A ≃ P GU (3, q). It follows that |A| = q 3 (q 2 -1) (q 3 + 1), so

|S(A)p| g = 2q 2 q-1 > 2 p p-1 and |S(A)p| g 2 = 4 q (q-1) 2 , where S(A) p denotes any p-Sylow subgroup of A. Thus, (C H , S(A) p ) is a big action with G ′ = G 2 ≃ (Z/pZ) s . It satisfies condition ( * ) if and only if 1 ≤ s ≤ 3.
(b) The Deligne-Lusztig curve arising from the Suzuki group

C S : W q + W = X q0 (X q + X)
with p = 2, q 0 = 2 s , s ≥ 1 and q = 2 2s+1 . In this case, g = q 0 (q -1) and A ≃ Sz(q) the Suzuki group. It follows that |A| = q 2 (q -1) (q 2 + 1), so

|S(A)p| g = q 2 q0(q-1) > 2 p p-1
and

|S(A)p| g 2 = q 2 q 2 0 (q-1) 2 < 4 (p 2 -1) 2 , for all s ≥ 1. Thus, (C S , S(A) p ) is a big action with G ′ = G 2 ≃ (Z/pZ) 2s+1 but it never satisfies condition ( * ).
(c) The Deligne-Lusztig curve arising from the Ree group C R : W q 1 -W 1 = X q0 (X q + X) and W q 2 -W 2 = X 2q0 (X q + X) with p = 3, q 0 = 3 s , s ≥ 1 and q = 3 2s+1 . Then, g = 3 2 q 0 (q-1) (q+q 0 +1) and A ≃ Ree(q) the Ree group. It follows that |A| = q 3 (q -1) (q 3 + 1), so

|S(A)p| g = 2q 3 3q0(q-1)(q+q0+1) > 2 p p-1 and |S(A)p| g 2 = 4q 3 9q 2 0 (q-1) 2 (q+q0+1) 2 < 4 (p 2 -1) 2 for all s ≥ 1. Thus, (C R , S(A) p )) is a big action with G ′ = G 2 ≃ (Z/pZ) 2(2s+1) but it never satisfies condition ( * ).
In each of these three cases, the group A is simple, so it has more than one p-Sylow subgroups. Now, fix C a connected nonsingular projective curve over k, with genus g ≥ 2. We highlight the link between the groups G satisfying G(C) (resp. G M (C)) and the p-Sylow subgroup(s) of A.

Proposition 3.2. Let C be a connected nonsingular projective curve over k, with genus g ≥ 2.

1. Let G satisfy G(C).

(a) Then, there exists a point of C, say ∞, such that G is included in A ∞,1 . For all i ≥ 0, we denote by

G i the i-th ramification group of G at ∞ in lower notation. Then, A ∞,1 satisfies G(C) and A ∞,2 = G 2 , i.e. (A ∞,1 ) ′ = G ′
. Thus, we obtain the following diagram:

0 -→ A ∞,2 -→ A ∞,1 π -→ W ⊂ k -→ 0 || ∪ ∪ 0 -→ G 2 -→ G = G 1 π -→ V -→ 0 In particular, G = π -1 (V ) where V is an F p -subvector space of W . (b) A ∞,1 is a p-Sylow subgroup of A. Moreover, except in the three special cases mentionned in Remark 3.1, A ∞,1 is the unique p-Sylow subgroup of A. (c) Let M be a positive real such that G satisfies G M (C). Then, A ∞,1 also satifies G M (C).
2. Conversely, let ∞ be a point of the curve C such that A ∞,1 satisfies G(C). Consider V an F p -vector space of W , defined as above, and put G := π -1 (V ).

(a) Then, the group G satisfies G(C) if and only if Remark 3.3. Except in the three special cases mentionned in Remark 3.1, the point ∞ of C defined in Proposition 3.2 is uniquely determined. In particular, except for the three special cases, if P is a point of C such that A P,1 satisfies G(C), then P = ∞. Lemma 4.1. Let M > 0 be a positive real such that (C, G) is a big action satisfying G M . Then, the order of G ′ is bounded as follows:

|W | ≥ |V | > 2 p p -1 g |A ∞,2 | (b) Let M be a positive real such that A ∞,1 satisfies G M (C). Then, G satisfies G M (C) if and only if |W | ≥ |V | ≥ M g 2 |A ∞,

As a conclusion, if

G satisfyies G(C) (resp. G M (C)) and if A ∞,1 is a (actually "the", in most cases) p-Sylow subgroup of A containing G, then A ∞,1 also satisfies G(C) (resp. G M (C))
p ≤ |G ′ | ≤ 4 p (p -1) 2 2 + M + 2 √ 1 + M M 2
Thus, |G ′ | only takes a finite number of values for (C, G) a big action satisfying G M .

Proof: We first recall that G ′ = G 2 is a non-trivial p-group (see e.g. [START_REF] Lehr | Automorphism groups for p-cyclic covers of the affine line[END_REF] Prop. 8.5). Now, let i 0 ≥ 2 be the integer such that the lower ramification filtration of G at ∞ reads:

G = G 0 = G 1 G 2 = • • • = G i0 G i0+1 = • • • Put |G 2 /G i0+1 | = p m , with m ≥ 1, and B m := 4 M |G2/Gi 0 +1| (|G2/Gi 0 +1|-1) 2 = 4 M p m (p m -1) 2 . By [LM05] (Thm. 8.6), M ≤ |G| g 2 implies 1 < |G 2 | ≤ 4 M |G2/Gi 0 +1| 2 (|G2/Gi 0 +1|-1) 2 = p m B m . From |G 2 | = p m |G i0+1 |, we infer 1 ≤ |G i0+1 | ≤ B m . Since (B m
) m≥1 is a decreasing sequence which tends to 0 as m grows large, we conclude that m is bounded. More precisely, m < m 0 where m 0 is the smallest integer such that B m0 < 1. As M ≤ 4 p (p-1) 2 ≤ 8 (see Remark 2.5), computation shows that

B m < 1 ⇔ p m > φ(M ) := 2+M+2 √ 1+M M . As (B m ) m≥1 is decreasing, |G 2 | ≤ p m B m ≤ φ 1 (M ) B 1 = φ(M ) M 4 p (p -1) 2
The claim follows.

We deduce that, for big actions (C, G) satisfying G M , an upper bound on |V | induces an upper bound on the genus g of C.

Corollary 4.2. Let M > 0 be a positive real such that (C, G) is a big action satisfying G M . Then,

g < |G ′ | |V | 2 p ≤ 2 p -1 2 + M + 2 √ 1 + M M 2 |V |
This raises the following question. Let (C, G) be a big action satisfying G M ; in which cases is |V | (and then g) bounded from above? In other words, in which cases, does the quotient |G| g take a finite number of values when (C, G) satisfy G M ? We begin with preliminary results on big actions leading to a purely group-theoretic discussion leading to compare the Frattini subgroup of G ′ with the commutator subgroup of G ′ and G.

∀ i ∈ {1, • • • , n}, W p i -W i = f i (X) = X S i (X) + c i X ∈ k[X]
where S i is an additive polynomial of k[X] with degree s i ≥ 1 in F and

s 1 ≤ s 2 • • • ≤ s n . Moreover, V ⊂ ∩ 1≤i≤n Z(Ad fi )
where Ad fi denotes the palindromic polynomial related to f i as defined in [START_REF] Rocher | Large p-groups actions with a p-elementary abelian second ramification group[END_REF] (Prop. 2.13)

Proof: The hypothesis first requires G ′ = G 2 to be abelian. Now, assume that G 2 has exponent strictly greater than p. Then, there exists a surjective map φ : Corollary 4.5. Let (C, G) be a big action. Let F := F ratt(G ′ ) be the Frattini subgroup of G ′ . 1. Then, F is trivial if and only if G ′ is an elementary abelian p-group.

G 2 → Z/p 2 Z. So H := Kerφ G 2 ⊂ Z(G)

We have the following inclusions

: F ⊂ [G ′ , G] G ′ .
3. The pair (C/F, G/F ) is a big action. Moreover, its second ramification group (G/F ) 2 = (G/F ) ′ = G 2 /F is p-elementary abelian.

4. Let M be a positive real. If (C, G) satisfies G M , then (C/F, G/F ) also satisfies G M .

Proof:

1. As G ′ is a p-group, F = (G ′ ) ′ (G ′ ) p ,
F ⊂ [G ′ , G]. As G ′ /[G ′ , G] is abelian , (G ′ ) ′ ⊂ [G ′ , G]. As G ′ /[G ′ , G] has exponent p, (G ′ ) p ⊂ [G ′ , G]. The claim follows. 3. Since F G ′ = G 2 is normal in G, we deduce from [MR08] (Lemma 2.4
) that the pair (C/F, G/F ) is a big action with second ramification group: (G/F ) 2 = G 2 /F = (G/F ) ′ . Furthermore, as G 2 is a p-group, G 2 /F is an elementary abelian p-group (see above).

4. This derives from [START_REF] Lehr | Automorphism groups for p-cyclic covers of the affine line[END_REF] (Prop. 8.5 (ii)). This leads us to discuss according to whether

F ratt(G ′ ) [G ′ , G] or F ratt(G ′ ) = [G ′ , G].

Case

: F ratt(G ′ ) [G ′ , G]
We start with the special case {e} = F ratt(G ′ ) [G ′ , G], i.e. G ′ is p-elementary abelian and G ′ ⊂ Z(G).

|V | ≤ 4 M |G 2 | (p -1) 2 ≤ 16 p (p -1) 4 2 + M + 2 √ 1 + M M 3
(1)

and p -1 2 |V | ≤ g < 32 p (p -1) 5 (2 + M + 2 √ 1 + M ) 2 M 5
(2)

Thus, under these conditions, g, |V | and so the quotient |G| g only take a finite number of values.

Proof: Write G ′ = G 2 ≃ (Z/pZ) n , with n ≥ 1. As G 2 ⊂ Z(G), [ Ro08a
] (Prop. 2.13) ensures the existence of a smaller integer j 0 ≥ 1 such that f j0+1 (X) cannot be written as c X + XS(X), with S in k{F }. If j 0 ≥ 2, it follows that, for all y in V , the coefficients of the matrix L(y) satisfy ℓ j,i (y) = 0 for all 2 ≤ i ≤ j 0 and 1 ≤ j ≤ i -1. Moreover, the matricial multiplication proves that, for all i in {1, • • • , j 0 }, the functions ℓ i,j0+1 are nonzero linear forms from V to F p . Put W := 1≤i≤j0 ker ℓ i,j0+1 . Let C fj 0 +1 be the curve parametrized by W p -W = f j0+1 (X). It defines an étale cover of the affine line with group Γ 0 ≃ Z/pZ. Since, for all y in W,

f i0+1 (X +y) = f i0+1 (X) mod ℘(k[X]
), the group of translations of the affine line: {X → X + y, y ∈ W} can be extended to a p-group of automorphisms of the curve C fj 0 +1 , say Γ, with the following exact sequence:

0 -→ Γ 0 ≃ Z/p Z -→ Γ -→ W -→ 0
The pair (C fj 0 +1 , Γ) is not a big action. Otherwise, its second ramification group would be p-cyclic, which contradicts the form of the function f j0+1 (X), as compared with [START_REF] Matignon | On smooth curves endowed with a large automorphism p-group in characteristic p[END_REF] (Prop. 2.5). Thus,

|Γ| gC f j 0 +1 = 2 p p-1 |W| (mj 0 +1-1) ≤ 2 p p-1 . The inequality |V | p j 0 ≤ |W| ≤ (m j0+1 -1
) combined with the formula given in [START_REF] Rocher | Large p-groups actions with a p-elementary abelian second ramification group[END_REF] (Cor. 2.7) yields a lower bound on the genus, namely:

g = p -1 2 n i=1 p i-1 (m i -1) ≥ p -1 2 p j0 (m j0+1 -1) ≥ p -1 2 |V |. It follows that M ≤ |G| g 2 = |G2| |V | g 2 ≤ 4 |G2| (p-1) 2 |V | .
Using Lemma 4.1, we gather inequality (1). Inequality (2) then derives from Corollary 4.2.

The following corollary generalizes the finiteness result of Proposition 4.6 to all big actions satisfying

G M such that F ratt(G ′ ) [G ′ , G]. Corollary 4.7. Let M > 0 be a positive real such that (C, G) is a big action satisfying G M . Suppose that F ratt(G ′ ) [G ′ , G].
(G/F ) 2 = G 2 /F is p-elementary abelian. From F [G 2 : G], we gather {e} [G 2 /F : G/F ], which implies (G/F ) 2 = (G/F ) ′ ⊂ Z(G/F ).
We deduce that |V | is bounded from above as in Proposition 4.6. The claim follows.

Case

: F ratt(G ′ ) = [G ′ , G] It remains to investigate the case where F ratt(G ′ ) = [G ′ , G]. In particular, this equality is satisfied when G ′ is included in the center of G and so is p-elementary abelian (cf. Lemma 3.3), i.e. {e} = F ratt(G ′ ) = [G ′ , G].
The finiteness result on g obtained in the preceding section is no more true in this case, as illustrated by the remark below.

Remark 4.8. For any integer s ≥ 1, Proposition 2.5 in [START_REF] Matignon | On smooth curves endowed with a large automorphism p-group in characteristic p[END_REF] exhibits an example of big actions

(C, G) with C : W p -W = X S(X) where S is an additive polynomial of k[X] with degree p s . In this case, g = p-1 2 p s , V = Z(Ad f ) ≃ (Z/pZ) 2s and G ′ = G 2 ≃ Z/pZ ⊂ Z(G). It follows that |G| g 2 = 4 p (p-1) 2 . So, for all M ≤ 4 p (p-1) 2 , (C, G) satisfies G M , with {e} = F ratt(G ′ ) = [G ′ , G], whereas g = p-1
2 p s grows arbitrary large with s. Therefore, in this case, neither g nor |V | are bounded. Nevertheless, the following section shows that, under these conditions, the quotient |G| g 2 take a finite number of values.

Case

: F ratt(G ′ ) = [G ′ , G] = {e}. Proposition 4.9. Let M > 0 be a positive real such that (C, G) is a big action satisfying G M . Assume that [G ′ , G] = F ratt(G ′ ) = {e}. Let s 1 be the integer in Lemma 4.3. Then, p 2s 1 g 2 and |V | p 2s 1
are bounded as follows:

p 2s1 g 2 ≤ (p -1) 2 4 p M 3 2 + M + 2 √ 1 + M (3)
and

1 ≤ |V | p 2s1 ≤ (p -1) 4 16 p M 3 2 + M + 2 √ 1 + M (4)
Thus, the quotient |G| g 2 takes a finite number of values.

Proof: Write G ′ = G 2 ≃ (Z/pZ) n , with n ≥ 1. Lemma 4.1 first implies that p n can only take a finite number of values. Moreover, as recalled in Lemma 4.3, V ⊂ n i=1 Z(Ad fi ) and |G| = |G 2 ||V | ≤ p n+2 s1
. We compute the genus by means of [START_REF] Rocher | Large p-groups actions with a p-elementary abelian second ramification group[END_REF] (Cor. 2.7):

g = p -1 2 n i=1 p i-1 (m i -1) = p -1 2 p s1 ( n i=1 p i-1 p si-s1 ) It follows that: 0 < M ≤ |G| g 2 ≤ 4 p n (p-1) 2 ( n i=1 p i-1 p s i -s 1 ) 2 This implies ( n i=1 p i-1 p si-s1 ) 2 ≤ 4 p n M (p-1) 2 . As p n is bounded from above, the set {s i -s 1 , i ∈ [1, n]} ⊂ N is
also bounded, and then finite. More precisely, we gather that

g 2 p 2s1 = (p -1) 2 4 ( n i=1 p i-1 p si-s1 ) 2 ) ≤ p n M Combined with Lemma 4.1, this gives inequality (3). Besides, from M ≤ |G| g 2 = |V | p n g 2 , we infer that 1 |V | ≤ p n M g 2 , which involves: 1 ≤ p 2s1 |V | ≤ p 2 s1 p n M g 2 = 4 p n M (p -1) 2 ( n i=1 p i-1 p si-s1 ) 2 ≤ 4 p n M (p -1) 2
This, together with Lemma 4.1, yields inequality (4). In particular, the set { p 2s 1

|V | } ⊂ N is bounded, and then finite, as well as the set

{ |V | p 2s 1 }. Therefore, the quotient |G| g 2 = p n |V | p 2s 1 p 2s 1
g 2 can only take a finite number of values.

The last remaining case is

F ratt(G ′ ) = [G ′ , G] = {e}. 4.4.2 Case: F ratt(G ′ ) = [G ′ , G] = {e}.
As shown below, this case can only occur for G ′ (= G 2 ) non abelian. Note that we do not know yet examples of big actions with a non abelian

G ′ (= G 2 ). Theorem 4.10. Assume that p > 2. Let (C, G) be a big action with F ratt(G ′ ) = [G ′ , G] = {e}. Then, G ′ (= G 2 ) is non abelian.
We deduce the following Corollary 4.11. Assume that p > 2. Let M > 0 be a positive real. Let (C, G) be a big action satisfying G M with G ′ abelian. Then, |G| g 2 only takes a finite number of values.

Remark 4.12. Theorem 4.10 is no more true for p = 2. A counterexample is given by [START_REF] Matignon | On smooth curves endowed with a large automorphism p-group in characteristic p[END_REF] (Prop. 6.9) applied with p = 2. Indeed, when keeping the notations of [MR08] (Prop. 6.9), take q = p e with p = 2, e = 2s -1 and s ≥ 2.

Put K = F q (X). Let L := F q (X, W 1 , V 1 , W 2 ) be the extension of K parametrized by W 2 2s-1 1 -W 1 = X 2 s-1 (X 2 2s-1 -X) V 2 2s-1 1 -V 1 = X 2 s-2 (X 2 2s-1 -X) [W 1 , W 2 ] 2 -[W 1 , W 2 ] = [X 1+2 s , 0] -[X 1+2 s-1 , 0]
Let G be the p-group of F q -automorphisms of L constructed as in [START_REF] Matignon | On smooth curves endowed with a large automorphism p-group in characteristic p[END_REF] (Prop. 6.9.3). Then, the formula established for g L in [MR08] (Prop. 6.9.4) shows that the pair (C, G) is a big action as soon as s ≥ 4. In this case,

G ′ = G 2 ≃ Z/2 2 Z × (Z/2Z) 6s-4 (cf.
[MR08] Prop. 6.7.2). As the functions X 2 s-1 (X 2 2s-1 -X) and X 2 s-2 (X 2 2s-1 -X) are products of two additive polynomials, it follows from next proof (cf. point 6) that

[G ′ , G] = F ratt(G ′ ) = {e}.
Proof of Theorem 4.10:

1. Preliminary remarks: the link with Theorem 5.1 in [START_REF] Matignon | On smooth curves endowed with a large automorphism p-group in characteristic p[END_REF].

(a) One first remarks that Theorem 4.10 implies Theorem 5.1 in [START_REF] Matignon | On smooth curves endowed with a large automorphism p-group in characteristic p[END_REF]. The latter states that there is no big action (C, G) with G 2 cyclic of exponent strictly greater than p. Indeed, assume that there exists one. Then,

G ′ = G 2 is abelian and F ratt(G ′ ) = (G ′ ) p = {e}.
To contradicts Theorem 4.10, it remains to show that

F := F ratt(G ′ ) = [G ′ , G]. From Corollary 4.5, we infer that (C/F, G/F ) is a big action whose second ramification group G 2 /F is cyclic of order p. Then, (G/F ) ′ = (G/F ) 2 = G 2 /F ⊂ Z(G/F ) (cf. [MR08] Prop. 2.5 and [Ro08a] Prop. 2.13). It follows that F ratt((G/F ) ′ ) = [(G/F ) ′ , G/F ] = {e}. As F ⊂ G ′ , this imposes F = [G ′ , G].
Then, Theorem 4.10 contradicts the fact that G ′ = G 2 is abelian.

(b) The object of Theorem 4.10 is to prove that there exists no big action (C, G) with G ′ = G 2 abelian of exponent strictly greater than p such that F ratt

(G ′ ) = [G ′ , G].
The proof follows the same canvas as the proof of [MR08] (Thm. 5.1). Nevertheless, to refine the arguments, we use the formalism related to the ring filtration of k[X] linked with the additive polynomials as introduced in [Ro08a] (cf. section 3). More precisely, we recall that, for any t ≥ 1, we define Σ t as the k-subvector space of k[X] generated by 1 and the products of at most t additive polynomials of k[X] (cf. [Ro08a] Def. 3.1). In what follows, we assume that there exists a big action (C, G) with G ′ = G 2 abelian of exponent strictly greater than p such that F ratt

(G ′ ) = [G ′ , G]. 2. One can suppose that G ′ = G 2 ≃ Z/p 2 Z × (Z/pZ) r , with r ≥ 1. Indeed, write G ′ /(G ′ ) p 2 ≃ (Z/p 2 Z) a × (Z/pZ) b . By assumption, a ≥ 1. Using [Su82] (Chap.2, Thm. 19), one can find an index p-subgroup of (G ′ ) p , normal in G, such that (G ′ ) p 2 ⊂ H (G ′ ) p G ′ = G 2 . Then, we infer from [MR08] (Lemma 2.4) that (C/H, G/H) is a big action with second ramification group (G/H) ′ = (G/H) 2 = G 2 /H ≃ (Z/p 2 Z) × (Z/pZ) a+b-1 . Furthermore, as G ′ is abelian, F ratt(G ′ ) = (G ′ ) p (resp. F ratt((G/H) ′ ) = ((G/H) ′ ) p ). From H ⊂ (G ′ ) p with H normal in G and F ratt(G ′ ) = [G ′ , G], we gather that F ratt((G/H) ′ ) = (G ′ ) p /H = F ratt(G ′ )/H = [(G/H) ′ , G/H].

Notation.

In what follows, we denote by L := k(C) the function field of C and by k(X) := L G2 the subfield of L fixed by G 2 . Following Artin-Schreier-Witt theory as already used in [START_REF] Matignon | On smooth curves endowed with a large automorphism p-group in characteristic p[END_REF] (proof of Thm. 5.1, point 2), we introduce the W 2 (F p )-module

A := ℘(W 2 (L)) ∩ W 2 (k[X]) ℘(W 2 (k[X]))
where W 2 (L) means the ring of Witt vectors of length 2 with coordinates in L and ℘ = F -id.

One can prove that A is isomorphic to the dual of G 2 with respect to the Artin-Schreier-Witt pairing (cf. [START_REF] Bourbaki | Algèbre commutative. Eléments de Mathématiques[END_REF] Chap. IX, ex. 19). Moreover, as a Z-module, A is generated by the classes mod ℘(k[X]) of (f 0 (X), g 0 (X)) and {(0,

f i (X))} 1≤i≤r in W 2 (k[X]
). In other words, L = k(X, W i , V 0 ) 0≤i≤r is parametrized by the following system of Artin-Schreier-Witt equations:

℘([W 0 , V 0 ]) = [f 0 (X), g 0 (X)] ∈ W 2 (k[X]) and ∀ i ∈ {1, • • • , r}, ℘(W i ) = f i (X) ∈ k[X]
An exercise left to the reader shows that one can choose g 0 (X) and each f i (X), with 0

≤ i ≤ r, reduced mod ℘(k[X]).
4. We prove that f 0 ∈ Σ 2 .

As a Z-module, p A is generated by the class of (0, f 0 (X)) in A. By the Artin-Schreier-Witt pairing, p A corresponds to the kernel G 2 [p] of the map:

G 2 → G 2 g → g p Thus, G 2 [p] G 2 is a normal subgroup of G. Then, it follows from [MR08] (Lemma 2.4) that the pair (C/G 2 [p], G/G 2 [p]
) is a big action parametrized by W p -W = f 0 (X) and with second ramification group

G 2 /G 2 [p] ≃ Z/pZ. Then, f 0 (X) = X S(X) + c X ∈ k[X] (cf. [MR08] Prop. 2.5
), where S is an additive polynomial of k{F } with degree s ≥ 1 in F .

The embedding problem.

For any y ∈ V , the classes mod ℘(k[X]) of (f 0 (X + y), g 0 (X + y)) and {(0, f i (X + y))} 1≤i≤r induces a new generating system of A. As in [START_REF] Matignon | On smooth curves endowed with a large automorphism p-group in characteristic p[END_REF] (proof of Thm 5.1, point 3), this is expressed by the following equation:

∀ y ∈ V, (f 0 (X + y), g 0 (X + y)) = (f 0 (X), g 0 (X) + r i=0 ℓ i (y) f i (X)) mod ℘(W 2 (k[X])) (5)
where, for all i in {0, • • • , r}, ℓ i is a linear form from V to F p . On the second coordinate, (5) reads:

∀ y ∈ V, ∆ y (g 0 ) := g 0 (X + y) -g 0 (X) = r i=0 ℓ i (y) f i (X) + c mod ℘(k[X]) (6)
where

c = p-1 i=1 (-1) i i y p-i X i+p s+1 + lower degree terms in X (7)
For more details on calculation, we refer to [START_REF] Matignon | On smooth curves endowed with a large automorphism p-group in characteristic p[END_REF] (proof of Thm 5.1, point 3 and Lemma 5.2).

6. We prove that

f i lies in Σ 2 , for all i in {0, • • • , r}, if and only if F ratt(G ′ ) = [G ′ , G].
Put F := F ratt(G ′ ). We deduce from Corollary 4.5 that (C/F, G/F ) is a big action whose second ramification group (G/F ) ′ = (G/F ) 2 = G 2 /F is p-elementary abelian. The function field of the curve C/F is now parametrized by the Artin-Schreier equations:

∀ i ∈ {0, • • • , r}, ℘(W i ) = f i (X) ∈ k[X] As F ⊂ [G ′ , G] (cf. Lemma 4.5), F = [G ′ , G] = [G 2 : G] ⇔ {e} = [G 2 /F, G/F ] = [(G/F ) ′ , G/F ] ⇔ (G/F ) ′ ⊂ Z(G/F )
By [START_REF] Rocher | Large p-groups actions with a p-elementary abelian second ramification group[END_REF] (Prop.2.13), this occurs if and only if for all i in {0,

• • • , r}, f i (X) = X S i (X) + c i X ∈ Σ 2 .
7. We prove that g 0 does not belong to Σ p . We first notice that the right-hand side of (6) does not belong to Σ p-1 : indeed, the monomial

X p-1+p s+1 ∈ Σ p -Σ p-1 occurs once in c but not in r i=0 ℓ i (y) f i (X) which lies in Σ 2 ⊂ Σ p-1
, for p ≥ 3. Now, assume that g 0 ∈ Σ p . Then, by [START_REF] Rocher | Large p-groups actions with a p-elementary abelian second ramification group[END_REF] (Lemma 3.9), the left-hand side of (6), namely ∆ y (g 0 ), lies in Σ p-1 , hence a contradiction. Therefore, one can define an integer a such that X a is the monomial of g 0 (X) with highest degree among those that do not belong to Σ p . Note that since g 0 is reduced mod ℘(k[X]), a = 0 mod p.

8. We prove that a -1 ≥ p -1 + p s+1 .

We have already seen that the monomial X p-1+p s+1 occurs in the right hand side of (6). In the left-hand side of (6), X p-1+p s+1 is produced by monomials

X b of g 0 with b > p -1 + p s+1 . If b > a, X b ∈ Σ p , so ∆ y (X b ) ∈ Σ p-1
, which is not the case of X p-1+p s+1 . It follows that X p-1+p s+1 comes from monomials X b with a ≥ b > p-1+p s+1 . Hence the expected inequality. 9. We prove that p divides a -1. Assume that p does not divide a -1. In this case, the monomial X a-1 is reduced mod ℘(k[X]) and (6) reads as follows:

∀ y ∈ V, c a (g 0 ) a y X a-1 + S p-1 (X) + R a-2 (X) = c + r i=0 ℓ i (y) f i (X) mod ℘ (k[X])
where c a (g 0 ) = 0 denotes the coefficient of X a in g 0 , S p-1 (X) is a polynomial in Σ p-1 produced by monomials X b of g 0 with b > a and R a-2 (X) is a polynomial of k[X] with degree lower than a -2 produced by monomials X b of g 0 with b ≤ a. We first notice that X a-1 does not occur in S p-1 (X). Otherwise, X a-1 ∈ Σ p-1 and X a = X a-1 X ∈ Σ p , hence a contradiction. Likewise, X a-1 does not occur in

r i=0 ℓ i (y) f i (X) ∈ Σ 2 . Otherwise, X a = X a-1 X ∈ Σ 3 ⊂ Σ p , as p ≥ 3. It follows that X a-1 occurs in c, which requires a -1 ≤ deg b = p -1 + p s+1 .
Then, the previous point implies a -1 = p -1 + p s+1 , which contradicts a = 0 mod p. Thus, p divides a -1. So, we can write a = 1 + λ p t , with t > 0, λ prime to p and λ ≥ 2 because of the definition of a. We also define j 0 := a -p t = 1 + (λ -1) p t . 10. We search for the coefficient of the monomial X j0 in the left-hand side of (6).

Since p does not divide j 0 , the monomial X j0 is reduced mod ℘(k[X]). In the left-hand side of (6), namely ∆ y (g 0 ) mod ℘(k[X]), the monomial X j0 comes from monomials of g 0 (X) of the form: X b , with b ≥ j 0 + 1. However, as seen above, the monomials X b with b > a produce in ∆ y (g 0 ) elements that belong to Σ p-1 , whereas X j0 ∈ Σ p-1 . Otherwise, X a = X j0 X p t ∈ Σ p , which contradicts the definition of a. So we only have to consider the monomials X b of g 0 (X) with b ∈ {j 0 + 1, • • • , a}. Then, the same arguments as those used in [START_REF] Matignon | On smooth curves endowed with a large automorphism p-group in characteristic p[END_REF] (proof of Thm. 5.1, point 6) allow to conclude that the coefficient of X j0 in the left-hand side of (6) is T (y) where T (Y ) denotes a polynomial of k[X] with degree p t .

11. We identify with the coefficient of X j0 in the right-hand side of (6) and gather a contradiction.

As mentionned above, the monomial X j0 does not occur in r i=0 ℓ i (y) f i (X) ∈ Σ 2 ⊂ Σ p-1 , for p ≥ 3. Assume that the monomial X j0 appears in c, which implies that j 0 ≤ p -1 + p s+1 . Using the same arguments as in [START_REF] Matignon | On smooth curves endowed with a large automorphism p-group in characteristic p[END_REF] (proof of Thm. 5.1, point 7), we gather that j 0 = 1 + (λ -1) p t = 1 + p s+1 . Then, X j0 lies in Σ 2 , which leads to the same contradiction as above. Therefore, the monomial X j0 does not occur in the right-hand side of (6). Then, T (y) = 0 for all y in V , which means that |V | ≤ p t . Call C 0 the curve whose function field is parametrized by ℘([W 0 , V 0 ]) = [f 0 (X), g 0 (X)]. The same calculation as in [START_REF] Matignon | On smooth curves endowed with a large automorphism p-group in characteristic p[END_REF] (proof of Theorem 5.1, point 7) shows that g C0 ≥ p t+1 (p -1). Furthermore, g ≥ p r g C0 (see e.g. [START_REF] Lehr | Automorphism groups for p-cyclic covers of the affine line[END_REF] Prop. 8.5, formula (8)). As |G| = |G 2 ||V | ≤ p 2+r+t , it follows that |G| g = p p-1 < 2 p p-1 , hence a contradiction.

Classification of big actions under condition ( * ).

We now pursue the classification of big actions initiated by Lehr and Matignon who characterize big actions (C, G) satisfying |G| g 2 ≥ 4 (p-1) 2 (cf. [START_REF] Lehr | Automorphism groups for p-cyclic covers of the affine line[END_REF]). In this section, we exhibit a parametrization for big actions (C, G) satisfying condition ( * ), namely:

|G| g 2 ≥ 4 (p 2 -1) 2 ( * )
As proved in [START_REF] Matignon | On smooth curves endowed with a large automorphism p-group in characteristic p[END_REF] (Prop. 4.1 and Prop. 4.2), this condition requires G ′ (= G 2 ) to be an elementary abelian p-group with order dividing p 3 . Since G 2 cannot be trivial (cf. [MR08] Prop. 2.2), this leaves three possibilities. This motivates the following Definition 5.1. Let (C, G) be abig action. Let i ≥ 1 be an integer. We say that

1. (C, G) satisfies G * if (C, G) satisfies condition ( * ) 2. (C, G) satisfies G p i * if (C, G) satisfies G * with G ′ ≃ (Z/pZ) i .
5.1 Preliminaries: big actions with a p-elementary abelian G ′ (= G 2 ).

To start with, we fix the notations and recall some necessary results on big actions with a pelementary abelian G 2 drawn from [START_REF] Rocher | Large p-groups actions with a p-elementary abelian second ramification group[END_REF].

Recall 5.2. Let (C, G) be a big action such that G ′ (= G 2 ) ≃ (Z/pZ) n , n ≥ 1. Write the exact sequence: 0 -→ G 2 ≃ (Z/pZ) n -→ G π -→ V ≃ (Z/pZ) v -→ 0
1. We denote by L be the function field of the curve C and by k(X) := L G2 the subfield of L fixed by G 2 . Then, the extension L/k(X) can be parametrized by n Artin-Schreier equations:

W p i -W i = f i (X) ∈ k[X] with 1 ≤ i ≤ n. Following [Ro08a] (Def. 2.
3), one can choose an "adapted basis" {f 1 (X), • • • , f n (X)} with some specific properties:

(a) For all i ∈ {1, • • • , n}, each function f i is assumed to be reduced mod ℘(k[X]) (b) For all i ∈ {1, • • • , n}, put m i := deg f i . Then, m 1 ≤ m 2 ≤ • • • ≤ m n . (c) ∀ (λ 1 , • • • λ n ) ∈ F n p not all zeros, deg ( n i=1 λ i f i (X)) = max i∈{1,••• ,n} {deg λ i f i (X)}.
In this case, the genus of the curve C is given by the following formula (cf. [START_REF] Rocher | Large p-groups actions with a p-elementary abelian second ramification group[END_REF] Cor. 2.7):

g = p -1 2 n i=1 p i-1 (m i -1) (8) 2. Now, consider the F p -subvector space of k[X] generated by the classes of {f 1 (X), • • • , f n (X)} mod ℘(k[X]): A := ℘(L) ∩ k[X] ℘(k[X])
Recall that A is isomorphic to the dual of G 2 with respect to the Artin-Schreier pairing (cf.

[Ro08a] section 2.1). As seen in [Ro08a] (section 2.2), V acts on G 2 via conjugation. This induces a representation φ: V → Aut(G 2 ). The representation ρ : V → Aut(A), which is dual with respect to the Artin-Schreier pairing, expresses the action of V on A by translation. More precisely, for all y in V , the automorphism ρ(y) is defined as follows:

ρ(y) : A → A f (X) → f (X + y)
where f (X) means the class in A of f (X) ∈ k[X] For all y in V , the matrix of the automorphism ρ(y) in the adapted basis fixed for A is an upper triangular matrix of Gl n (F p ) with identity on the diagonal, namely

L(y) :=       1 ℓ 1,2 (y) ℓ 1,3 (y) • • • ℓ 1,n (y) 0 1 ℓ 2,3 (y) • • • ℓ 2,n (y) 0 0 • • • • • • ℓ i,n (y) 0 0 0 1 ℓ n-1,n (y) 0 0 0 0 1       ∈ Gl n (F p )
where, for all i in {1, • • • , n -1}, ℓ i,i+1 is a nonzero linear form from V to F p (see [START_REF] Rocher | Large p-groups actions with a p-elementary abelian second ramification group[END_REF] section 2.4). In other words,

∀ y ∈ V, f 1 (X + y) -f 1 (X) = 0 mod ℘(k[X]) ∀ i ∈ {2, • • • , n}, ∀ y ∈ V, f i (X + y) -f i (X) = i-1 j=1 ℓ j,i (y) f j (X) mod ℘(k[X]) (9)
For all map ℓ, we write ℓ = 0 if ℓ is identically zero and ℓ = 0 otherwise.

3. The case of a trivial representation can be charactrized as follows (see [START_REF] Rocher | Large p-groups actions with a p-elementary abelian second ramification group[END_REF] Prop. 2.13). Indeed, the following assertions are equivalent:

(a) The representation ρ is trivial, i.e.

∀ i ∈ {1, • • • , n}, ∀ y ∈ V, f i (X + y) -f i (X) = 0 mod ℘(k[X]) (b) The commutator subgroup of G ′ and G is trivial, i.e. G ′ ⊂ Z(G). (c) For all i in {1, • • • , n}, f i (X) = X S i (X) + c i X ∈ k[X]
where each S i ∈ k{F } is an additive polynomial with degree s i ≥ 1 in F . So, write S i (F ) = si j=0 a i,j F j with a i,si = 0. Then, one defines an additive polynomial related to f i , called the "palindromic polynomial" of f i :

Ad fi := 1 a p s i i,si F si ( si j=0 a i,j F j + F -j a i,j ) In this case, V ⊂ n i=1 Z(Ad fi )
Since, under condition ( * ), G ′ is p-elementary abelian, we deduce from point (b) that the case of a trivial representation corresponds to the case {e}

= F ratt(G ′ ) = [G ′ , G].
4. To conclude, we recall that for all t ≥ 1, Σ t means the k-subvector space of k[X] generated by 1 and the products of at most t additive polynomials of k[X] (cf. [Ro08a] Def. 3.1). As proved in [START_REF] Rocher | Large p-groups actions with a p-elementary abelian second ramification group[END_REF] (Thm. 3.13), for all i in {1, • • • , n}, f i lies in Σ i+1 .

5.2 First case: big actions satisfying G p * .

Proposition 5.3. We keep the notations of section 5.1.

(C, G) is a big action with G 2 ≃ Z/pZ if and only if C is birational to a curve C f parametrized by W p -W = f (X) = X S(X) ∈ k[X],
where S is a (monic) additive polynomial with degree s ≥ 1 in F .

2.

In what follows, we assume that C is birational to a curve C f as described in the first point.

(a) If s ≥ 2, A ∞,1 is the unique p-Sylow subgroup of A, where ∞ denotes the point of C corresponding to X = ∞.

(b) If s = 1, there exists r := p 3 + 1 points of C: P 0 := ∞, P 1 , • • • , P r such that (A Pi,1 ) 0≤i≤r are the p-Sylow subgroups of A. In this case, for all i in {1, • • • , r}, there exists σ i ∈ A such that σ i (P i ) = ∞.

In both cases, A ∞,1 is an extraspecial group (see [START_REF] Suzuki | Group theory. II. Grundlehren der Mathematischen Wissenschaften[END_REF] Def. 4.14) with exponent p (resp. p 2 ) if p > 2 (resp. p = 2) and order p 2s+1 . More precisely, A ∞,1 is a central extension of its center Z(A ∞,1 ) = (A ∞,1 ) ′ by the elementary abelian p-group Z(Ad f ), i.e.

0 -→ Z(A ∞,1 ) = (A ∞,1 ) ′ ≃ Z/pZ -→ A ∞,1 π -→ Z(Ad f ) ≃ (Z/pZ) 2s -→ 0
Furthermore, (C, A ∞,1 ), and so each (C, A Pi,1 ), with 1 ≤ i ≤ r, are big actions satisfying G p * .

3. Let V be a subvector space of Z(Ad f ) with dimension v over F p . Then, (C, π -1 (V )) is also a big action satisfying G p * if and only if

if p = 2, 2s ≥ v ≥ max{s + 1, 2s -3} if p = 2, 2s ≥ v ≥ max{s + 1, 2s -4}
We collect the different possibilities in the table below:

case v s V G 1- 2s s ≥ 1 † Z(Ad f ) † A † ∞,1 2 2s -1 s ≥ 2 index p subgroup of Z(Ad f ) index p subgroup of A ∞,1 3 2s -2 s ≥ 3 index p 2 subgroup of Z(Ad f ) index p 2 subgroup of A ∞,1 4 2s -3 s ≥ 4 index p 3 subgroup of Z(Ad f ) index p 3 subgroup of A ∞,1 5 (p=2) 2s -4 s ≥ 5 index p 4 subgroup of Z(Ad f ) index p 4 subgroup of A ∞,1 case |G|/g |G|/g 2 1 2 p p-1 p s 4 (p 2 -1) 2 (p + 1) 2 p 2 2 p p-1 p s-1 4 (p 2 -1) 2 (p + 1) 2 3 2 p p-1 p s-2 4 (p 2 -1) 2 (p+1) 2 p 4 2 p p-1 p s-3 4 (p 2 -1) 2 (p+1) 2 p 2 5 (p=2) 2 p p-1 p s-4 4 (p 2 -1) 2 (p+1) 2
p 3 † Note: In the case s = 1, this result is true up to conjugation by σ i as defined in Proposition 5.3.

Proof:

1 3. This essentially derives from Proposition 3.2 which implies (p+1) 2 ≥ p 2s-v-1 . If 2s-v -1 ≥ 3, it implies p 2 + 2 p + 1 ≥ p 3 , which is impossible for p > 2. Accordingly, if p > 2, we obtain 2s -v -1 ≤ 2, which means v ≥ 2s -3. If p = 2, (p + 1) 2 ≥ p 2s-v-1 is satisfied if and only if 2s -v -1 ≤ 3, i.e. v ≥ 2s -4. The claim follows.

Remark 5.4. Note that, for p > 2, the solutions can be parametrized by s algebraically independent variables over F p , namely the s coefficients of S assumed monic after an homothety on the variable X. Note that s ∼ log g.

5.

3 Second case: big actions satisfying G p 2 * .

Case

: [G ′ , G] = F ratt(G ′ ) = {e}.
Proposition 5.5. Let (C, G) be a big action satisfying G p 2 * . Assume that [G ′ , G] = {e} and keep the notations of section 5.1.

1. The pair (C, A ∞,1 ) is a big action satisfying G p 2 * . Moreover, A ∞,1 is a special group (see [START_REF] Suzuki | Group theory. II. Grundlehren der Mathematischen Wissenschaften[END_REF] Def. 4.14) with exponent p (resp. p 2 ) (for p > 2 (resp. p = 2) and order p 2+2 s1 . More precisely, A ∞,1 is a central extension of its center Z(A ∞,1 ) = (A ∞,1 ) ′ by the elementary abelian p-group Z(Ad f1 ), i.e.

0 -→ Z(A ∞,1 ) = (A ∞,1 ) ′ ≃ (Z/pZ) 2 -→ A ∞,1 π -→ Z(Ad f1 ) ≃ (Z/pZ) 2s1 -→ 0 2. Furthermore, s 2 = s 1 or s 2 = s 1 + 1. (a) If s 2 = s 1 , G = π -1 (V ), where V is a subvector space of Z(Ad f1 ) with dimension v over F p such that 2 s 1 -2 ≤ v ≤ 2 s 1 . Then, A ∞,1 is a p-Sylow subgroup of A. It is normal except if C is birationnal to the Hermitian curve: W q -W = X 1+q with q = p 2 . (b) If s 2 = s 1 + 1, V = Z(Ad f1 ) and G = A ∞,1 is the unique p-Sylow subgroup of A.
The different possibilities are listed in the table below:

case s 1 s 2 v V G (a)-1 s ≥ 2 s 2s Z(Ad f1 ) = Z(Ad f2 ) A ∞,1 (a)-2 s ≥ 2 s 2s -1 index p subgroup of Z(Ad f1 ) index p subgroup of A ∞,1 (a)-3 s ≥ 3 s 2s -2 index p 2 subgroup of Z(Ad f1 ) index p 2 subgroup of A ∞,1 (b) s ≥ 3 s + 1 2s Z(Ad f1 ) A ∞,1 case |G|/g |G|/g 2 (a)-1 2 p p-1 p 1+s 1+p 4 (p 2 -1) 2 p 2 (a)-2 2 p p-1 p s 1+p 4 (p 2 -1) 2 p (a)-3 2 p p-1 p s-1 1+p 4 (p 2 -1) 2 (b) 2 p p-1 p 1+s 1+p 4 (p 2 -1) 2 p 2 (p+1) 2 (1+p 2 ) 2 Proof:
1. Use Proposition 3.2 to prove that the pair (C, A ∞,1 ) is a big action satisfying G p 2 * with the following exact sequence:

0 -→ A ∞,2 -→ A ∞,1 π -→ Z(Ad f1 ) ≃ (Z/ p Z) 2 s1 -→ 0 The proof to show that A ∞,1 is a special group, i.e. satisfies Z(A ∞,1 ) = (A ∞,1 ) ′ = F ratt(A ∞,1 ) ≃ (Z/pZ) 2 ,
is the same that the one exposed in [START_REF] Rocher | Large p-groups actions with a p-elementary abelian second ramification group[END_REF] (Prop. 4.3.3). Nevertheless, one has to choose H an index p-subgroup of G 2 such that C/H is the curve parametrized by

W p 1 -W 1 = f 1 (X).
2. Assume that s 2 -s 1 ≥ 2. Then, |G| = p 2+v ≤ p 2+2 s1 and g = p-1 2 p s1 (1 + p 1+s2-s1 ) ≥ p-1

2 p s1 (1 + p 3 ). So, |G| g 2 ≤ 4 (p 2 -1) 2 (1+p) 2 p 2 (1+p 3 ) 2 < 4 (p 2 -1) 2 ,
which contradicts condition ( * ). So, 0 ≤ s 2 -s 1 ≤ 1. In each case, the description of A ∞,1 and G derive from Proposition 3.2 combined with Remark 3.1.

To go further in the description of the functions f ′ i s in each case, we introduce two additive polynomials V and T defined as follows:

∀ i ∈ {1, 2}, V := y∈V (X -y) divides T := gcd{Ad f1 , Ad f2 } divides Ad fi
In what follows, we work in the Ore ring k{F } and write the additive polynomials as polynomials in F .

case deg F V deg F T deg F (Ad f1 ) deg F (Ad f2 ) T (a)-1 2s 2s 2s 2s V = T = Ad f1 = Ad f2 (a)-2-i 2s -1 2s 2s 2s V divides T = Ad f1 = Ad f2 (a)-2-ii 2s -1 2s -1 2s 2s V = T divides Ad f1 (a)-3-i 2s -2 2s 2s 2s V divides T = Ad f1 = Ad f2 (a)-3-ii 2s -2 2s -1 2s 2s V divides T divides Ad f1 (a)-3-iii 2s -2 2s -2 2s 2s V = T divides Ad f1 (b) 2s 2s 2s 2s + 2 V = T = Ad f1 divides Ad f2
The three cases where Ad f1 = Ad f2 can be parametrized in the same way as in [START_REF] Rocher | Large p-groups actions with a p-elementary abelian second ramification group[END_REF] (Prop. 4.2).

case

S 1 or Ad f1 S 2 or Ad f2 (a)-1 S 1 = s/d j=0 α jd F jd , α s = 1 S 2 = γ S 1 , γ ∈ F p d -F p , d ≥ 2 (a)-2-i S 1 = s/d j=0 α jd F jd , α s = 1 S 2 = γ S 1 , γ ∈ F p d -F p , d ≥ 2 (a)-2-ii Ad f1 = (α 1 F + β 1 I) T, α 1 = 0 Ad f2 = (α 2 F + β 2 I) T, α 2 = 0 (a)-3-i S 1 = s/d j=0 α jd F jd , α s = 1 S 2 = γ S 1 , γ ∈ F p d -F p , d ≥ 2 (a)-3-ii Ad f1 = (α 1 F + β 1 I) T, α 1 = 0 Ad f2 = (α 2 F + β 2 I) T, α 2 = 0 (a)-3-iii Ad f1 = (α 1 F 2 + β 1 F + δ 1 I) T, α 1 = 0 Ad f2 = (α 2 F 2 + β 2 F + δ 2 I) T, α 2 = 0 (b) Ad f1 = v∈V (X -v) Ad f2 = (α 2 F 2 + β 2 F + δ 2 I) Ad f1 , α 2 = 0
We display the parametrization of the functions f i 's in the case (a)-2-ii for the smallest values of s, namely s = 2 and s = 3.

Cas (a)-2-ii with s = 2 for p > 2.

f 1 f 1 (X) = X 1+p 2 + a 1+p X 1+p + 1 2 a 2 X 2 a 1+p a 1+p ∈ k a 2 a 2 ∈ k f 2 f 2 (X) = b p 2 1+p 2 X 1+p 2 + b 1+p X 1+p + b 2 X 2 + b 1 X b 1+p 2 b 1+p 2 ∈ Z(w p 2 X p 3 + w p , (-a p 2 + a p 1+p w p 2 -w p 2 +p 3 ) X p 2 + (a 1+p -w p 2 ) X p -w -1 X) with b 1+p 2 ∈ F p 2 . w w ∈ Z(X 1+p+p 2 +p 3 -a p 1+p X 1+p+p 2 + a p 2 X 1+p -a 1+p X + 1) b 1+p b 1+p = w p 2 (b p 2 1+p 2 -b 1+p 2 ) p + b p 1+p 2 a 1+p b 2 2 b 2 = w p (b p 2 1+p 2 -b 1+p 2 ) (a 1+p -w p 2 ) + b 1+p 2 2 b 1 b 1 ∈ k Case (a)-2-ii with s = 3 for p > 2. f 1 f 1 (X) = X 1+p 3 + a 1+p 2 X 1+p 2 + a 1+p X 1+p + 1 2 a 2 X 2 a 1+p 2 a 1+p 2 ∈ k a 1+p a 1+p ∈ k a 2 a 2 ∈ k f 2 f 2 (X) = b p 3 1+p 3 X 1+p 3 + b 1+p 2 X 1+p 2 + b 1+p X 1+p + b 2 X 2 + b 1 X w w ∈ Z(X 1+p+p 2 +p 3 +p 4 +p 5 -a p 2 1+p 2 X 1+p+p 2 +p 3 +p 4 +a p 2 1+p X 1+p+p 2 +p 3 -a p 2 2 X 1+p+p 2 + a p 1+p X 1+p -a 1+p 2 X + 1) b 1+p 3 b 1+p 3 ∈ Z(P 1 ) ∩ Z(P 2 ) -F p 3 with P 1 (X) = w p 3 +1 X p 5 + (1 -w a 1+p 2 )X p 3 + (w a 1+p 2 -w p 3 +1 ) X p 2 -X with P 2 (X) = w p 2 (a 1+p 2 -w p 3 ) X p 4 + w p (-a p 2 + a p 1+p w p 2 -a p 1+p 2 w p 2 +p 3 + w p 2 +p 3 +p 4 ) X p 3 +(a 1+p + w p 2 +p 3 -a 1+p 2 w p2 ) X p +(-a 1+p + a p 2 w p -a p 1+p w p+p 2 + a p 1+p 2 w p+p 2 +p 3 -w p+p 2 +p 3 +p 4 ) X b 1+p 2 b 1+p 2 = w p 3 (b p 3 1+p 3 -b 1+p 3 ) p 2 + b p 2 1+p 3 a 1+p 2 b 1+p b 1+p = w p 2 (b p 3 1+p 3 -b 1+p 3 ) p (a 1+p 2 -w p 3 ) + b p 1+p 3 a 1+p b 2 2 b 2 = w p (b p 3 1+p 3 -b 1+p 3 ) (a 1+p -a 1+p 2 w p 2 + w p 2 +p 3 ) + b 1+p 3 a 2 b 1 b 1 ∈ k
The calculation of the case s = 3 already raises a problem as the parameter b 1+p 3 has to lie in the set of zeroes of two polynomials.

For the remaining last two cases (a)-3-iii and (b), we merely display examples of realization so as to prove the effectiveness of these cases.

An example of realization for the case (a)-3-iii.

T T = F 2s-2 + I V V = Z(F 2s-2 + I) Ad f1 Ad f1 = (F 2 + I) T f 1 f 1 (X) = X 1+p s + X 1+p s-2 Ad f2 Ad f2 = (F 2 + F + I) T f 2 f 2 (X) = X 1+p s + X 1+p s-1 + X 1+p s-2
An example of realization for the case (b).

f 1 f 1 (X) = X 1+p s f 2 f 2 (X) = α 2 X 1+p s+1 + β 2 X 1+p s + δ 2 X 1+p s-1 α 2 α 2 ∈ F p 2s β 2 β 2 ∈ F p s δ 2 δ 2 ∈ F p 2s 5.3.2 Case: [G ′ , G] F ratt(G ′ ) = {e}.
Proposition 5.6. Let (C, G) be a big action satisfying G p 2 * such that [G ′ , G] = {e}. We keep the notations introduced in section 5.1.

(a) Then

, G = A ∞,1 is the unique p-Sylow subgroup of A. (b) For all i in {1, 2}, f i ∈ Σ i+1 -Σ i and m i = 1 + i p s , with p ≥ 3 and s ∈ {1, 2}. (c) Moreover, v = s + 1. More precisely, y ∈ V if and only if ℓ 1,2 (y) p -ℓ 1,2 (y) = 0 .
2. There exists a coordinate X for the projective line C/G 2 such that the functions f i 's are parametrized as follows:

(a) If s = 1, p > 3 p = 3 f 1 f 1 (X) = X 1+p + a 2 X 2 f 1 (X) = X 4 + a 2 X V V = Z(Ad f1 ) = Z(X p 2 + 2 a p 2 X p + X) V = Z(Ad f1 ) = Z(X 9 + 2 a 3 2 X 3 + X) f 2 f 2 (X) = b 1+2 p X 1+2 p + b 2+p X 2+p + b 3 X 3 + b 1 X f 2 (X) = b 7 X 7 + b 5 X 5 + b 1 X b 1+2 p b 1+2 p ∈ k × b 16 7 = 1 a 2 2 a p 2 = -b -p 1+2 p (b p 2 1+2 p + b 1+2 p ) ⇔ b 1+2 p ∈ V 2 a 3 2 = -b 6 7 -b -2 7 b 2+p b 2+p = -b p 1+2 p b 5 = -b 3 7 b 3 3 b p 3 = b -p 1+2 p (b 2 p 2 1+2 p -b 2 1+2 p ) b 1 b 1 ∈ k b 1 ∈ k ℓ 1,2 ℓ 1,2 (y) = 2 (b 1+2 p y p -b p 1+2 p y) ℓ 1,2 (y) = 2 (b 7 y 3 -b 3 7 y)
Therefore, for p > 3, the solutions are parametrized by 2 algebraically independent variables over F p , namely b 1+2 p ∈ k × and b 1 ∈ k. For p = 3, as the monomial X 3 can be reduced mod ℘(k[X]), the parameter b 1+2 p satisfies an additional algebraic relation: b 16 7 = 1. Then, b 7 takes a finite number of values.

In both cases (p = 3 or p > 3),

|G| g = 2 p p -1 p 2 1 + 2 p and |G| g 2 = 4 (p 2 -1) 2 p 2 (p + 1) 2 (1 + 2 p) 2 (b) If s = 2 and p > 3, f 1 f 1 (X) = X 1+p 2 + a 1+p X 1+p + a 2 X 2 Ad f1 X p 4 + a p 2 1+p X p 3 + 2 a p 2 2 X p 2 + a p 1+p X p + X f 2 f 2 (X) = b 1+2 p 2 X 1+2 p 2 + b 1+p+p 2 X 1+p+p 2 + b 2+p 2 X 2+p 2 + b 1+p 2 X 1+p 2 + b 1+2 p X 1+2 p +b 2+p X 2+p + b 1+p X 1+p + b 3 X 3 + b 2 X 2 + b 1 X b 1+2 p b 1+2 p ∈ k × b 2+p 2 b 2+p 2 ∈ k × b 1+p+p 2 b p 1+p+p 2 = -2 b p 1+2 p (b p 2+p 2 b -p 2 1+2 p + b p-1 2+p 2 ) ℓ 1,2 ∀ y ∈ V, ℓ 1,2 (y) = 2 b 1+2 p y p 2 + b 1+p+p 2 y p + 2 b 2+p 2 y V V is an index p-subgroup of Z(Ad f1 ) V = Z(2 b p 1+2 p X p 3 + (b p 1+p+p 2 -2 b 1+2 p ) X p 2 + (2b p 2+p 2 -b 1+p+p 2 )X p -2b 2+p 2 X) a 1+p a p 2 1+p = -b p-p 2 1+2 p -b p 1+2 p b -1 2+p 2 -b p 2 2+p 2 b -p 3 1+2 p -b p 2 -p 2+p 2 a 2 2 a p 2 2 = b p 2 2+p 2 b -p 2 1+2 p + b 1+2 p b -1 2+p 2 + b p 2+p 2 b p-2 p 2 1+2 p + 2 b p-1 2+p 2 b p-p 2 1+2 p + b p 1+2 p b p-2 2+p 2 b 1+2 p b p 2 1+2 p = -b 2 p-p 2 1+2 p -b 2 p 1+2 p b -1 2+p 2 + b 2 p 2 2+p 2 b p 2 -2 p 3 1+2 p + 2 b 2 p 2 -p 2+p 2 b p 2 -p 3 1+2 p + b p 2 1+2 p b 2 p 2 -2 p 2+p 2 b 2+p b p 2 2+p = b p 2+p 2 b 2 p-2 p 2 1+2 p + 2 b p-1 2+p 2 b 2 p-p 2 1+2 p + b 2 p 1+2 p b p-2 2+p 2 -b 2 p 2 2+p 2 b -p 3 1+2 p -b 2 p 2 -p 2+p 2 b 3 3 b p 2 3 = b 2 p 2 2+p 2 b -p 2 1+2 p -b 2 p 2+p 2 b 2 p-3p 2 1+2 p -3 b 2 p-2 2+p 2 b 2 p-p 2 1+2 p -3 b 2 p-1 2+p 2 b 2 p-2 p 2 1+2 p -b 2 p 1+2 p b 2 p-3 2+p 2 + b 2 1+2 p b -1 2+p 2 b 1+p 2 b 1+p 2 ∈ Z(b p 2 2+p 2 b -p 3 1+2 p X p 3 -(b p 2 2+p 2 b -p 3 1+2 p + b p-p 2 1+2 p + b p 2 -p 2+p 2 ) X p 2 + (b p-p 2 1+2 p + b p 1+2 p b -1 2+p 2 + b p 2 -p 2+p 2 ) X p -b p 1+2 p b -1 2+p 2 X) b 1+p b p 2 1+p = -(b p-p 2 1+2 p + b p 2 2+p 2 b -p 3 1+2 p + b p 2 -p 2+p 2 ) b p 2 1+p 2 -b p 1+2 p b -1 2+p 2 b 1+p 2 b 2 2 b p 2 2 = (b p 2+p 2 b p-2 p 2 1+2 p + b p-1 2+p 2 b p-p 2 -p 1+2 p + b p 2 2+p 2 b -p 2 1+2 p )b p 2 1+p 2 + (b p 1+2 p b p-2 2+p 2 + b p-1 2+p 2 b p-p 2 1+2 p + b 1+2 p b -1 2+p 2 ) b 1+p 2 b 1 b 1 ∈ k
Therefore, for p > 3, the solutions can be parametrized by 3 algebraically independent variables over

F p , namely b 1+2 p 2 ∈ k × , b 2+p 2 ∈ k × and b 1 ∈ k.
One also finds a fourth parameter b 1+p 2 which runs over an F p -subvector space of k, namely the set of zeroes of an additive separable polynomial whose coefficients are rational functions in b 1+2 p 2 and b 2+p 2 . So, for given b 1+2 p and b 2+p 2 , the parameter b 1+p 2 takes a finite number of values. In both cases (p = 3 or p > 3),

For p = 3, f 1 (X) = X 10 + a 4 X 4 + a 2 X 2 f 2 (X) = b 19 X 19 + b 13 X 13 + b 11 X 11 + b 10 X 10 + b 7 X 7 + b 5 X 5 + b 4 X 4 + b 2 X 2 + b 1 X with a 4 ,
|G| g = 2 p p -1 p 2 1 + 2 p and |G| g 2 = 4 (p 2 -1) 2 p (p + 1) 2 (1 + 2 p) 2
Remark 5.7. One can now answer the second problem raised in [START_REF] Rocher | Large p-groups actions with a p-elementary abelian second ramification group[END_REF] (section 6). Indeed, one notices that the family obtained for s = 2 is larger than the one obtained after the additive base change: X = Z p + c Z, c ∈ k -{0} (see [START_REF] Matignon | On smooth curves endowed with a large automorphism p-group in characteristic p[END_REF] Prop. 3.1) applied to the case s = 1. Indeed, such a base change does not produce any monomial Z 1+p 2 in f 2 (Z).

A few special cases. 1. When s = 1 and p > 3, the special case a 2 = 0 corresponds to the parametrization of the extension K m S /K given by Auer (cf . [Au99] Prop. 8.9 or [MR08] section 6), namely f 1 (X) = a X 1+p with a p + a = 0, a = 0.

f 2 (X) = a 2 X 2 p (X -X p 2 ). 2. When s = 2, the special case b 1+p 2 ∈ F p leads to b 1+p = b 1+p 2 a 1+p and b 2 = b 1+p 2 a 2 . So, one can replace f 2 by f 2 (X) -b 1+p 2 f 1 (X)
, which eliminates the monomials X 1+p 2 , X 1+p and X 2 .

Proof of Proposition 5.6:

1. As ℓ 1,2 = 0, the group G satisfies the third condition of [Ro08a] (Prop. 5.2). Then, the equality G = A ∞,1 derives from [Ro08a] (Cor. 5.7). The unicity of the p-Sylow subgroup is explained in Remark 3.1. The second and third assertions come from [Ro08a] (Thm. 5.6). Moreover, the description of V displayed in (c) is due to[Ro08a] (Prop 2.9.2). It remains to show that s = 1 or s = 2. Using formula (8), we compute g = (p-1) 2

(p s + p (m 2 -1)) = (p-1) 2 p s (1 + 2 p).
As |G| = p 3+s , condition ( * ) requires:

4 (p 2 -1) 2 ≤ |G| g 2 = 4 (p 2 -1) 2 (p+1) 2 p s-3 (1+2 p) 2 . It follows that 3 -s > 0, i.e. 1 ≤ s ≤ 2.
2. We merely explain the case s = 1. One can find a coordinate X of the projective line

C/G 2 such that f 1 (X) = X S 1 (X) = X (X p + a 2 X) (cf. [Ro08a]
Cor. 2.12). Then,

Ad f1 = F 2 + 2 a p 2 F + I (cf. [Ro08a] Prop. 2.13). As V ⊂ Z(Ad f1 ) and dim Fp Z(Ad f1 ) = 2 = s + 1 = v, we deduce that V = Z(Ad f1 ). As f 2 ∈ Σ 3 -Σ 2 with deg f 2 = 1 + 2 p s
and as the functions f i 's are supposed to be reduced mod ℘(k[X]), equation (9) reads:

∀ y ∈ V, f 2 (X + y) -f 2 (X) = ℓ 1,2 (y) f 1 (X) mod ℘(k[X]) with f 1 (X) = X 1+p + a 2 X 2 and f 2 (X) = b 1+2 p X 1+2 p + b 2+p X 2+p + b 1+p X 1+p + b 3 X 3 + b 2 X 2 + b 1 X for p > 3 (resp. f 2 (X) = b 1+2 p X 1+2 p + b 2+p X 2+p + b 1+p X 1+p + b 2 X 2 + b 1 X for p = 3)
Then, calculation gives the relations gathered in the table. In particular, we find:

f 2 (X) = b 1+2 p X 1+2 p + b 2+p X 2+p + b 3 X 3 + b 1 X + b 1+p f 1 (X) with b 1+p ∈ F p .
Since we are working in the F p -space generated by f 1 (X) and f 2 (X), we can replace f 2 (X) with f 2 (X) -b 1+p f 1 (X), hence the expected formula. We solve the case s = 2 in the same way.

5.4 Third case: big actions satisfying G p 3 * .

Preliminaries.

The idea is to use, as often as possible, the results obtained in the preceding section.

Remark 5.8. Let (C, G) be a big action with G ′ (= G 2 ) ≃ (Z/pZ) 3 . We keep the notations introduced in section 5.1.

1. Let C 1,2 be the curve parametrized by the two equations: W p i -W i = f i (X), with i ∈ {1, 2}, and let K 1,2 := k(C 1,2 ) be the function field of this curve. Then, K 1,2 /k(X) is a Galois extension with group Γ 1,2 ≃ (Z/pZ) 2 . Moreover, the group of translations by V : {X → X + y, y ∈ V } extends to an automorphism p-group of C 1,2 say G 1,2 , with the following exact sequence:

0 -→ Γ 1,2 -→ G 1,2 -→ V -→ 0
Let A 1,2 be the F p -subvector space of A generated by the classes of f 1 (X) and f 2 (X). Let H 1,2 G 2 be the orthogonal of A 1,2 with respect to the Artin-Schreier pairing. Then, C 1,2 = C/H 1,2 and G 1,2 = G/H 1,2 . Furthermore, as A 1,2 is stable under the action of ρ, its dual H 1,2 is stable by the dual representation φ, i.e. by conjugation by the elements of G (see section 5.1). It follows that H 1,2 G 2 is a normal subgroup in G. So, by [START_REF] Matignon | On smooth curves endowed with a large automorphism p-group in characteristic p[END_REF] (Lemma 2.4), the pair (C 1,2 , G 1,2 ) is a big action with second ramification group isomorphic to (Z/pZ) 2 . 2. Likewise, if ℓ 2,3 = 0, the F p -subvector space of A generated by the classes of f 1 (X) and f 3 (X) is also stable by ρ (see matrix L(y) in section 5.1). So, the two equations: W p i -W i = f i (X), with i ∈ {1, 3}, also parametrize a big action, say (C 1,3 , G 1,3 ), with second ramification group isomorphic to (Z/pZ) 2 .

3. Similarly, if ℓ 1,2 = ℓ 1,3 = 0, the F p -subvector space of A generated by the classes of f 2 (X) and f 3 (X) is stable by ρ (see matrix L(y) in section 5.1). So, the two equations: W p i -W i = f i (X), with i ∈ {2, 3}, also parametrize a big action, say (C 2,3 , G 2,3 ), with second ramification group isomorphic to (Z/pZ) 2 . Lemma 5.9. Let (C, G) be a big action satisfying G p 3 * . Let (C 1,2 , G 1,2 ) be defined as in Remark 5.8. We keep the notations introduced in section 5.1. 1. Then, (C 1,2 , G 1,2 ) is a big action satisfying G p 2 * .

2. If ℓ 1,2 = 0, then m 1 = m 2 = 1 + p s , with s ≥ 2.

3. If ℓ 1,2 = 0, then m 1 = 1 + p s , m 2 = 1 + 2 p s , with s ∈ {1, 2} and p ≥ 3. In this case, v = s + 1.

Proof:

1. Use Remark 5.8 and [START_REF] Lehr | Automorphism groups for p-cyclic covers of the affine line[END_REF] (Prop. 8.5 (ii)) to see that condition ( * ) is still satisfied.

2. We deduce from Proposition 5.5 that m 1 = 1+p s1 and m 2 = 1+p s2 with s 2 = s 1 or s 2 = s 1 +1. Assume that s 2 = s 1 + 1. Then, m 3 ≥ m 2 = 1 + p s1+1 . We compute the genus by means of (8): g = p-1 2 (p s1 + p 1+s2 + p 2 (m 3 -1)) ≥ p-1 2 p s1 (1 + p 2 + p 3 ). Besides, by [START_REF] Matignon | On smooth curves endowed with a large automorphism p-group in characteristic p[END_REF] (Thm. 2.6), V ⊂ Z(Ad f1 ), so |G| = p 3+v ≤ p 3+2s1 . Thus,

|G| g 2 ≤ 4 (p 2 -1) 2 p 3 (p+1) 2 (1+p 2 +p 3 ) 2 < 4 (p 2 -1) 2 , which contradicts condition ( * ). It follows that s 2 = s 1 ≥ 2.
3. Apply Proposition 5.6 to (C 12 , G 12 ).

Remark 5.10. Let (C, G) be a big action satisfying G p 3 * . Assume that ℓ 1,2 = ℓ 1,3 = 0. Then, the results of Lemma 5.9 also hold for the big action (C 2,3 , G 2,3 ) as defined in Remark 5.8. Lemma 5.11. Let (C, G) be a big action satisfying G p 3 * . We keep the notations introduced in section 5.1. Assume that ℓ 2,3 = 0. Let (C 1,3 , G 1,3 ) be defined as in Remark 5.8. 1. Then, (C 1,3 , G 1,3 ) is a big action satisfying G p 2 * .

2. If ℓ 1,3 = 0, then ℓ 1,2 = 0 and m 1 = m 2 = m 3 = 1 + p s with s ≥ 2. In this case, v = 2 s.

3. If ℓ 1,3 = 0, then m 1 = 1 + p s , m 3 = 1 + 2 p s , with s ∈ {1, 2} and p ≥ 3. In this case, v = s + 1.

Proof:

1. Use Remark 5.8 and [LM05] (Prop. 8.5 (ii)).

2. As ℓ 1,3 = 0, we deduce from Proposition 5.5 that m 1 = 1 + p s and m 3 = 1 + p s3 with s 3 = s or s 3 = s + 1.

(a) We show that ℓ 1,2 = 0. Assume that ℓ 1,2 = 0. Then, Lemma 5.9 applied to (C 1,2 , G 1,2 ) implies m 2 = 1 + 2 p s with s ∈ {1, 2} and p ≥ 3. Moreover, v = s + 1. As m 2 ≤ m 3 , there are two possibilities: i. s = 1 and s 3 = s + 1 = 2., i.e. m 1 = 1 + p, m 2 = 1 + 2 p, m 3 = 1 + p 2 and v = 2. Then, |G| g 2 = 4 (p 2 -1) 2 p 3 (p+1) 2 (1+2 p+p 3 ) 2 < 4 (p 2 -1) 2 , which contradicts condition ( * ). ii. s = 2 and s 3 = s + 1 = 3. i.e. m 1 = 1 + p 2 , m 2 = 1 + 2 p 2 , m 3 = 1 + p 3 and v = 3.

Then, |G| g 2 = 4 (p 2 -1) 2 p 2 (p+1) 2 (1+2 p+p 3 ) 2 < 4 (p 2 -1) 2 , which also contradicts condition ( * ). As a consequence, ℓ 1,2 = 0. (b) We deduce that m 1 = m 2 = 1 + p s with s ≥ 2.

Lemma 5.9 applied to (C 1,2 , G 1,2 ) implies m 1 = m 2 = 1 + p s with s ≥ 2. In particular,

g = p-1 2 p s (1 + p + p 2+s3-s ) and |G| g 2 = 4 (p 2 -1) 2 p 3+v-2s (p+1) 2 (1+p+p 2+s 3 -s ) 2 . (c) We show that v = 2 s 3 and conclude that s 3 = s. Assume that v ≤ 2 s 3 -3. Then, |G| g 2 < 4 (p 2 -1) 2 p 2s 3 -2s (p+1) 2 p 4+2s 3 -2s < 4 (p 2 -1) 2 which contradicts condition ( * ). Therefore, 2 s 3 -2 ≤ v ≤ 2 s ≤ 2 s 3 . Assume that v ≤ 2 s 3 -3. Then, |G| g 2 < 4 (p 2 -1) 2 p 2s 3 -2s (p+1) 2 p 4+2s 3 -2s < 4 (p 2 -1) 2 which contradicts condition ( * ). Assume that v = 2 s 3 -1. So, v is odd and 2 s 3 -2 < v ≤ 2s ≤ 2 s 3 implies s 3 = s and v = 2s -1. Then, |G| g 2 = 4 (p 2 -1) 2 p 2 (p+1) 2 (1+p+p 2 ) 2 < 4 (p 2 -1) 2 , which is excluded. Now, assume that v = 2 s 3 -2. Then, 2 s 3 -2 = v ≤ 2s ≤ 2 s 3 implies s 3 = s or s 3 = s + 1. In the first case, v = 2s -2 and |G| g 2 = 4 (p 2 -1) 2 p (p+1) 2 (1+p+p 2 ) 2 < 4 (p 2 -1) 2 . In the second case, v = 2s and |G| g 2 = 4 (p 2 -1) 2 p 3 (p+1) 2 (1+p+p 3 ) 2 < 4 (p 2 -1) 2 .
In both cases, we obtain a contradiction. We gather that v = 2 s 3 . Applying [START_REF] Rocher | Large p-groups actions with a p-elementary abelian second ramification group[END_REF] (Prop. 4.2), we conclude that s = s 3 . 3. Apply Proposition 5.6 to (C 13 , G 13 ).

Case

: [G ′ , G] = F ratt(G ′ ) = {e}.
Proposition 5.12. Let (C, G) be a big action satisfying G p 3 * such that [G ′ , G] = {e}. We keep the notations introduced in section 5.1.

1. Then, G = A ∞,1 is a special group of exponent p (resp. p 2 ) for p > 2 (resp. p = 2) and order p 3+2 s1 . More precisely, G is a central extension of its center Z(G) = G ′ by the elementary abelian p-group V = Z(Ad f1 ) = Z(Ad f2 ) = Z(Ad f3 ):

0 -→ Z(G) = G ′ ≃ (Z/pZ) 3 -→ G π -→ Z(Ad f1 ) = Z(Ad f2 ) = Z(Ad f3 ) ≃ (Z/pZ) 2s1 -→ 0
Furthermore, G is a p-Sylow subgroup of A, which is normal except when C is birational to the Hermitian curve: W q -W = X 1+q , with q = p 3 .

2. There exists a coordinate X for the projective line C/G 2 , s ≥ 2, d ≥ 2 dividing s, and

γ 2 , γ 3 in F p d -F p linearly independent over F p , b 1 ∈ k, c 1 ∈ k such that: f 1 f 1 (X) = X S 1 (X) with S 1 (F ) = s/d j=0 a jd F jd ∈ k{F } a s = 1 f 2 f 2 (X) = X S 2 (X) + b 1 X with S 2 = γ 2 S 1 f 3 f 3 (X) = X S 3 (X) + c 1 X with S 3 = γ 3 S 1 V V = Z(Ad f1 ) = Z(Ad f2 ) = Z(Ad f3 )
Therefore, the solutions can be parametrized by s + 4 algebraically independent variables over F p , namely the s coefficients of S, γ

2 ∈ F p d -F p , γ 3 ∈ F p d -F p , b 1 ∈ k and c 1 ∈ k. Moreover, |G| g = 2 p p -1 p s 1 + p + p 2 and |G| g 2 = 4 (p 2 -1) 2 p 3 (p + 1) 2 (1 + p + p 2 ) 2 Proof: As ℓ 1,2 = ℓ 2,3 = ℓ 1,3 = 0, the second point of Lemma 5.11 first implies v = 2 s 3 . Applying [Ro08a] (Prop. 4.2), we gather that s 1 = s 2 = s 3 , that V = Z(Ad f1 ) = Z(Ad f2 ) = Z(Ad f3
) and we get the expected formulas for the functions f ′ i s. Moreover, it follows from [Ro08a] (Prop. 4.3 and Rem. 4.5) that G = A ∞,1 is a special group. The unicity of the p-Sylow subgroup is discussed in Remark 3.1.

Case

: [G ′ , G] F ratt(G ′ ) = {e}.
Lemma 5.13. Let (C, G) be a big action satisfying G p 3 * such that [G ′ , G] = {e}. We keep the notations introduced in section 5.1. Then, one cannot have ℓ 1,2 = ℓ 2,3 = 0.

Proof: Assume that ℓ 1,2 = 0 and ℓ 2,3 = 0. Since the representation ρ is non trivial, ℓ 1,3 = 0. The second point of Lemma 5.9 shows that m 1 = m 2 = 1 + p s with s ≥ 2. The third point of Lemma 5.11 implies that m 3 = 1 + 2 p s with p ≥ 3 and s ∈ {1, 2}. Moreover, v = s + 1. As s ≥ 2, we obtain:

|G| g 2 = 4 (p 2 -1) 2 (p+1) 2 p 2 (1+p+2 p 2 ) 2 < 4 (p 2 -1) 2 , hence a contradiction. As a conclusion, either ℓ 1,2 = 0 or ℓ 2,3 = 0.
As a consequence, there are 3 cases to study: ℓ 1,2 = 0 and ℓ 2,3 = 0 (cf. Proposition 5.14). ℓ 1,2 = 0 or ℓ 2,3 = 0 (cf. Proposition 5.15). ℓ 1,2 = 0 or ℓ 2,3 = 0 (cf. Proposition 5.16).

Proposition 5.14. Let (C, G) be a big action satisfying G p 3 * such that [G ′ , G] = {e}. We keep the notations introduced in section 5.1. Assume that ℓ 1,2 = 0 and ℓ 2,3 = 0.

1. Then, p ≥ 5 and there exists a coordinate X for the projective line C/G 2 such that the functions f i 's can be parametrized as follows:

f 1 f 1 (X) = X 1+p + a 2 X 2 V V = Z(Ad f1 ) = Z(X p 2 + 2 a p 2 X p + X) f 2 f 2 (X) = b 1+2 p X 1+2 p + b 2+p X 2+p + b 3 X 3 + b 1 X b 1+2 p b 1+2 p ∈ k × a 2 2 a p 2 = -b -p 1+2 p (b 1+2 p + b p 2 1+2 p ) ⇔ b 1+2 p ∈ V V V = Z(X p 2 -b -p 1+2 p (b 1+2 p + b p 2 1+2 p ) X p + X) b 2+p b 2+p = -b p 1+2 p b 3 3 b p 3 = b -p 1+2 p (b 2 p 2 1+2 p -b 2 1+2 p ) b 1 b 1 ∈ k ℓ 1,2 ℓ 1,2 (y) = 2 (b 1+2 p y p -b p 1+2 p y) f 3 f 3 (X) = c 1+2 p X 1+2 p + c 2+p X 2+p + c 3 X 3 + c 1 X c 1+2 p c 1+2 p ∈ k × c 1+2 p c 1+2 p ∈ V , c 1+2 p and b 1+2 p F p -independent c 2+p c 2+p = -c p 1+2 p c 3 3 c p 3 = -c -p 1+2 p (c 2 p 2 1+2 p + c 2 1+2 p ) c 1 c 1 ∈ k ℓ 1,3 ℓ 1,3 (y) = 2 (c 1+2 p y p -c p 1+2 p y) ℓ 2,3 ℓ 2,3 (y) = 0
Therefore, the solutions are parametrized by 4 algebraically independent variables over

F p , namely b 1+2 p ∈ k × , c 1+2 p ∈ k × , b 1 ∈ k and c 1 ∈ k. Moreover, |G| g = 2 p p -1 p 3 1 + 2 p + 2 p 2 and |G| g 2 = 4 (p 2 -1) 2 p 3 (p + 1) 2 (1 + 2 p + 2 p 2 ) 2 2. In this case, G = A ∞,1 is the unique p-Sylow subgroup of A.

Proof:

1. Lemma 5.9 first shows that m 1 = 1 + p s , m 2 = 1 + 2 p s , with p ≥ 3 and s ∈ {1, 2}. Moreover, v = s + 1. As ℓ 1,2 = 0 and ℓ 2,3 = 0, the second point of Lemma 5.11 imposes ℓ 1,3 = 0. Then, Lemma 5.11 shows that m

3 = 1 + 2 p s . If s = 2, m 1 = 1 + p 2 , m 2 = m 3 = 1 + 2 p 3 and v = 3. So |G| g 2 = 4 (p 2 -1) 2 p 2 (p+1) 2 (1+2 p+2 p 2 ) 2 < 4 (p 2 -1) 2 , which contradicts condition ( * ). It follows that s = 1. In this case, m 1 = 1 + p, m 2 = m 3 = 1 + 2 p, v = 2 and |G| g 2 = 4 (p 2 -1) 2 p 3 (p+1) 2 (1+2 p+2 p 2 ) 2 . Therefore, condition (
* ) is satisfied as soon as p ≥ 5. The parametrization of the functions f i 's then derives from Proposition 5.6. Furthermore, the third condition (cf. Recall 4.2.1-c) imposed on the degree of the functions f i 's requires that the parameters b 1+2 p and c 1+2 p are linearly independent over F p .

The equality

G = A ∞,1 derives from the maximality of V = Z(Ad f1 ) (see Proposition 3.2).
The unicity of the p-Sylow subgroup is due to Remark 3.1.

Proposition 5.15. Let (C, G) be a big action satisfying G p 3 * such that [G ′ , G] = {e}. We keep the notations introduced in section 5.1. Assume that ℓ 1,2 = 0 and ℓ 2,3 = 0.

1. Then, p ≥ 5 and there exists a coordinate X for the projective line C/G 2 such that the functions f i 's can be parametrized as follows:

f 1 f 1 (X) = X 1+p 2 + a 2 X 2 f 2 f 2 (X) = γ 2 ( X 1+p 2 + a 2 X 2 ) + b 1 X b 1 b 1 ∈ k γ 2 γ 2 ∈ F p 2 -F p V V = Z(Ad f1 ) = Z(Ad f2 ) = Z(X p 4 + 2 a p 2 2 X p 2 + X) First case: b 1 = 0 f 3 f 3 (X) = c 1+2 p 2 X 1+2 p 2 + c 2+p 2 X 2+p 2 + c 1+p 2 X 1+p 2 +c 1+p X 1+p + c 3 X 3 + c 2 X 2 + c 1 X c 1+2 p 2 c 1+2 p 2 ∈ k × a 2 2 a p 2 2 = -c -p 2 1+2 p 2 (c p 4 1+2 p 2 + c 1+2 p 2 ) ⇔ c 1+2 p 2 ∈ V V V = Z(X p 2 -c -p 1+2 p 2 (c 1+2 p 2 + c p 2 1+2 p 2 ) X p + X) c 2+p 2 c 2+p 2 = -c p 2 1+2 p 2 c 3 3 c p 2 3 = -c p 2 1+2 p 2 (3 c 2 p 4 1+2 p 2 + 4 c 1+p 4 1+2 p 2 + c 2 1+2 p 2 ) e := c 1+p 2 -c p 2
1+p 2 e ∈ Z ((c p 7 -p 3 1+2 p 2 + 1 + c p-p 5 1+2 p 2 + c p 7 +p-p 5 -p 3 1+2 p 2 ) X 1+p 4 -X 1+p 2 -X p 2 -X -1) +(c p 7 -p 3 1+2 p 2 + 1 + c p-p 5 1+2 p 2 + c p 7 +p-p 5 -p 3 1+2 p 2

) (c 1+p 2 -c p 2 1+p 2 ) p 3 +p 2 +p-1 c 1 c 1 ∈ k Therefore, the solutions can be parametrized by 3 algebraically independent variables over F p , namely c 1+2 p 2 ∈ k × , c 1 ∈ k and γ 2 ∈ F p 2 -F p . One also finds a fourth parameter e := c 1+p 2 -c p 2 1+p 2 which runs over the set of zeroes of a polynomial whose coefficients are rational functions in c 1+2 p 2 . So, for a given c 1+2 p 2 , the parameter e takes a finite number of values.

Second case: b 1 = 0 2. Moreover, G = A ∞,1 is the unique p-Sylow subgroup of A.

f 3 f 3 (X) = c 1+2 p 2 X 1+2 p 2 + c 2+p 2 X 2+p 2 + c 3 X 3 c 1+2 p 2 c 1+2 p 2 ∈ k × a 2 2 a p 2 2 = -c -p 2 1+2 p 2 (c p 4 1+2 p 2 + c 1+2 p 2 ) ⇔ c 1+2 p 2 ∈ V V V = Z(X

Proof:

1. (a) We describe f 1 , f 2 and V .

Lemma 5.9 first implies that m 1 = m 2 = 1 + p s , with s ≥ 2. More precisely, we deduce from Proposition 5.5 that f 1 (X) = X S 1 (X) and f 2 (X) = γ 2 X S 1 (X) + b 1 X, where S 1 is a monic additive polynomial with degree s in F , b 1 ∈ k and γ 2 ∈ F p d -F p with d an integer dividing s. Moreover, v = 2 s and V = Z(Ad f1 ) = Z(Ad f2 ).

(b) We show that ℓ 1,3 = 0. Indeed, assume that ℓ 1,3 = 0. Then, we deduce from Remark 5.10 that m 3 = 1 + 2 p s , with s ∈ {2, 3} and p ≥ 3. Moreover, v = s + 1. As s = 1, it follows that s = 2 and (k) We gather the parametrization of f 1 , f 2 and V .

As s = d = 2, f 1 reads f 1 (X) = X S 1 (X) with S 1 (F ) = s/d j a jd F jd = a 0 I + F 2 , since S 1 is assumed to be monic. Then, f 1 (X) = X (X p 2 + a 2 X 2 ) and f 2 (X) = γ 2 X (X p 2 + a 2 X 2 ) + b 1 X with a 2 ∈ k, b 1 ∈ k and γ 2 ∈ F p 2 -F p . In this case, V = Z(Ad f1 ) = Z(X p 4 + +2 a p 2 2 X p 2 + X) (l) We show that f 3 ∈ Σ 4 but f 3 ∈ Σ 4 -Σ 3 . By [START_REF] Rocher | Large p-groups actions with a p-elementary abelian second ramification group[END_REF] (Thm. 3.13), f 3 ∈ Σ 4 . We now show that f 3 does not have any monomial in Σ 4 -Σ 3 . Indeed, as m 3 = 1 + 2 p 2 , the possible monomials of f 3 in Σ 4 -Σ 3 are X 1+2 p+p 2 , X 2+p+p 2 , X 3+p 2 , X 1+3 p , X 2+2 p , X 3+p and X 4 . Now, equate the coefficients of the monomial X 1+p+p 2 ∈ Σ 3 in each side of (10). In the left-hand side, i.e. in ∆ y (f 3 ) mod ℘(k[X]), X 1+p+p 2 is produced by monomials X b of f 3 that belong to Σ 4 -Σ 3 and satisfy b > 1 + p + p 2 . This leaves only two possibilities: X 1+2 p+p 2 and X 2+p+p 2 . In the righthand side of (10), X 1+p+p 2 ∈ Σ 3 -Σ 2 does not occur since ℓ 1,3 (y) f 1 (X) + ℓ 2,3 (y) f 2 (X) lies in Σ 2 . It follows that, for all y in V , 2 c 2+p+p 2 y p + 2 c 1+2 p+p 2 y = 0, where c t denotes the coefficient of the monomial X t in f 3 . As v = dim Fp V = 4, we deduce that c 2+p+p 2 = c 1+2 p+p 2 = 0. We go on this way and equate successively the coefficients of X 2+p 2 , X 1+2 p , X 2+p and X 3 to prove that f 3 does not contain any monomial in Σ 4 -Σ 3 . Therefore, f 3 reads as follows:

f 3 (X) = c 1+2 p 2 X 1+2 p 2 + c 1+p+p 2 X 1+p+p 2 + c 2+p 2 X 2+p 2 + c 1+p 2 X 1+p 2 + c 1+2 p 2 X 1+2 p + c 2+p X 2+p + c 1+p X 1+p + c 3 X 3 + c 2 X 2 + c 1 X (m)
We determine f 3 . We finally have to solve (10) with f 1 , f 2 and f 3 as described above. Calculation show that c 1+p+p 2 = c 1+2 p 2 = c 2+p = 0 and that the coefficients a 2 , c 2+p 2 and c 3 can be expressed as rational functions in c 1+2 p 2 (see formulas in the table given in the proposition). To conclude, one has to distinguish the cases b 1 = 0 and b 1 = 0. In the first case, b 1 , c 1+p and c 2 can be expressed as rational functions in c 1+p 2 whereas e := c 1+p 2 -c p 2 1+p 2 belongs to the set of zeroes of a polynomial whose coefficients are rational functions in c 1+2 p 2 (see table ). When b 1 = 0, then c 1+p = 0, c 1 = c 1+p 2 a 2 and c 1+p 2 ∈ F p 2 . It follows that f 3 (X) = c 1+2 p 2 X 1+2 p 2 + c 2+p 2 X 2+p 2 + c 3 X 3 + c 1 X + c 1+p 2 f 1 (X). As γ 2 ∈ F p 2 -F p , {1, γ 2 } is a basis of F p 2 over F p . Write ǫ = ǫ 1 + ǫ 2 γ 2 , with ǫ 1 and ǫ 2 in F p . By replacing f 3 with f 3 -(ǫ 1 f 1 + ǫ 2 f 2 ), one obtains the expected formula.

2. The equality G = A ∞,1 derives from the maximality of V = Z(Ad f1 ) (see Proposition 3.2).

The unicity of the p-Sylow subgroup is due to Remark 3.1.

The last case: ℓ 1,2 = 0 and ℓ 2,3 = 0, generalizes the results obtained in [Ro08a] (section 6.2).

Proposition 5.16. Let (C, G) be a big action satisfying G p 3 * such that [G ′ , G] = {e}. We keep the notations introduced in section 5.1. Assume that ℓ 1,2 = 0 and ℓ 2,3 = 0.

1. Then, p ≥ 11 and there exists a coordinate X for the projective line C/G 2 such that the functions f i 's can be parametrized as follows:

2. The equality G = A ∞,1 derives from [START_REF] Matignon | On smooth curves endowed with a large automorphism p-group in characteristic p[END_REF] (Cor. 5.7). The unicity of the p-Sylow subgroup comes from Remark 3.1. The description of the center is due to [START_REF] Matignon | On smooth curves endowed with a large automorphism p-group in characteristic p[END_REF] (Prop. 6.15).

  2 | Proof: The first assertion (1.a) derives from [LM05] (Prop 8.5) and [MR08] (Cor. 2.10). The second point (1.b) comes from [MR08] (Rem 2.11) together with Remark 3.1. The other claims are obtained via calculation.

  and has the same derived subgroup. So, in our attempt to classify the big actions (C, G) satisfying G M , this leads us to focus on the derived subgroup G ′ of G. 4 Finiteness results for big actions satisfying G M . 4.1 An upper bound on |G ′ |.

4. 2

 2 Preliminaries to a group-theoretic discussion.

Lemma 4. 3 .

 3 Let (C, G) be a big action. If G ′ ⊂ Z(G), then G ′ (= G 2 ) is p-elementary abelian, say G ′ ≃ (Z/pZ) n , with n ≥ 1. In this case, the function field L = k(C) is parametrized by n equations:

Proposition 4. 6 .

 6 Let M > 0 be a positive real such that (C, G) is a big action satisfying G M . Suppose that {e} = F ratt(G ′ ) [G ′ , G]. Then, |V | and g are bounded as follows:

  Then, |V | and g are bounded as in Proposition 4.6. So the quotients |G| g and |G| g 2 only take a finite number of values. Proof: Put F := F ratt(G ′ ). Corollary 4.5 asserts that the pair (C/F, G/F ) is a big action satisfying G M whose second ramification group:

  . See [LM05] (Thm. 1.1 I) 2. See Remark 3.1, [LM05] (Thm. 3.1) and [MR08] (Prop. 2.5).

  a 2 , b 13 , b 7 , b 5 , b 3 and b 2 satisfying the same relations as above. But, this time, the parameters b 19 and b 11 are linked through an algebraic relation, namely:

b 1 b p 5 -p 4 +p 3 -p 2 1 = -e p 3

 13

|G| g 2

 2 = 4 (p 2 -1) 2 p 2 (p+1) 2 (1+p+2 p 2 ) 2 < 4 (p 2 -1) 2 ,which contradicts condition ( * ).

  G/H) is a big action with second ramification group (G/H) 2 = G 2 /H. From H = [G 2 : G],we gather that G 2 /H ⊂ Z(G/H). Therefore, we deduce from Lemma 4.3 that (G/H) 2 is p-elementary abelian.

	3. As H	G ′ = G 2 is normal in G, it follows from [MR08] (Lemma 2.4 and Thm. 2.6) that
	the pair (C/H,
	is a normal subgroup of G. It follows from [MR08] (Lemma 2.4) that the pair (C/H, G/H)
	is a big action with second ramification group (G/H) 2 ≃ Z/p 2 Z. This contradicts [MR08] (Thm. 5.1). The last part of the lemma comes from [Ro08a] (Prop. 2.13).
	Corollary 4.4. Let (C, G) be a big action. Let H := [G ′ , G] be the commutator subgroup of G ′ and
	G.
	1. Then, H is trivial if and only if G ′ ⊂ Z(G).
	2. The group H is strictly included in G ′ .
	3. The pair (C/H, G/H) is a big action. Moreover, its second ramification group (G/H) 2 = (G/H) ′ = G 2 /H ⊂ Z(G/H) is p-elementary abelian.
	Proof:
	1. The first assertion is clear.
	2. As G ′ is normal in G, then H ⊂ G ′ . Assume that G ′ = H. Then, the lower central series of G is stationnary, which contradicts the fact that the p-group G is nilpotent (see e.g. [Su86]
	Chap.4). So H G ′ .

  where (G ′ ) ′ means the derived subgroup of G ′ and (G ′ ) p

	the subgroup generated by the p powers of elements of G ′ (cf. [LGMK02] Prop. 1.2.4). This
	proves that if G ′ is p-elementary abelian, then F is trivial. The converse derives from the fact
	that G ′ /F is p-elementary abelian (cf. [LGMK02] Prop. 1.2.4).
	2. Using Corollary 4.4, the only inclusion that remains to show is

  p 2 -c -p 1+2 p 2 (c 1+2 p 2 + c p 2 1+2 p 2 ) X p + X) c 2+p 2 c 2+p 2 = -c p 2In this case, the solutions can be parametrized by 3 algebraically independent variables overF p , namely c 1+2 p 2 ∈ k × , c 1 ∈ k and γ 2 ∈ F p 2 -F p .

	c 3 c 1	1+2 p 2 1+2 p 2 + 4 c 1+p 4 1+2 p 2 (3 c 2 p 4 1+2 p 2 + c 2 1+2 p 2 ) 3 = -c p 2 3 c p 2 c 1 ∈ k
	In both cases,
		|G| g	=	2 p p -1	p 4 1 + p + 2 p 2 and	|G| g 2 =	4 (p 2 -1) 2	p 3 (p + 1) 2 (1 + p + 2 p 2 ) 2

Acknowledgements.

I wish to express my gratitude to my supervisor, M. Matignon, for his valuable advice, his precious help and his permanent support all along the elaboration of this paper.

(c) We show that f 3 ∈ Σ 2 .

If f 3 ∈ Σ 2 , the representation ρ is trivial, hence a contradiction. Therefore, f 3 ∈ Σ 2 and one can define an integer a ≤ m 3 such that X a is the monomial of f 3 with highest degree among those that do not belong to Σ 2 . Since f 3 is assumed to be reduced mod ℘(k[X]), then a = 0 mod p.

(d) We show that p divides a -1. Consider the equation:

where ℓ 1,3 and ℓ 2,3 are non zero linear forms from V to F p . The monomials of f 3 with degree strictly lower than a belong to Σ 2 . So they give linear contributions in ∆ y (f 3 ) mod ℘(k[X]) (cf. [Ro08a] Lemma 3.9). Assume that p does not divide a -1. Then, for all y in V , equation (10) gives the following equality mod ℘(k[X]):

which is another contradiction. So, a -1 = 1 + p s and by equating the corresponding coefficients in (10), one gets: a y = ℓ 1,3 (y) + γ 2 ℓ 2,3 (y), for all y in V . So, V ⊂ F p + γ 2 F p and v ≤ 2. As v = 2 s, we deduce that s = 1, which is a contradiction. Thus, p divides a -1 and one can write a = 1 + λ p t with t ≥ 1 and λ ≥ 2, because of the definition of a.

We also define j 0 := a -p t .

(e) We show that v ≥ t + 1. By [START_REF] Rocher | Large p-groups actions with a p-elementary abelian second ramification group[END_REF] (Lemma 3.11),

(f) We show that j 0 = 1 + p s . If j 0 < 1 + p s , we gather the same contradiction as the one found in [START_REF] Rocher | Large p-groups actions with a p-elementary abelian second ramification group[END_REF] [proof of Theorem 5.6, point 4, with i = 2]. Now, assume that j 0 > 1 + p s . As in [START_REF] Matignon | On smooth curves endowed with a large automorphism p-group in characteristic p[END_REF] [proof of Theorem 5.1, point 6], we prove that the coefficient of X j0 in the left-hand side of (10) is T (y), where T is a polynomial of k[X] with degree p t . If j 0 > 1 + p s , then T (y) = 0, for all y in V . This implies V ⊂ Z(T ) and v ≤ t, which contradicts the previous point.

(g) We show that either v = t + 1 or v = t + 2.

We have already seen that v ≥ t. As j 0 = 1 + p s , we equate the corresponding coefficients in (10) and obtain T (y) = ℓ 1,3 (y) + γ 2 ℓ 2,3 (y), for all y in V . As ℓ 1,3 (y) ∈ F p and ℓ 2,3 (y) ∈ F p , we get T (y) p -T (y) = ℓ 2,3 (y) (γ p 2 -γ 2 ), with γ 2 ∈ F p . Then, for all y in V , R(y) := T (y) p -T (y)

Assume that m 3 > a. Then, by definition of a, m 3 = 1 + p s3 with s 3 ≥ s. Note that s 3 ≥ s + 1. Otherwise, m 3 = 1 + p s = j 0 < a. On the one hand, |G| = p 3+v = p 3+2s . On the other hand,

We already know that s ≥ 2 and v = 2 s ≥ 4. So, |G| = p 3+v ≤ p 7 . Assume that s ≥ 3. Then, as t ≥ 1, we get:

which is a contradiction. So s = 2 and v = 4. We have previously mentionned that γ 2 ∈ F p d -F p , where d is an integer dividing s. As s = 2, the only possibility is d = 2.

(j) We deduce that t = s = 2, so m 3 = 1 + 2 p 2 and p ≥ 5.

We have seen v = t + 1 or v = t + 2, with t ≥ 1. As v = 4, there are two possibilities either t = 2 or t = 3. If t = 3, |G| = p 7 and g = p-1

and condition ( * ) requires p ≥ 5.

Therefore, the solutions can be parametrized by 3 algebraically independent variables over 2. G = A ∞,1 is the unique p-Sylow subgroup of A. Furthermore, Z(G) is cyclic of order p.

Proof:

1. In this case, the group G satisfies the third condition of [Ro08a] (Prop. 5.2). So, we deduce from [Ro08a] (Thm. 5.6) that m 1 = 1 + p s , m 2 = 1 + 2 p s , m 3 = 1 + 3 p s with p ≥ 5 and v = s + 1. Furthermore, it follows from Lemma 5.9 that s ∈ {1, 2}. Assume that s = 2. Then,

(1+2 p+3 p 2 ) 2 and condition ( * ) is satisfied as soon as p ≥ 11. Then, we deduce from Proposition 5.6 the parametrization of f 1 , V and f 2 mentionned in the table. Besides, we deduce from [Ro08a] (Thm. 5.6) that f 3 is in Σ 4 -Σ 3 with m 3 = 1 + 3 p. This means that f 3 reads as follows:

We determine the expressions of the coefficient by solving the equation:

with ℓ 1,2 (y) = ℓ 2,3 (y) = 2 (b 1+2 p y p -b p 1+2 p y) (cf. [START_REF] Rocher | Large p-groups actions with a p-elementary abelian second ramification group[END_REF], Prop. 5.4.1). The results are gathered in the table above.


Yohann Wittrant

Laurent L Leotoing

Fabien F Wauquier

Claire C Philippe

Véronique Coxam

Laurent Léotoing

GPR40: a new relevant target for nutraceutical

Keywords: GPR40, fatty acids, osteoporosis, osteoclast, NF-κB

come

Background:

Industrialized countries face an amplified prevalence of metabolic disorders as a consequence of increased amount of fat in daily food intake. The main social and economic burdens of these pathologies lay in their complications including cardio-vascular diseases, cancers and osteoporosis. However, from a "bone" point of view, mechanisms linking fat intake to bone alteration remain quite controversial.

Objectives:

By demonstrating the expression of the free fatty acid receptor G-coupled Protein Receptor 40 (GPR40) in bone cells, we hypothesized that this receptor may play a role in mediating the effects of fatty acids on bone remodeling.

Methods and Results:

Using µCT analysis, we showed that GPR40-/-mice exhibit osteoporotic features suggesting a positive role of GPR40 on bone density. In primary cultures of bone marrow, we revealed that GW9508, a GRP40 agonist, abolished bone resorbing cell differentiation. This alteration of the Receptor Activator of NF-κB Ligand (RANKL)-induced osteoclast differentiation occurred via the inhibition of the Nuclear Factor κB (NF-κB) signalling pathway as demonstrated by decrease in gene reporter activity, inhibitor of κB kinase (IKKα/β) activation, inhibitor of κB (IkBα) phosphorylation and Nuclear Factor of Activated T cells 1 (NFATc1) expression. The GPR40-dependent effect of GW9508 was confirmed using shRNA interference in osteoclast precursors and GPR40-/-primary cell cultures. In addition, in vivo administration of GW9508 counteracted ovariectomy-induced bone loss in wild-type but not GPR40-/-mice enlightening the obligatory role of the GPR40 receptor.

Conclusions:

In a context of growing prevalence of metabolic and age-related bone disorders, our results demonstrate for the first time in translational approaches that GPR40 is a relevant target for the design of new nutritional and therapeutic strategies to counter bone complications.

Funding:

N/A


A Kajma
email: a.kajma@nencki.gov.pl

P Bednarczyk

A Szewczyk

Marcin M Kamiński
email: m.kaminski@dkfz.de

Daniel Röth

Sabine Sass

Sven W Sauer

Peter H Krammer

Karsten Gülow
email: k.guelow@dkfz.de

A novel mitochondrial potassium channel in embryonic hippocampal mitochondra

Since 1991, when in Nature Inoue and co-workers described the first potassium channel in the inner mitochondrial membrane, ATPregulated (mitoK ATP ), four other channels were found. ATP-regulated (mitoK ATP ), large-conductance calcium activated (mitoBK Ca ), voltagedependent (mitoKv1.3), and intermediate-conductance calciumactivated potassium channel (mitoIK Ca ), were identified by the electrophysiological techniques. The last, TWIK-related acid-sensitive potassium channel (TASK-3) was identified with the use of immunofluorescence methods.

Patch-clamp single channel studies on mitochondria isolated from embryonic rat hippocampus revealed the presence of the potassium channel which has outwardly rectifying activity at the symmetrical conditions (150 mM/150 mM KCl). The channel displayed a conductance of 61 pS, at positive voltages, and also strong voltage dependence. Patch-clamp studies at the mitoplast-attached mode showed that this channel was not sensitive to the classical activators and inhibitors of the mitochondrial potassium channels, but regulated by the pH and arachidonic acid.

In summary, by the single channel recordings, we characterized for the first time an ion channel which was cation selective, permeable to potassium ions and displayed voltage sensitivity. The channel does not correspond to the potassium ion channels described earlier in the inner mitochondrial membrane.

T cell receptor (TCR)-mediated generation of mitochondrial reactive oxygen species (ROS) plays a crucial role for expression of interleukin 2 (IL-2) and CD95 ligand (CD95L, FasL/Apo-1L) genes, and consequently, for T cell activation and activation-induced cell death (AICD). Our study demonstrates that a major mitochondrial antioxidative enzyme, manganese superoxide dismutase (MnSOD/SOD2), acts as an important control switch in the process of T activation-induced oxidative signal generation. Higher abundance and activity of MnSOD in the late phase of TCR-triggered response temporally associates with a shutdown phase of mitochondrial oxidative signal generation. Transient or inducible MnSOD over-expression inhibited mitochondrial ROS production. In turn, lower oxidative signal resulted in an abrogated NF-κB-and AP-1-mediated transcription of IL-2 and CD95L genes and decreased IL-2 secretion as well as CD95L-dependent AICD. Moreover, TCR-mediated transcriptional upregulation of MnSOD demonstrates a negative feedback regulation being itself dependent on oxidation-sensitive transcription. Our finding stresses a critical role for MnSOD and mitochondria as regulators of T cell activation.

AMP-activated protein kinase (AMPK) is a central cellular signaling hub involved in energy homeostasis, growth and proliferation. The kinase is considered as a promising target for pharmacological intervention in several energy-related pathologies like diabetes type II and cancer. However, its signaling network is still incompletely understood. Here we have applied a novel, two-dimensional in vitro screen to search for novel AMPK substrates [1]. By combining biophysical interaction based on surface 7pasmon resonance with in vitro phosphorylation, this screen can identify AMPK substrates that are specific for a given AMPK isoform. Application of this screen to full-length AMPK α2β2γ1 and soluble rat liver proteins identified the tumor suppressor fumarate hydratase (FH). FH was confirmed to interact with and to be preferentially phosphorylated by the AMPKα2 isoform using yeast-two-hybrid [2] and in vitro phosphorylation assays. AMPK-mediated phosphorylation of FH significantly increased enzyme activity in vitro and in vivo, suggesting that it is a bona fide AMPK substrate. In vivo, AMPKα2 may target the cytosolic/ nuclear pools of FH, whose tumor suppressor functions rely on DNA damage repair and stabilization of HIF-1α-signaling that induces pseudohypoxia.

Mitochondria are crucial not only in energy metabolism but also in regulation of cell senescence and apoptosis. The strict control of inner mitochondrial membrane permeability and selective ion transport is essential for mitochondria functioning. Potassium ion homeostasis is an important process for mitochondrial optimal functioning. Potassium channels such as ATP-regulated, large conductance calcium activated and voltage dependent channels were observed in inner mitochondrial membrane in various mammalian tissues. Recently, we have identified potassium channels in inner mitochondrial membrane of potato Solanum tuberosum and Acanthamoeba castellanii. Currently we characterize mitochondrial potassium channels from one of Dictyostelium species. It is commonly used as a model organism to study cell differentiation, metabolism and programmed cell death. Preliminary experiments are focused on biophysical and pharmacological characterization of mitochondrial ion channels. Purified inner mitochondrial membranes (submitochondrial particles) were reconstituted into planar lipid bilayer. To form model membranes asolectin from soybean mixture of phospholipids was used. We observed two types of potassium selective ion channels in submitochondrial particle samples: a large-and small-conductance channels. Experiments were performed both in gradient solution 50/150 mM KCl (cis-trans) and in symmetrical solution 150/150 mM KCl at voltages from -50 to 50 mV. Regulation of the channel activity by divalent cations such as Ca 2+ and Mg 2+ was explored. Additionally, interaction of the ATP with mitochondrial potassium channels was characterized. The knowledge on mitochondrial ion channels may contribute to understanding molecular mechanism of Dictyostelium discoideum functioning.

This Endothelium is a thin layer of cells lining all cardiovascular system. It plays crucial role in such diseases as atherosclerosis, hypertension and diabetes. It is well documented that the inflammation is responsible for these diseases, however very little is known about the role of mitochondria in development of proinflammatory state. It is known that reactive oxygen species (ROS) produced by mitochondria can be involved in the proinflammatory process.

As a model of our study we used endothelial immortalized cell line EA.hy 926 and as a marker molecule the ICAM-1 (intracellular adhesion molecule 1) expression was measured using flow cytometry method. The inflammation was induced by cytokine TNF-α (tumor necrosis factor α). We examined the role of reactive oxygen species (ROS) in the proinflammatory process using fluorescent probe DCF-DA. The main source of ROS production in cells is mitochondria, therefore we checked the effect of rotenone, the complex I of respiratory chain inhibitor, on ROS level in EA.hy 926 cells.

In our study TNF-α caused time and dose dependent increase of ICAM-1 from hardly detected residual level. Additionally our results show that TNF-α increases ROS production in EA.hy 926 cells in dose dependent manner. Rotenone was ineffective in changing the ROS production level in EA,hy 926 cells. Our results related to rotenone are slightly different to literature data that suggests that different cellular models can response to rotenone in different ways.

This work was supported by the European Union from the resources of the European Regional Development Fund under the Innovative Economy Programme (POIG.01.01.02-00-069/09). doi:10.1016/j.bbabio.2012.06.235 11P10 NCLX, but not Letm1, mediates matrix Ca 2+ extrusion and modulates the mitochondrial redox state during HeLa cell stimulation Umberto De Marchi 1,2 , Jaime Santo-Domingo 2 , Israel Sekler3 , Andreas Wiederkehr 1 , Nicolas Demaurex 2 Mitochondria sense and shape intracellular Ca 2+ signals, acting as a cell signaling hub. The uptake of Ca 2+ into the mitochondrial matrix activates intermediary and energy metabolism, and Ca 2+ extrusion mechanisms ensure that this Ca 2+ signal is transient. After a long quest, the proteins promoting Ca 2+ uptake and release have been discovered: a mitochondrial Ca 2+ uniporter was shown to mediate Ca 2+ uptake, and two ion exchangers, NCLX and Letm1, were proposed to exchange Ca 2+ against Na + or H + respectively.

To relate mitochondrial Ca 2+ extrusion to mitochondrial function, we have manipulated the expression levels of NCLX and Letm1 and measured by single cell imaging their impact on matrix Ca 2+ , matrix redox state, and NAD(P)H concentration evoked by Ca 2+ mobilizing agonists.

We find that the histamine stimulated mitochondrial Ca 2+ rise is highly variable in individual HeLa cells. The rate of Ca 2+ extrusion is a function of this amplitude, being highest for large matrix Ca 2+

Laboratory of Intracellular Ion Channels, Nencki Institute of Experimental Biology,

Pasteura St., 02-093 Warsaw, Poland E-mail: agnieszka_lojek@sggw.pl


Cecile Desbiez
email: cecile.desbiez@avignon.inra.fr

Catherine Wipf-Scheibel

Charlotte Chandeysson

Hervé Lecoq

Cécile Desbiez

Do recombinants appearing in natural populations of watermelon mosaic virus represent new agronomic threats?

Watermelon mosaic virus (WMV, Potyvirus) is very common in France where it has been present for at least 40 years. Since the early 2000s, new "emerging" (EM) strains of WMV, highly divergent molecularly from the "classic" (CL) strains present before and probably originating from recent introductions, were detected in southeastern France. Since both types of strains did not appear to be present in the same geographic locations before, this situation constituted a unique opportunity to study the frequency of appearance and the potential spread of recombinants in the few years following the introduction of the new strains. Analyzing isolates from experimental plots in Montfavet (Southeastern France) as well as from epidemiological surveys performed from 2004 to 2008 all around France (about 2000 WMV isolates) revealed at least 7 independent recombination events, either between CL and EM strains or between different EM subgroups. Most recombinants were found in a few plants from the same field, but, with one notable exception, did not seem to spread or be maintained locally for several years. Mixed infections of CL and EM isolates were also performed in experimental conditions in order to compare the frequency and nature of recombination events to those of natural situation. The fitness of natural recombinants relative to potential "parental" strains was also tested in controlled conditions. The epidemiological and evolutionary consequences of recombination in WMV populations will be discussed. This work represents one of the first estimation for the frequency of appearance of recombinants in natural populations of a plant RNA virus.

About ICPVE

The International Committee for Plant Virus Epidemiology (ICPVE) is a subject committee of the International Society for Plant Pathology (ISPP). The ISPP was founded in 1968 in the United Kingdom, for worldwide development of plant pathology. The ISPP sponsors International Congress of Plant Pathology, and International Meetings of its Subject Committees. ICPVE, since formation in 1979, has conducted eleven international symposia in different parts of the world. This 12th IPVE Symposium in Arusha, Tanzania, is the first to be held in the Africa.

Institute of Tropical Agriculture in partnership with

Mikocheni Agricultural Research Institute (MARI), Tanzania National Agricultural Research Organization (NARO), Uganda West and Central African Council for Agricultural Research and Development (CORAF/WECARD) Bioversity International AVRDC -The World Vegetable Center

Supported by

Africa's Genomics Company

ISBN 978-978-8444-08-4

Published by

International Institute of Tropical Agriculture (IITA) 2013 IITA, PMB 5320, Ibadan, Nigeria Telephone: +1-201-6336094, +234 2 7517472 Mobile: +234 8034035281-3 E-mail: iita@cgiar.org web: www.iita.org

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This book contains the abstracts of the papers in the 12 th International Symposium on Plant Virus Epidemiology on the theme 'Evolution, Ecology and Control of Plant Viruses' organized by the ICPVE and IITA. They reflect authors' opinions and are published, after formatting, as submitted without any formal review. Their inclusion in the publication does not necessarily constitute endorsement by the organizers. Enquires concerning the technical content of Abstracts should be addressed directly to the Authors.

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