Automorphism groups for p-cyclic covers of the affine line ^{∗}

Claus Lehr and Michel Matignon

May 21, 2004

Abstract

Let k be an algebraically closed field of positive characteristic p > 0 and C → ℙ_{k}^{1} a p-cyclic cover of the projective line ramified in exactly one point. We are interested in the p-Sylow subgroups of the full automorphism group Aut_{k}C. We prove that for curves C with genus ≥ 2 these groups are exactly the extensions of a p-cyclic group by an elementary abelian p-group. The main tool is an efficient algorithm to compute the p-Sylow subgroups of Aut_{k}C starting from an Artin-Schreier equation for the cover C → ℙ_{k}^{1}. We also characterize curves C with genus g_{C} ≥ 2 and a p-group action G ⊂ Aut_{k}C such that \frac{2p}{p−1} < \frac{|G|}{g_{C}} and \frac{4}{(p−1)^{2}} ≤ \frac{|G|}{g_{C}^{2}}.

Our methods rely on previous work by Stichtenoth [St1], [St2] whose approach we have adopted.

1 Introduction

When considering semi-stable models for p-cyclic covers of the projective line over a p-adic field K with an algebraically closed residue field k, we get as irreducible components of the special fiber p-cyclic étale covers of the affine line (see [Le1], [Ma]). An interesting object in arithmetic geometry is the monodromy, i.e. the minimal Galois extension K′/K such that a semi-stable model is defined over K′. A general result (see [De], [Liu, theorem 4.44, p.551]) asserts that Gal(K′/K) acts faithfully on the special fiber of the semi-stable model as an automorphism group; so the complexity of the monodromy group is intimately related with the automorphism group of p-cyclic covers of the affine line over the residue field k. The aim of this paper is first to report on and then to complete the literature on the subject (cf. [St1], [St2],[Si], [He],[Br-St], [Br], [Na], [vdG-vdV],[El]). We do this with the objective to use the results to study the monodromy group which occurs when considering p-cyclic covers of the projective line over a p-adic field (see [Le-Ma1], [Le-Ma2]).

∗Work supported in part by the European Community’s Human Potential Programme under Contract HPRN-CT-2000-00114, GTEM

Our choice to disjoin this paper from the one on monodromy came from the new interest in automorphism groups of curves (see [Co-Ka-Ko],[Co-Ka],[Gu], [Ko], [Leo], [Po1], [Po2], [vdP-V1] , [vdP-V2]...).

In the following theorem we have gathered the main results proved in the paper.

Theorem 1.1. Let k be an algebraically closed field of characteristic p > 0 and f(X) ∈ k[X] a polynomial of degree m := deg f prime to p. Let k(X, W)/k(X) be an extension of degree p defined by W^{p} − W = f(X) and denote by ∞ the place in k(X, W) above X = ∞. We denote by g(f) = \frac{(m−1)(p−1)}{2} the genus of k(X, W)/k and assume g(f) ≥ 2. Let Aut_{k}k(X, W) be the full automorphism group, G_{∞}(f) ⊂ Aut_{k}k(X, W) the inertia group at ∞ and G_{∞,1}(f) the wild inertia group at ∞. Let ρ ∈ G(f) be such that ρ(X) = X and ρ(W) = W + 1; then ρ generates a p-cyclic subgroup in G_{∞}(f). I. a) [Section 3] The group G_{∞,1}(f) is a p-Sylow subgroup of Aut_{k}k(X, W); moreover outside a small number of cases it is normal.

b) [Section 3] The group G_{∞,1}(f) is an extension (necessarily central) of < ρ >≃ ℤ/pℤ by a p-elementary abelian group (ℤ/pℤ)^{t} which is a subgroup of translations X → X + y of the affine line 𝔸_{k}^{1}.

c) [Theorem 7.1] Reciprocally any extension of ℤ/pℤ by an elementary abelian group (ℤ/pℤ)^{t} with t ≥ 0 is obtained in this way. II. a) [St2, Satz 4], [Prop. 6.6 ] |G_{∞,1}(f)| ≤ p(m − 1)^{2}, i.e. \frac{|G_{∞,1}(f)|}{g(f)^{2}} ≤ \frac{4p}{(p−1)^{2}} .

b) [St2, Satz 4], [Prop. 6.6 ] If (m − 1, p) = 1 then |G_{∞,1}(f)| = p.

c) [St2, Satz 5], [Prop. 6.6] If f(X) = X^{m} then |G_{∞,1}(f)| = p for m − 1 ≠ p^{s} and |G_{∞,1}(f)| = p^{2s+1} for m − 1 = p^{s}.

d) [Prop. 6.6] If m − 1 = ℓp^{s}, s > 0, ℓ > 1, (ℓ, p) = 1, then |G_{∞,1}(f)| ≤ p^{s+1} for p > 2 and |G_{∞,1}(f)| ≤ 2^{s} for p = 2; moreover these bounds are optimal.

e) [Prop. 8.3] One has \frac{p}{p−1} < \frac{|G_{∞,1}(f)|}{g(f)} (2/3 for p = 2) iff f(X) = XS(X), where S(X) − S(0) is an additive polynomial.

f) [Theorem 8.6] Let C be a non singular projective curve over k of genus g_{C} ≥ 2 and G a p-subgroup of Aut_{k}C. We assume that \frac{2p}{p−1} < \frac{|G|}{g_{C}}. Then \frac{4}{(p−1)^{2}} ≤ \frac{|G|}{g_{C}^{2}} iff C is birational to a curve W^{p} − W = f(X) where f(X) = XS(X) and S(X) − S(0) is an additive polynomial. Moreover either \frac{|G|}{g_{C}^{2}} = \frac{4p}{(p−1)^{2}} and the group G is then equal to the

group G_{∞,1}(f) or \frac{|G|}{g_{C}^{2}} = \frac{4}{(p−1)^{2}} and the group G is then equal to an index p-subgroup of the group G_{∞,1}(f).

The paper is organized as follows. In section 3 we give the structure of p-Sylow subgroups of Aut_{k}k(X, W). Section 4 is a group theoretic one: we give a description of extensions of a p-cyclic group by any p-elementary abelian group as a saturated subgroup of those extensions for which the center is p-cyclic: the latter extensions are the so-called extraspecial groups. This description seems to be missing in literature and is adapted to the realization result we have in mind (see I. c. above). Section 5 gives a lemma for the calculation of the group of translations of the affine line occurring in Th. 1.1. I.b) and for the structure of G_{∞,1}(f). An algorithm is presented and then applied to a concrete example f(X) ∈ 𝔽_{2}[X] for which G_{∞,1}(f) is the dihedral group D_{8} of order 8. In section 6 we develop the notion of modification of covers. One type, which corresponds to a base change by a cover of the affine line by itself, has the effect to enlarge the group G_{∞,1}(f). A second type of modification is presented which has the property to limit the order of the group. We then give bounds for the size of the group G_{∞,1}(f) and show that they are sharp. In section 7 we realize extensions of p-cyclic by p-elementary abelian groups as groups G_{∞,1}(f). It is noticeable that one type of extraspecial group (namely the central product of copies of D_{8} in characteristic 2) is particularly difficult to realize explicitly. We apply the method to give a realization of D_{8} with minimal conductor. Section 8 describes the pcyclic covers of the line which maximalize the ratio |G_{∞,1}(f)|/g(f)^{2}. They correspond to a special locus in the moduli spaces of curves describing the ”big” actions of pgroups on curves in characteristic p > 0 - a first step towards a classification.

2 Notation

Throughout this paper we use the following notation:

• k is an algebraically closed field of characteristic p > 0.

• F denotes the Frobenius endomorphism for a k-algebra.

• k{F} denotes the k-subspace of k[X] generated by the polynomials F^{i}(X), i = 0, 1, 2, ...; it is a ring under the composition and for α ∈ k, Fα = α^{p}F. The elements in k{F} are the additive polynomials, i.e. those P ∈ k[X] such that P (α + β) = P (α) + P (β) for α, β ∈ k. Moreover a separable polynomial in k[X] is additive iff the set of its roots Z(P) is a subgroup of k ([Go], Chap.1).

• For f(X) ∈ k[X] the equation W^{p} − W = f(X) defines an étale cover of the affine line that we denote by C_{f} and each étale cover of the affine line can be presented like this. Let f(X) ∈ k[X] then there is a unique polynomial red(f)(X) ∈ k[X] called the reduced representative of f which is p-power free, i.e. red(f)(X) ∈ ⨁_{(i,p)=1} kX^{i} and such that red(f)(X) = f(X) mod (F − Id)k[X]. By Artin-Schreier Theory the covers C_{f} and C_{red(f)} are the same pcyclic cover of the affine line. The curve C_{f} is irreducible iff red(f) ≠ 0. In what follows we shall always assume that red(f) ≠ 0. The degree of the reduced polynomial red(f) is called the conductor of the cover. It is prime to p and equal to the degree of f if (deg f, p) = 1.

• By C_{f} we also denote the non singular projective curve with function field k(X, W). If m = deg f is prime to p then the genus of C_{f} is g(C_{f}) = \frac{(m−1)(p−1)}{2}.

• For M ∈ C_{f} and n ∈ ℕ let L(nM) = {φ ∈ k(C_{f}) | (φ) + nM ≥ 0}. Then P(M) := {−v_{M}(φ) | φ ∈ ∪_{n∈ℕ}L(nM) − {0}} is called the polar semi-group at M.

• We denote by ∞ ∈ C_{f} the point X = ∞, W = ∞ and by M_{x,w} the point X = x, W = w with w^{p} − w = f(x).

• Let G(f) = Aut_{k}C_{f}, G_{∞}(f) the inertia group at ∞ and G_{∞,1}(f) the wild inertia group at ∞. Let ρ ∈ G(f) be such that ρ(X) = X and ρ(W) = W + 1. Then ρ generates a p-cyclic subgroup in G_{∞}(f).

3 p-Sylow subgroups of the group G(f)

With the above notation Stichtenoth proves the following:

Theorem 3.1 ([St2], Satz 6, 7). Assume the genus g(C_{f}) ≥ 2, i.e. m ≥ 2 and {m, p} ≠ {2, 3}. Then ∞ is the only point M ∈ C_{f} such that P(M) = deg fℕ + pℕ; there are though the following two exceptions:

a) m < p, m|1 + p, f(X) = X^{m}. In this case in addition to ∞ exactly the p zeroes of X have the same semi-group P(M). More precisely for i ∈ 𝔽_{p} let σ_{i} be given by σ_{i}(X) = \frac{X}{(W −i)^{δ}} and σ_{i}(W) = −\frac{1}{W −i}, where δ = \frac{1+p}{m}. Then σ_{i} ∈ G(f) and σ_{i}(M_{0,i}) = ∞. In particular G(f) acts transitively on the p + 1 points M such that P(M) = P(∞); |G(f)| = pm(p^{2} − 1) and G_{∞,1}(f) is a p-Sylow of G(f).

b) f(X) = X^{1+p}. Now exactly the p^{3} points M_{α,β} where α^{p^{2}} = −α and β^{p} − β = α^{1+p} have the same semi-group as the point ∞. Let a ∈ k with a^{p^{2}−1} = 1,

σ_{α,β}(X) = α + \frac{aX}{W} and σ_{α,β}(W) = β − \frac{a^{1+p}}{W} − aα^{p} \frac{X}{W} .

Then σ_{α,β} ∈ G(f) and σ_{α,β}(∞) = M_{α,β}. In particular G(f) acts transitively on the p^{3} +1 points M such that P(M) = P(∞); |G(f)| = p^{3}(p^{3} +1)(p^{2} −1) and G_{∞,1}(f) is a p-Sylow of G(f).

Remark 3.2. Stichtenoth considers equations of the type W^{p} + W = f(X). The expressions of the automorphisms σ_{i} and σ_{α,β} in cases a) and b) are then simpler. For a description of the full automorphism group in case b) we refer to [Leo].

We deduce the following

Proposition 3.3. Let f ∈ Xk[X], g ∈ T k[T] and C_{g} : V^{p} − V = g(T) such that deg f, deg g are prime to p and g(C_{f}) ≥ 2. We assume ∃φ : C_{f} → C_{g} a kisomorphism. Then ∃σ ∈ G(f) such that φ ∘ σ(∞) = ∞ and φ ∘ σ descends to ℙ^{1}, i.e. ∃φ̃ : ℙ^{1} → ℙ^{1} and a commutative diagram

C_{f} ⟶^{φ∘σ} C_{g}
X ↓  ↓ T
ℙ^{1} ⟶^{φ̃} ℙ^{1}

Moreover ∃(a, b) ∈ (k^{×}, k) and c ∈ 𝔽_{p}^{×} such that φ̃^{♯}(T) = aX + b and cf(X) − g(aX +b) ∈ (F−Id)k[X]. If f is not in case a) or b) of Theorem 3.1 we can take σ = Id; then G(f) = G_{∞}(f) and so G_{∞,1}(f) is the unique p-Sylow subgroup. Reciprocally any triple (a, b, c) ∈ k^{×} ×k ×𝔽_{p}^{×} such that cf(X)−g(aX +b) ∈ (F−Id)(k[X]) induces such a commutative diagram.

Proof. Let m = deg f then it is classical that g(C_{f}) = \frac{(m−1)(p−1)}{2}. Therefore as g(C_{f}) = g(C_{g}) it follows that m = deg f = deg g. The homomorphism φ^{♯} : k(C_{g}) → k(C_{f}) is a k-isomorphism which induces a graded isomorphism between the linear spaces L(∞φ(M)) and L(∞M) and so the polar semi-groups are preserved. It follows that φ^{−1}(∞) ∈ C_{f} has a polar semi-group equal to that of ∞ ∈ C_{f}. We deduce from Theorem 3.1. the existence of σ and can assume from now on that φ(∞) = ∞.

Now we follow the discussion in Stichtenoth’s paper [St2, Satz 4].

Case 1. m > p. Then 1, T, resp. 1, T, ..., T^{[ \frac{m}{p}]}, V is a k-basis for L(p∞) ⊂ k(C_{g}) resp. L(m∞) ⊂ k(C_{g}). Idem 1, X and 1, X, ..., X^{[ \frac{m}{p}]}, W are a basis for L(p∞) ⊂ k(C_{f}) resp. L(m∞) ⊂ k(C_{f}). As φ^{♯}(L(p∞)) = L(p∞) resp. φ^{♯}(L(m∞)) = L(m∞) we get the existence of a ∈ k^{×}, b ∈ k, such that φ^{♯}(T) = aX + b and φ^{♯}(V) = cW + Q(X) where c ∈ k^{×} and Q(X) ∈ k[X] with deg Q(X) ≤ [\frac{m}{p}]. Then (c^{p} − c)W + c^{p}f(X)+(Q(X)^{p}−Q(X)) = g(aX+b), so c^{p} = c and cf(X)−g(aX+b) ∈ (F−Id)k[X]. Reciprocally these relations define an isomorphism.

Case 2. m < p. It works similarly; namely 1, V, resp. 1, V, ..., V^{r}, T with rm < p < (r +1)m is a basis for L(m∞) resp. L(p∞) in k(C_{g}). Then φ^{♯}(T) = aX +P (W), where P (W) ∈ k[W] with m deg P < p and φ^{♯}(V) = cW + Q where c ∈ k^{×} and Q ∈ k. We get the equation (c^{p} − c)W + c^{p}f(X) + (Q^{p} − Q) = g(aX + P (W)) = a^{m}X^{m}+(ma^{m−1}P (W)+b_{m−1})X^{m−1}+...+P (W)^{m}, where g(T) = T^{m}+b_{m−1}T^{m−1}+.... Comparing the W -degrees it follows that P (W) = b ∈ k and c^{p} = c. Now we again get the condition cf(X) − g(aX + b) ∈ (F − Id)k[X].

Corollary 3.4. Let f ∈ Xk[X] be such that g(C_{f}) ≥ 1 and S(f) := {(a, b, c) ∈ k^{×} ×k ×𝔽_{p}^{×} | ∃P_{a,b,c}(X) ∈ k[X], cf(X)−f(aX +b) = P_{a,b,c}(X)^{p} −P_{a,b,c}(X)}. Such a polynomial P_{a,b,c} in k[X] is determined by these relation up to addition by an element in 𝔽_{p}. For (a, b, c) ∈ S(f) the formulas σ_{a,b,c}(X) = aX +b, σ_{a,b,c}(W) = cW +P_{a,b,c}(X) define an automorphism of C_{f} which lies in G_{∞}(f). Further G_{∞}(f) =< ρ, σ_{a,b,c} > for (a, b, c) ∈ S(f).

Proof. If g(C_{f}) > 1 this follows from the proposition. If g(C_{f}) = 1 then {p, m} = {2, 3}. For p = 2 we have f(X) = aX + X^{3} and if p = 3 then f(X) = aX + X^{2}. In each case we can give a basis for L(m∞) and L(p∞) of the same kind as given in the two cases of the proof of the proposition and we conclude in the same way. Note that if g(C_{f}) = 0 then G_{∞}(f) = k^{×}X + k.

Now we can give the general structure of the wild automorphism group G_{∞,1}(f). In order to simplify the notations for (1, y, 1) ∈ S(f) we denote by σ_{y} the element σ_{1,y,1} and by P_{f}(X, y) the corresponding polynomial P_{1,y,1}(X). For (1, y, 1) and (1, z, 1) in S(f) we define the following functions T r_{f}(y) := P_{f}(X, y)) + P_{f}(X + y, y) + P_{f}(X + 2y, y) + ... + P_{f}(X + (p − 1)y, y) ϵ_{f}(y, z) := P_{f}(X, y) + P_{f}(X + y, z) − P_{f}(X, z) − P_{f}(X + z, y).

Corollary 3.5. With the above notations assume g(C_{f}) > 0. Then ρ ∈ Z(G_{∞,1}(f)) and G_{∞,1}(f) =< ρ, σ_{y}, y ∈ S(f) >. Further Tr_{f}(y) ∈ ℤ/pℤ and σ_{y}^{p} = ρ^{Tr_{f} (y)}. The commutation rule is given by [σ_{y}, σ_{z}] = ρ^{ϵ_{f} (y,z)} where ϵ_{f}(y, z) = P_{f}(X, y) + P_{f}(X + y, z)−P_{f}(X, z)−P_{f}(X +z, y) ∈ ℤ/pℤ. Moreover we have the following exact sequence 0 →< ρ >≃ ℤ/pℤ → G_{∞,1}(f) →^{π} k, where π(σ_{y}) = y ∈ k and the image of π is finite dimensional as 𝔽_{p}-vector space, i.e. a finite p-elementary abelian group.

Proof. From σ_{y}(X) = X +y and σ_{y}(W) = W +P_{f}(X, y) we get σ_{y}^{p}(W) = W +Tr_{f}(y). As σ_{y}^{p}(X) = X it follows that Tr_{f}(y) ∈ ℤ/pℤ and σ_{y}^{p} = ρ^{Tr_{f} (y)}. For the commutation rule we remark that [σ_{y}, σ_{z}](X) = X. From this it follows that [σ_{y}, σ_{z}] = ρ^{ϵ_{f} (y,z)} for some ϵ_{f}(y, z) ∈ ℤ/pℤ. In order to determine ϵ_{f}(y, z) we calculate σ_{y} ∘ σ_{z}(W) = W +P_{f}(X, y)+P_{f}(X+y, z) which is equal to W +P_{f}(X, z)+P_{f}(X+z, y)+ϵ_{f}(y, z).

4 Extensions of a p-cyclic group by a p-elementary abelian group

In Corollary 3.5 we saw that the groups G_{∞,1}(f) belong to the following class of p-groups:

Definition 4.1. We denote by C_{1} the set of finite groups G which are extensions of a p-cyclic group N by a finite p-elementary abelian group G/N. We remark that the induced homomorphism G/N → AutN is necessarily trivial for order reasons and so such an extension is central.

Reciprocally, in section 7, we prove that any group in C_{1} is a group G_{∞,1}(f). To this end we need a group theoretic classification in terms of a special class of such groups.

4.1 Extraspecial groups

We refer to [Hu], [Su1] and [Su2] for the structure of finite p-groups.

If G lies in C_{1} then necessarily N ⊂ Z(G). Those non abelian groups in C_{1} for which Z(G) is itself p-cyclic are called extraspecial. In particular a non abelian group G of order p^{3} is extraspecial:

If p > 2 then G is isomorphic to one of the following two groups:

E(p^{3}) =< x, y|x^{p} = y^{p} = [x, y]^{p} = 1, [x, y] ∈ Z(E(p^{3})) > with exponent p, or M(p^{3}) =< x, y|x^{p^{2}} = y^{p} = 1, y^{−1}xy = x^{1+p} > with exponent p^{2}. If p = 2 then G is either isomorphic to the dihedral group D_{8} or to the quaternion

group Q_{8}. These two groups both have exponent 2^{2}.

More generally let G be an extraspecial p-group then (see [Su2] Th. 4.18) |G| = p^{2n+1} for some n > 0 and the following 4 types occur.

I. If p > 2 then either

a. exponent(G) = p and G is a central product of n groups E(p^{3}) (i.e. the direct product where the p-cyclic centers of each factor are amalgamated).

b. exponent(G) = p^{2} and G is a central product of M(p^{3}) and n−1 groups E(p^{3}). II. If p = 2 then either a. G is a central product of n groups D_{8}. Let q_{a}(n) (resp. d_{a}(n)) be the number

of elements of order 4 (resp. ≤ 2). Or

b. G is a central product of a group Q_{8} and n − 1 groups D_{8}. Let q_{b}(n) (resp. d_{b}(n)) be the number of elements of order 4 (resp. ≤ 2).

We have the following:

q_{a}(n) = −2^{n} + 2^{2n} and so d_{a}(n) = 2^{n} + 2^{2n}

q_{b}(n) = 2^{n} + 2^{2n} and so d_{b}(n) = −2^{n} + 2^{2n}

In each case the structure of the central product is uniquely determined by the structure of the factors. In the sequel when speaking of the isomorphism type of an extraspecial group we will refer to the 4 types above, namely I.a ,I.b , II.a and II.b.

The class C_{1} has been described in the following result (cf. [Su2, chap.4. th. 4.16]):

Proposition 4.2. Any G ∈ C_{1} is isomorphic to one of the groups in the following list.

a) An elementary abelian p-group.

b) An abelian group of type (p, p, …, p, p^{2}).

c) A central product of an extraspecial p-group E and an abelian group A. If A is not elementary abelian, then

E ∩ A = Z(E) = A^{p}

4.2 Saturated subgroups of extraspecial groups

Definition 4.3. Let E be an extraspecial group and π : E → \frac{E}{Z(E)} the canonical homomorphism. A saturated subgroup of E is a group π^{−1}(V), for V a subgroup of \frac{E}{Z(E)}. We denote

C_{2} := {G which are saturated subgroup of some extraspecial group}.

From the description of an extraspecial group as a central product of copies of those of order p^{3} given in section 4.1, it follows that for p > 2 (resp. p = 2) any extraspecial group of type I.a (resp. II.a) is a saturated subgroup of an extraspecial group of type I.b (resp. II.b).

We will need the following result from group theory. Although it is possible to give a proof using the classification of extraspecial groups and Prop. 4.2, we give a direct proof which deals with factor systems and fits well with the realization result we intend to prove.

Proposition 4.4. The 2 classes C_{i} for i = 1, 2 are equal.

Proof. Let G ∈ C_{1}, then G′ ⊂ N ⊂ Z(G) (see Def. 4.1). This last condition allows to define a skew-symmetric bilinear form on the 𝔽_{p}-vector space G/N: If \overline{x}, \overline{y} ∈ G/N then [x, y] ∈ G′ =< ρ >⊂ N, i.e. [x, y] = ρ^{ϵ} where ϵ ∈ ℤ/pℤ. Note that [x, y] is independent of the lifts of \overline{x} and \overline{y} to G as N ⊂ Z(G). We define < \overline{x}, \overline{y} >:= ϵ. Note that x ∈ Z(G) iff < \overline{x}, \overline{y} >= 0 for all y ∈ G. Therefore < ., . > is non degenerate iff G′ = N = Z(G), i.e. G is extraspecial. We call < ., . > the form associated to G ∈ C_{1}.

Consider the extension of groups

1 → N → G →^{↶^{s}} V := G/N → 1.

Let s be a set theoretical section. To any two v_{1}, v_{2} ∈ V we have a c(v_{1}, v_{2}) ∈ N such that s(v_{1})s(v_{2}) = s(v_{1}v_{2})c(v_{1}, v_{2}) and c(., .) is the 2-cocycle corresponding to the equivalence class of the above extension in H^{2}(V, N) = H^{2}(V, 𝔽_{p}). Note that N has trivial action by V (cf. Def.4.1). From c(., .) we recover G in the following way: On the set V × 𝔽_{p} one defines a group structure via:

(v_{1}, α)(v_{2}, β) = (v_{1} + v_{2}, α + β + c(v_{1}, v_{2})).

The form associated to G, i.e. < ., . > on V, can be expressed in terms of c: < v_{1}, v_{2} >= [s(v_{1}), s(v_{2})] = s(v_{1})^{−1}s(v_{2})^{−1}s(v_{1})s(v_{2}) = (s(v_{2})s(v_{1}))^{−1}s(v_{1})s(v_{2}) = c(v_{2}, v_{1})^{−1}s(v_{2}v_{1})^{−1}s(v_{1}v_{2})c(v_{1}, v_{2}) = c(v_{2}, v_{1})^{−1}c(v_{1}, v_{2}). Identifying N with 𝔽_{p} we write < v_{1}, v_{2} >= c(v_{1}, v_{2}) − c(v_{2}, v_{1}). We distinguish the cases p > 2 and p = 2.

Case p > 2. Let V ↪ W := V ⨁ V be the embedding that identifies V with the first factor of W and pr_{1} : W → V the projection on the first factor. We denote by c a cocycle corresponding to the given group extension and we extend the corresponding 2-form < v_{1}, v_{2} >= c(v_{1}, v_{2}) − c(v_{2}, v_{1}) from V to W to a non degenerate skew form given by the 2n × 2n-block matrix

( \begin{matrix}A & −𝕀 \\ 𝕀 & 0 \end{matrix} ) ∈ M_{2n}(𝔽_{p})

where A ∈ M_{n}(𝔽_{p}) is the matrix of < ., . > on the n dimensional vector space V and 𝕀 ∈ M_{n}(𝔽_{p}) the identity.

Now we obtain a 2-cocycle d : W × W → 𝔽_{p} via (w_{1}, w_{2}) → < w_{1}, w_{2} > +c(pr_{1}(w_{2}), pr_{1}(w_{1})). We remark that d_{|V} maps (v_{1}, v_{2}) to < v_{1}, v_{2} > +c(v_{2}, v_{1}) = c(v_{1}, v_{2}) − c(v_{2}, v_{1}) + c(v_{2}, v_{1}) = c(v_{1}, v_{2}). So d_{|V} = c and the group extension E corresponding to d therefore contains G as a subgroup.

It remains to show that |Z(E)| = p. This amounts to the skew-form << ., . >> associated to d on W to be non degenerate. We compute on W:

<< w_{1}, w_{2} >>= d(w_{1}, w_{2}) − d(w_{2}, w_{1})
=< w_{1}, w_{2} > +c(pr_{1}(w_{2}), pr_{1}(w_{1}))− < w_{2}, w_{1} > −c(pr_{1}(w_{1}), pr_{1}(w_{2}))
= 2 < w_{1}, w_{2} > − < pr_{1}(w_{1}), pr_{1}(w_{2}) > .

Therefore << ., . >> has the 2n × 2n-block matrix

( \begin{matrix}2A & −2𝕀 \\ 2𝕀 & 0 \end{matrix} ) − ( \begin{matrix}A & 0 \\ 0 & 0 \end{matrix} ) = ( \begin{matrix}A & −2𝕀 \\ 2𝕀 & 0 \end{matrix} ) ∈ M_{2n}(𝔽_{p})

which has maximal rank as p > 2. We conclude E is extraspecial. We have obtained G as subgroup of the extraspecial group E and N = Z(E) ⊂ G follows from the fact that in the construction above the second factor of V × 𝔽_{p} and W × 𝔽_{p} correspond to N and Z(E) respectively.

The case p = 2. Using the above notation, let n := dim V and M_{n}(𝔽_{2}) the 𝔽_{2}- vector space of n × n matrices. Any such matrix defines a bilinear form (hence a 2-cocycle) on V. Therefore we have a map of 𝔽_{2}-vector spaces

M_{n}(𝔽_{2}) →^{φ} H^{2}(V, 𝔽_{2}).  (1)

Moreover a matrix A is in the kernel K of φ iff its associated 2-cocycle c defines the split extension, which is the elementary abelian 2-group of rank n + 1.

This is equivalent to c(v_{1}, v_{2}) = c(v_{2}, v_{1}) for all v_{1}, v_{2} ∈ V and c(v, v) = 0 for all v ∈ V (here we use p = 2). In other words A = A^{t} and A has only zeroes on its diagonal. We conclude dim ker φ = n − 1 + n − 2 + ... + 1 = \frac{n(n−1)}{2}. It is known that dim H^{2}(V, 𝔽_{2}) = \frac{n(n+1)}{2} [Jo] p. 168.

Therefore

1 → K → M_{n}(𝔽_{2}) →^{φ} H^{2}(V, 𝔽_{2}) → 1

is exact. In particular φ is onto, so every element of H^{2}(V, 𝔽_{2}) can be represented by a 2-cocycle that is a bilinear form.

Again we let the given extension correspond to the cocycle c ∈ Z^{2}(V, 𝔽_{2}) and by the above we may assume c is bilinear corresponding to a matrix A.

Let V ↪ W := V ⨁ V be the first factor and consider d ∈ Z^{2}(V, 𝔽_{2}) corresponding to the block matrix

( \begin{matrix}A & 𝕀 \\ 0 & 0 \end{matrix} ) ∈ M_{2n}(𝔽_{p}).

Then d_{|V} = c, so the group defined by d contains G.

Claim: E is an extraspecial group, i.e. the 2-form << ., . >> defined by d on W is non degenerate.

We calculate << w_{1}, w_{2} >>= d(w_{1}, w_{2})−d(w_{2}, w_{1}) = w_{1}^{t}Bw_{2}+w_{2}^{t}Bw_{1} = w_{1}^{t}Bw_{2}+ (w_{2}^{t}Bw_{1})^{t} = w_{1}^{t}Bw_{2} + w_{1}^{t}B^{t}w_{2} = w_{1}^{t}(B + B^{t})w_{2} and

B + B^{t} = ( \begin{matrix}A + A^{t} & 𝕀 \\ 𝕀 & 0 \end{matrix} ) ∈ M_{2n}(𝔽_{p})

has rank 2n (independently of what A is).

Conversely (for any characteristic p > 0) let G ∈ C_{2}. Then G ↪ E is a saturated subgroup of an extraspecial group E. Therefore G/Z(E) ↪ E/Z(E), both are elementary abelian and hence we have an exact sequence

1 ⟶ Z(E) ⟶ G ⟶ G/Z(E) ⟶ 1

showing that G ∈ C_{1}.

Remark 4.5. Let p > 2 and let G ∈ C_{1} with exponent p. We have realized G as a saturated subgroup of an extraspecial group E. A calculation using the extended cocycle c to W then shows that E has the same exponent as G.

5 Computation of G_{∞,1}(f); an algorithm

5.1 Universal family

Notations. In order to be able to treat families of covers it is useful for a given conductor m, prime to p, to work over the ring A := 𝔽_{p}[t_{i}, 1 ≤ i ≤ m] and consider f(X) = ∑_{1≤i≤m} t_{i}X^{i}.

A specialization homomorphism is an homomorphism φ : A → k, where k is an algebraically closed field of characteristic p > 0; then φ(f)(X) = ∑_{1≤i≤m} φ(t_{i})X^{i} ∈ k[X]. Note that the conductor of the cover is preserved iff φ(t_{m}) ≠ 0.

Let i < m and (i, p) = 1 and denote by n(i) := max{n ∈ ℕ | ip^{n} < m}. The following lemma is the key tool; for this we introduce Δ(f)(X, Y) := f(X + Y) − f(X) − f(Y).

Lemma 5.1. (compare with [St2] p. 621) With the above notations there is a unique polynomial R(X, Y) ∈ ⨁_{1≤i<m, (i,p)=1} A[Y]X^{ip^{n(i)}} and a unique polynomial P_{f}(X, Y) ∈ XA[Y][X] such that

Δ(f)(X, Y) = R(X, Y) + (Id − F)P_{f}(X, Y).  (2)

The polynomial P_{f}(X, Y) is characterized by the following:

P_{f}(X, Y) = (Id + F + ... + F^{n−1})Δ(f) mod (X^{[ \frac{m−1}{p} ]+1})  (3)

for any n such that p^{n} > [\frac{m−1}{p}].

Proof. Existence. Note that deg_{X} Δ(f) = m − 1 and Δ(f) ∈ (X, Y)A[X, Y]. Let (i, p) = 1 with 1 ≤ i < m and 0 ≤ j ≤ n(i). For a monomial δ_{ip^{j}}(Y)X^{ip^{j}} of Δ(f) where δ_{ip^{j}}(Y) ∈ A[Y] and of total degree < m we write δ_{ip^{j}}(Y)X^{ip^{j}} = (δ_{ip^{j}}(Y))^{p^{n(i)−j}}X^{ip^{n(i)}}+(Id−F)(P_{ip^{j}}(X)) with P_{ip^{j}}(X) = (Id+F+....+F^{n(i)−j−1})(δ_{ip^{j}}(Y)X^{ip^{j}}).

For the unicity we remark that if P_{f}(X, Y) satisfies the formula (2) then deg_{X} P_{f}(X, Y) ≤ [\frac{m−1}{p}] and so it suffices to prove formula (3) in the lemma. We have the identity:

(Id + F + ... + F^{n−1})Δ(f) = (Id + F + ... + F^{n−1})R(X, Y) + (Id − F^{n})P_{f}(X, Y)

for any n. Now R(X, Y) ∈ (X^{[ \frac{m−1}{p}]+1}) as ip^{n(i)} < m < ip^{n(i)+1} and P_{f}(X, Y) ∈ XA[Y][X]. Then for p^{n} > [\frac{m−1}{p}] we obtain the formula (3).

Definition 5.2. Let φ : A → k be a specialization homomorphism with φ(t_{m}) ≠ 0. We denote by the same letter the induced homomorphism on polynomials via the action on the coefficients. Let Ad_{φ(f)}(Y) ∈ k[Y] be the monic generator of the ideal generated by the coefficients of φ(R)(X, Y) ∈ k[Y]. As usual we denote by Z(Ad_{φ(f)}(Y)) the set of zeroes in the algebraically closed field k. For i = 1, 2 let φ_{i} be two specialization homomorphisms with φ_{i}(t_{m}) ≠ 0. If redφ_{1}(f) = redφ_{2}(f) then φ_{1}(R)(X, Y) = φ_{2}(R)(X, Y) and so Ad_{φ_{1}(f)} = Ad_{φ_{2}(f)}. It follows that for g ∈ k[x], deg g = m and φ : A → k a specialization homomorphism with φ(t_{m}) ≠ 0 and such that g = φ(f) we can define Ad_{g} := Ad_{φ(f)}. Moreover one has Ad_{g} = Ad_{redg}.

Corollary 5.3. Let m − 1 = ℓp^{s} with (ℓ, p) = 1 (i.e. s = v_{p}(m − 1)). Write R(X, Y) = ∑_{j} a_{j}(Y)X^{j} = ∑_{1≤i<m, (i,p)=1} a_{ip^{n(i)}}(Y)X^{ip^{n(i)}}.

i) The coefficient a_{m−1}(Y) ∈ mt_{m}Y + (Y A[Y])^{p} and deg a_{m−1}(Y) ≤ (m − 1)^{2}. ii) If ℓ > 1, let j_{0} = 1 + (ℓ − 1)p^{s}. Then j_{0} is prime to p (also if s = 0) and

n(j_{0}) = 0. Moreover the coefficient a_{j_{0}}(Y) = ℓt_{m}Y^{p^{s}} + ... + 2t_{j_{0}+1}Y if p > 2 and if p = 2 one has a_{j_{0}}(Y) = ℓt_{m}Y^{2^{s}} + ... + t_{j_{0}+2}Y^{2}. In particular if (p, m − 1) = 1 (so s = 0 and p > 2) one has j_{0} = m − 1 and a_{j_{0}}(Y) = mt_{m}Y.

Proof. For i) we remark that m−1 = ℓp^{s} = ℓp^{n(ℓ)} is the highest representative < m of ℓ modulo multiplication by a power of p. Now Δ(X^{m}) = mY X^{m−1}+lower degree terms and, as the lower degree monomials have coefficients in Y A[Y] which give contributions in the coefficient of the monomial X^{m−1} of R(X, Y) after we raise to some p-power, the result follows. Concerning the degree, we remark that Δ(f)(X, Y) = ∑_{t≤m−1} δ_{t}(Y)X^{t} and deg δ_{t}(Y) ≤ m − 1. Now like in the proof of lemma 4.1 we can write t = ip^{j} where (i, p) = 1, 1 ≤ i < m and 0 ≤ j ≤ n(i). Then the contribution of δ_{ip^{j}}(Y)X^{ip^{j}} in R(X, Y) is (δ_{ip^{j}}(Y))^{p^{n(i)−j}}X^{ip^{n(i)}} whose Y -degree is ≤ (m − 1)^{2}.

For ii) we remark that j_{0} > \frac{m−1}{p} + 1 and j_{0} is prime to p. So the coefficient of the monomial X^{j_{0}} in R(X, Y) is the same as that of Δ(f) and so equal to ∑_{j_{0}<i≤m,(i,p)=1} \binom{i}{j_{0}}t_{i}Y^{i−j_{0}} = ℓt_{m}Y^{p^{s}} + ⋯ + (j_{0} + 1)t_{j_{0}+1}Y. The result follows.

Remark 5.4. In [He] on p.114 one can find a discussion of the equation W^{p} − W = f(X) and our system of representatives modulo p^{ℕ} as well as j_{0} as it occurs above.

Proposition 5.5. Let φ : A → k be a specialization homomorphism with φ(t_{m}) ≠ 0. Then Ad_{φ(f)}(Y) is a separable and additive polynomial (section 2) and its set of zeroes Z(Ad_{φ(f)}) is equal to {y ∈ k | Δ(φ ∘ f)(X, y) ∈ (Id − F)k[X]}. Moreover if y ∈ Z(Ad_{φ(f)}) then there is a unique P (X) ∈ Xk[X] such that Δ(φ(f))(X, y) = (Id−F)P(X) and P (X) = φ(P_{f})(X, y). Further Z(Ad_{φ(f)}) = {y ∈ k | σ_{y} ∈ G_{∞,1}(f)} where σ_{y}(X) = X + y and σ_{y}(W) = W + φ(P_{f})(X, y) (for notations see 3.5). In particular |G_{∞,1}(φ(f))| = p deg Ad_{φ(f)}(Y) and for y, z ∈ Z(Ad_{φ(f)}) the commutation rule for σ_{y}, σ_{z} ∈ G_{∞,1}(φ(f)) is determined by ϵ_{φ(f)}(y, z) = φ(P_{f})(X, y) + φ(P_{f})(X + y, z) − φ(P_{f})(X, z) − φ(P_{f})(X + z, y).

Proof. Ad_{φ(f)}(Y) is separable because from Cor. 5.3 i) we know that it divides the polynomial φ(a_{m−1}(Y)) ∈ mφ(t_{m})Y + (Y k[Y])^{p} which is separable. To prove that it is additive, it suffices to show that its set of roots is stable under addition (section 2). We first remark that Z(Ad_{φ(f)}) ⊂ {y ∈ k | σ_{y} ∈ G_{∞,1}(f)} (cf. Cor. 3.4). For the reverse inclusion we remark that in the equality

Δφ(f)(X, y) = (Id − F)P(X),  (4)

for suitable P (X) ∈ k[X], we can assume that P (X) ∈ Xk[X] as Δ(φ(f))(0, y) = 0. Note that deg P ≤ [\frac{m−1}{p}]. Then for n >> 0 we have (Id + .... + F^{n})Δ(φ(f))(X, y) =

P(X) mod X^{[ \frac{m−1}{p} ]+1}, so P (X) = φ(P_{f})(X, y) (cf. Cor. 3.4). Then from Lem. 5.1 and (4) it follows that φ(R)(X, y) = 0, i.e. Ad_{φ(f)}(y) = 0.

Let y, z ∈ k such that Δ(φ(f))(X, y), Δ(φ(f))(X, z) ∈ (Id − F)k[X]. We have the following general identity: Δ(φ(f))(X, y + z) = Δ(φ(f))(X + z, y) + Δ(φ(f))(X, z) − Δ(φ(f))(y, z) and for our choice of y, z each term on the right hand side is in (Id − F)k[X], so Ad_{φ(f)}(y + z) = 0.

Remark 5.6. In Def. 5.2 we saw that if φ : A → k is a specialization homomorphism with φ(t_{m}) ≠ 0 and such that g = φ(f) then Ad_{φ(f)} doesn’t depend on φ. It follows from Prop.5.5 that ϵ_{φ(f)}(y, z) doesn’t depends on φ. This also is a consequence of the identity for h ∈ k[X]: Δ(h)(X, Y)+Δ(h)(X +Y, Z)−Δ(h)(X, Z)−Δ(h)(X +Z, Y) = 0.

Corollary 5.7. We denote by Id : A → FrA the inclusion homomorphism. A. i) If p = 2 and m = 3 then Ad_{Id(f)}(Y) = t_{3}^{−2}(t_{3}^{2}Y^{4} + t_{3}Y). ii) If p = 2 and m = 5 then Ad_{Id(f)}(Y) = t_{5}^{−4}(t_{5}^{4}Y^{16} + t_{3}^{4}Y^{8} + t_{3}^{2}Y^{2} + t_{5}Y). iii) If p = 3 and m = 4 then Ad_{Id(f)}(Y) = t_{4}^{−3}(t_{4}^{3}Y^{9} + 2t_{2}^{3}Y^{3} + t_{4}Y). B. Outside case A. One has Ad_{Id(f)}(Y) = Y. Moreover ∃D(\underline{t}) ∈ 𝔽_{p}[\underline{t}] − \{0\} such

that for any specialization φ : A → k with φ(D(\underline{t})) ≠ 0 one has Ad_{φ(f)}(Y) = Y and so G_{∞,1}(φ(f)) ≃ ℤ/pℤ.

Proof. A. A direct calculation gives the formulas.

B. In order to simplify the proof we can assume that t_{m} = 1. If p doesn’t divide m − 1 the result follows from Cor. 5.3 ii). Therefore we can write m = 1 + ℓp^{s} with (ℓ, p) = 1, s > 0. We distinguish two cases.

1. ℓ > 1. Let j_{0} = 1 + (ℓ − 1)p^{s} and write R(X, Y) = ∑_{j} a_{j}(Y)X^{j}. Then a_{j_{0}}(Y) = ℓY^{p^{s}}+ lower degree terms whose leading coefficient is a unit in A. The ring A is factorial hence it follows that Ad_{Id(f)}(Y) is also a gcd in A[Y] of the coefficients of R(X, Y) and so lies in A[Y]. Let φ : A → k be a specialization morphism. Then φ(Ad_{Id(f)}) is monic and deg φ(Ad_{Id(f)}) = deg Ad_{Id(f)}(Y). Now we remark that φ(AdId) divides φ(a_{m−1}(Y)) ∈ mY + (Y k[Y])^{p} and so is still separable in k[X] (cf. Cor. 5.3 i). Let us consider the specialization homomorphism φ_{0} : A → 𝔽_{p} defined by φ_{0}(t_{i}) = 0 for i = 1, ..., m − 1. Then φ_{0}(f)(X) = X^{m} and Ad_{φ_{0}(f)}(Y) divides ℓY^{p^{s}} (see the proof of Cor. 5.3 ii). It follows that Ad_{Id(f)}(Y) = Y.

2. ℓ = 1. Then m = 1 + p^{s} with s > 0. A look at the monomials in Δ(f)(X, Y) shows that the highest contribution in a_{m−1}(Y) comes from (X + Y)^{1+p^{s}} and more precisely from the linear term Y^{p^{s}}X which we need to raise to the power p^{s}. So finally a_{m−1}(Y) = Y^{p^{2s}}+ lower degree monomials and so Ad_{Id(f)}(Y) ∈ A[Y]. Now, like in case 1, we look for good specialization morphisms.

For p > 3 we consider φ_{0}(f)(X) = X^{3} + X^{m}; then Ad_{φ_{0}(f)}(Y)|φ_{0}(a_{2p^{s−1}}(Y)) = 3^{p^{s−1}}Y^{p^{s−1}} and we conclude as in case 1.

For p = 2 and s > 2 we consider φ_{0}(f)(X) = X^{7}+X^{m}; then Ad_{φ_{0}(f)}(Y)|φ_{0}(a_{3p^{s−2}}(Y)) = Y^{p^{s−2}} and we conclude as in case 1.

For p = 3 and s > 1 we consider φ_{0}(f)(X) = X^{5}+X^{m}; then Ad_{φ_{0}(f)}(Y)|φ_{0}(a_{4p^{s−2}}(Y)) = (2Y)^{p^{s−2}} and we conclude as in case 1.

In order to exhibit a polynomial D(\underline{t}) we recall that Y is the content of the polynomial R(X, Y) = ∑_{j} a_{j}(Y)X^{j} in A[Y][X]. Therefore there exist elements b_{j}(Y) ∈ (FrA)[Y] such that ∑_{j} b_{j}(Y)a_{j}(Y) = Y. Any D(\underline{t}) ∈ 𝔽_{p}[\underline{t}] − \{0\} such that D(\underline{t})b_{j}(Y) ∈ A[Y] for all j works.

Remark 5.8. As A is a UFD it is a natural question to ask for the best D(\underline{t}).

5.2 The algorithm

Let k be an algebraically closed field of characteristic p > 0 and f ∈ k[X] such that red(f) ≠ 0. We are interested in the set S(f) := {y ∈ k | Δ(f)(X, y) ∈ (F−Id)k[X]}. Note that the two endomorphisms τ_{y} : X → X +y and F of k[X] commute. From this it follows that S(f) only depends of the class of f mod (F−Id)k[X] and in particular S(f) = S(red(f)). If deg f is prime to p then f is a specialization (in many ways) of the universal family considered in section 5.1. Consequently Ad_{f} is well defined and S(f) = Z(Ad_{f}(Y)).

The following procedure has been written in Maple; we comment on some steps and then present a concrete example.

Let d := deg f and assume for simplicity that (d, p) = 1. Then d is the conductor m of the cover C_{f}. If (d, p) ≠ 1 the algorithm still works but needs to be adapted.

Step 1: Replace f by f − f(0).

Step 2: Define δ_{0} := f(X + Y) − f(X) − f(Y).

Step 3: Define inductively δ_{n+1} := δ_{0} + δ_{n}^{p} mod (p, X^{[ \frac{d−1}{p}]+1}).

Step 4: Iterate until it is stationary and define P (X, Y) to be the limit.

Step 5: Define R := δ_{0}−P (X, Y)+P (X, Y)^{p} ∈ k[Y][X]. This polynomial depends only on the class of f modulo (F−Id)k[X]; moreover the occurring X-monomials have degrees between [\frac{d−1}{p}] and d.

Step 6: Define Ad_{f}(Y) := gcd of the polynomials in k[Y] which are the coefficients of R ∈ k[Y][X].

Step 7: Calculate Tr_{f}(Y) := P (X, Y) + P (X + Y, Y) + ... + P (X + (p − 1)Y, Y) mod Ad_{f}(Y). Then for y ∈ Z(Ad_{f}) we have Tr_{f}(y) ∈ 𝔽_{p}. If Tr_{f}(y) = 0 then y induces automorphisms of order dividing p. Otherwise it induces automorphisms of order p^{2} (see Cor. 3.5).

Step 8: Calculate ϵ(Y, Z) := P (Y, Z) − P (Z, Y) mod (Ad_{f}(Y), Ad_{f}(Z)). For a given y ∈ Z(Ad_{f}(Y)) the roots of ϵ(y, Z) correspond to those automorphisms that commute with the ones corresponding to y. This is the commutation rule (see Cor. 3.5).

5.3 Illustration of the Algorithm

Here we illustrate the algorithm which, for a given f, gives the structure of the group G_{∞,1}(f). This example is a realization of D_{8} with f ∈ 𝔽_{2}[X]. We use Maple and include some comments:

> restart;
> f:=X^(1+2)+X^(1+2+2^2)+X^(1+2+2^4)+X^(1+2+2^5)+X^(1+2^3+2^5):

> delta0:=collect(subs(X=X+Y,f),X) -f-subs(X=Y,f),[X,Y]) mod 2:
> delta1:=rem(collect(delta0+delta1^2-subs(X=0,delta0+delta1^2),[X,Y])
mod 2,X^21,X) mod 2:

Note that 21 = 40/2 + 1. Here one reiterates the command until it is stationary.

> P:=delta1:
> F:=collect(delta0-P-P^2,[X,Y]) mod 2;
F :=
(Y^24+Y^80+Y^132+Y^528+Y^192+Y^64+Y^576+Y^1280+Y^1088+Y^6+Y^3+Y^16+Y^9
+Y^272)*X^32+(Y^256+Y^128+Y^4+Y^32)*X^24+(Y^128+Y^2)*X^36+(Y^4+Y)*X^34
+(Y+Y^16)*X^40+(Y^8+Y^2)*X^33

Here we remark that Ad_{f}(Y) divides the coefficient of X^{34}.

> F:=collect(rem(F,Y^4+Y,Y)mod 2,X);
F := 0

Conclusion: Ad_{f}(Y) = Y^{4} + Y.

> Tr_f(Y):=rem(collect(P+subs(X=X+Y,P),X),Y^4+Y,Y) mod 2;
Y^3+Y^2+Y
> Gcd(Y^4+Y,Y^3+Y^2+Y) mod 2;
Y^3+Y^2+Y

It follows that the 3 roots of Y^{3} + Y^{2} + Y induce 6 elements of order 2 and the last root Y = 1 induces 2 elements of order 4.

> epsilon:=subs([X=Y,Y=Z],P)-subs([X=Z],P) mod 2:
> epsilon:=collect(rem(epsilon,Y^4+Y,Y) mod 2,Z);
> epsilon:=collect(rem(epsilon,Z^4+Z,Z) mod 2,Y);
epsilon:= Z^2*Y+Z*Y^2

The group is non abelian of order 8 with 2 elements of order 4; this is D_{8}.

Note that f := X^{3} + X^{7} + X^{19} + X^{35} + X^{41} is reduced with conductor > 25. More generally it is a good question to ask for realizations over 𝔽_{2} (we mean f ∈ 𝔽_{2}[X]) for groups in the class C_{1}.

6 Bounds for |G_{∞,1}(f)|

6.1 Modifications of covers

In 4.1. we fixed the conductor which is equivalent to fixing the genus of the cover. Here we study how for f(X) ∈ k[X], the additive polynomial Ad_{redf}(Y) changes through natural algebraic transformations which do not necessarily preserve the genus. We will use these transformations in order to produce elements f ∈ k[X] for which the bounds on G_{∞,1}(f), given in section 6.2, are attained. In section 7 we use the same transformations to obtain groups G_{∞,1}(f) with a given structure. Definition 6.1. Let S(X) be an additive polynomial ∈ k[X]. We say that the cover C_{f∘S} is a modification of type 1 of C_{f}.

Note that if (deg f, p) = 1 in general (deg f ∘ S, p) ≠ 1. This explains why in the formulas below we work with red(f ∘ S). Geometrically the p-cyclic cover C_{f∘S} → ℙ^{1} is obtained from the p-cyclic cover C_{f} → ℙ^{1} after base change with the (resp. etale) cover 𝔸^{1} → 𝔸^{1} defined by the (resp. separable) additive polynomial S. Proposition 6.2. Let f ∈ k[X] be any polynomial with (deg f, p) = 1 and S(X) ∈ k[X] be a separable and additive polynomial. Then Ad_{f}(S(Y)) divides Ad_{red(f∘S)}(Y). Further for y, z ∈ Z(Ad_{f}(S(Y))) we have y, z ∈ Z(Ad_{red(f∘S)}(Y)) and ϵ_{f}(S(y), S(z)) = ϵ_{red(f∘S)}(y, z) (see Prop. 5.5).

Proof. One can write Δ(f)(X, Y) = Ad_{f}(Y)G(X, Y) + P_{f}(X, Y) − P_{f}(X, Y)^{p}, where G(X, Y) ∈ k[Y][X] has content equal to 1 and P_{f}(X, Y) ∈ k[Y][X]. Then Δ(f ∘ S)(X, Y) = Δ(f)(S(X), S(Y)) =

Ad_{f}(S(Y))G(S(X), S(Y)) + P_{f}(S(X), S(Y)) − P_{f}(S(X), S(Y))^{p}.

If y ∈ Z(Ad_{f}(S(Y))) then Δ(f ∘ S)(X, y) = P (S(X), S(y)) − P (S(X), S(y))^{p} and as red(f ∘ S)(X) = f ∘ S(X) + Q(X) − Q(X)^{p} for some Q(X) ∈ k[X] we can write Δ(red(f ∘ S))(X, y) = (Id−F)(P(S(X), S(y))+Δ(Q)(X, y)) which, by Prop. 5.5, gives Ad_{red(f∘S)}(y) = 0. The divisibility follows as S(Y) and so Ad_{f}(S(Y)) is separable. Moreover by the same proposition for y ∈ Z(Ad_{f}(S(Y))) we have P_{red(f∘S)}(X, y) = P (S(X), S(y)) + Δ(Q)(X, y). We remark that Δ(Q)(X, y) + Δ(Q)(X + y, z) − Δ(Q)(X, z) − Δ(Q)(X + z, y) = 0 implies that ϵ_{f}(S(y), S(z)) = ϵ_{red(f∘S)}(y, z) for y, z ∈ Z(Ad_{f}(S(Y))). Remark 6.3. In Proposition 6.2 it can happen that Ad_{f}(S(Y)) strictly divides Ad_{red(f∘S)}(Y). For example if f(X) = X^{1+p^{n}} then Ad_{f}(Y) = Y + Y^{p^{2n}} (see Prop. 8.1). If S(X) = X + X^{p} then f ∘ S(X) = (X + X^{p})(X^{p^{n}} + X^{p^{n+1}}) = X^{1+p^{n}} + X^{1+p^{n+1}} + X^{p+p^{n}} + X^{p+p^{n+1}} and so red(f ∘ S) = X^{1+p^{n−1}} + 2X^{1+p^{n}} + X^{1+p^{n+1}}. Further Ad_{red(f∘S)}(Y) = Y + ... + Y^{p^{2(n+1)}} (see Prop. 8.1) and Ad_{f}(S(Y)) = Y + Y^{p} + Y^{p^{2n}} + Y^{p^{2n+1}} strictly divides Ad_{red(f∘S)}(Y) .

Definition 6.4. Let g(X), h(X) ∈ k[X] and assume that none of red(g), red(h), red(g+ h) is zero. If Ad_{red(h)}(Y)|Ad_{red(g)}(Y) we say that the cover C_{g+h} is a modification of type 2 of C_{g}. Proposition 6.5. Let g, h be as above and assume that C_{g+h} is a modification of type 2 of C_{g}. Then Ad_{red(h)}(Y)|Ad_{red(g+h)}(Y). Further for y, z ∈ Z(Ad_{red(h)}(Y)) we have y, z ∈ Z(Ad_{red(g+h)}(Y)) and ϵ_{red(g+h)}(y, z) = ϵ_{red(g)}(y, z) + ϵ_{red(h)}(y, z) (see Prop. 5.5). Proof. Clear from Prop. 5.5.

6.2 Bounds

For fixed conductor we give bounds for |G_{∞,1}(f)|.

Proposition 6.6. Let k be an algebraically closed field of positive characteristic p and f(X) ∈ k[X] a polynomial of degree m prime to p. Let g(f) = \frac{(m−1)(p−1)}{2} be the genus of C_{f} and assume g(f) ≥ 2. We write m − 1 = ℓp^{s} with ℓ ≥ 1, (ℓ, p) = 1 and s ≥ 0. Then I. |G_{∞,1}(f)| ≤ p(m − 1)^{2}, i.e. \frac{|G_{∞,1}(f)|}{g(f)^{2}} ≤ \frac{4p}{(p−1)^{2}} and we have equality for f(X) = X^{1+p^{s}}. II. If s = 0, i.e. (m − 1, p) = 1 then |G_{∞,1}(f)| = p. III. If s > 0

i) Let ℓ > 1 and p = 2. Then |G_{∞,1}(f)| ≤ 2^{s} and \frac{|G_{∞,1}(f)|}{g(f)} ≤ \frac{2}{3}. The two inequalities are equalities for f(X) = red((X + X^{2^{s−1}})^{7}) and s > 1.

ii) Let ℓ > 1 and p > 2. Then |G_{∞,1}(f)| ≤ p^{s+1} and \frac{|G_{∞,1}(f)|}{g(f)} ≤ \frac{2}{ℓ} \frac{p}{p−1} ≤ \frac{p}{p−1}. The 3 inequalities are equalities for f(X) = X^{1+2p^{s}} − X^{2+p^{s}}.

iii) Let ℓ = 1, i.e. m = 1 + p^{s}. Then |G_{∞,1}(f)| ≤ p^{2s+1}, i.e. \frac{|G_{∞,1}(f)|}{g(f)} ≤ 2p^{s} \frac{p}{p−1} and the 2 inequalities are equalities for f(X) = X^{1+p^{s}}.

Proof. I. follows from Cor. 5.3 i). For the example f(X) = X^{1+p^{s}} we refer to III. iii). The last part of Cor. 5.3 gives II. For III. i) and ii) we can assume that f(X) = ∑_{1≤i<m} t_{i}X^{i} + X^{m} ∈ k[X]. Let j_{0} = 1 + (ℓ − 1)p^{s}. We have seen in Cor. 5.3 that a_{j_{0}}(Y) = ℓY^{p^{s}} + ... + 2t_{j_{0}+1}Y, so deg Ad_{f}(Y) ≤ deg a_{j_{0}}(Y) = p^{s} if p > 2 and deg Ad_{f}(Y) ≤ (1/2) deg a_{j_{0}}(Y) = 2^{s−1} if p = 2. The result follows from Prop. 5.5.

Now we discuss the examples in which the bounds are attained. Let p = 2, (ℓ, 2) = 1 and f(X) = X^{1+ℓ2}. In this case it is an easy consequence of Cor. 5.3 that Ad_{f}(Y) = Y. Let S(X) = X + X^{2^{s−1}} where s > 1. Then f ∘ S(X) = (X + X^{2^{s−1}})(X^{2} + X^{2^{s}})^{ℓ} = X(X^{2} + X^{2^{s}})^{ℓ} + X^{2^{s−2}}(X + X^{2^{s−1}})^{ℓ} mod (Id − F)k[X] has conductor 1 + ℓ2^{s} = m. From Prop. 6.2 we know that Ad_{f}(S(Y)) = Y + Y^{2^{s−1}}|Ad_{f∘S}(Y) and from the first part of the proposition we get deg Ad_{f∘S}(Y) ≤ 2^{s−1}. Consequently we have equality Ad_{f∘S}(Y) = Y + Y^{2^{s−1}} = Ad_{f}(S(Y)). Therefore we conclude \frac{|G_{∞,1}(f∘S)|}{g(f)} = \frac{2}{ℓ} ≤ \frac{2}{3}. This yields equality for l = 3, i.e. f(S(X)) = (X + X^{2^{s−1}})^{7}.

Let p > 2 we give an example for ℓ = 2. As a_{j_{0}}(Y) = ℓY^{p^{s}} + ... + 2t_{j_{0}+1}Y we need t_{j_{0}+1} ≠ 0. Now j_{0} + 1 = 2 + (ℓ − 1)p^{s} = 2 + p^{s}, so we consider f(X) = X^{1+2p^{s}} − X^{2+p^{s}} and show that Ad_{f}(Y) = Y^{p^{s}} − Y. The first part of the proposition yields deg Ad_{f}(Y) ≤ p^{s} so it suffices to prove a divisibility. Let y ∈ k be such that y^{p^{s}} = y. Then Δ(f)(X, y) = (X + y)(X^{p^{s}} + y)^{2} − (X + y)^{2}(X^{p^{s}} + y) − f(X) − f(y) = (X + y)(X^{2p^{s}} + 2yX^{p^{s}} + y^{2}) − (X^{2} + 2yX + y^{2})(X^{p^{s}} + y) − f(X) − f(y) = 0 mod (Id − F)k[X].

iii) The inequality is a special case of I. Let f(X) = X^{1+p^{s}} and write q = p^{s}. Then Δ(f)(X, Y) = Y^{q}X + Y X^{q} = (Y^{q^{2}} + Y)X^{q} + P (X, Y) − P (X, Y)^{p}. It follows that Ad_{f}(Y) = Y^{q^{2}} + Y and we obtain equalities.

Remark 6.7. In general for a given curve C in order to bound the automorphism group one considers a prime ℓ > 2 not equal to p and uses the injective homomorphism

AutC ↪ Gl(2g(C), 𝔽_{ℓ})

where g(C) is the genus. Let us for example consider the case p = 2 and C = C_{f} with deg f = 1 + 2^{k}. Then 2g(C_{f}) = 2^{k} and |G_{∞,1}(f)| divides the order of a 2-Sylow of Gl(2^{k}, 𝔽_{ℓ}) which is equal to 2^{g(C_{f})F (1)−1}, where F (1) = 1 + v_{2}(ℓ^{2} − 1) + v_{2}(ℓ − 1) (see [C-F] for the structure of 2-Sylow subgroups of Gl(2^{k}, 𝔽_{ℓ})). From this one also obtains bounds for AutC_{f} but they are far from being as good as the ones in Prop. 6.6.

7 Group theoretic characterization of G_{∞,1}(f)

7.1 Realization of groups as G_{∞,1}(f)

In Corollary 3.5 we saw that the groups G_{∞,1}(f) belong to the class C_{1} of p-groups (see Def. 4.1). Moreover in section 4.2. we saw that C_{1} = C_{2}, the class of saturated subgroups of the extraspecial groups. We define a third class

C_{3} := {G | ∃f ∈ Xk[X], (deg f, p) = 1 such that G ≃ G_{∞,1}(f)}.

Theorem 7.1. The 3 classes C_{i}, i = 1, 2, 3 are equal.

Proof. It is sufficient for G in C_{1} to give a recipe to get an f ∈ k[X] such that G ≃ G_{∞,1}(f).The method is the following: as G belongs to C_{2}, there is E an extraspecial group such that G ≃ π^{−1}(V ⊂ E/Z(E)), a saturated subgroup. Moreover if p > 2 (resp. p = 2), we can choose E of type I.b (resp. type II b); see Def. 4.3. Realizations of the extraspecial groups of type I.a and II.b occur naturally. In order to produce extraspecial groups of type I.b we first consider a realization of the extraspecial group with the same order but of type I.a. Then we do a modification using the cocycle which occurs for the Witt vectors of length 2. For E an extraspecial group of type I.b (resp. II.b) consider G_{∞,1}(f_{E}) a realization. Now to a subspace V ⊂ E/Z(E) we associate an additive polynomial S_{V} which we use in order to produce a convenient modification of the cover C_{f_{E}}. The key point is that our modification will not change the commutation rule.

• p > 2. Let E be an extraspecial group of order pq^{2} where q = p^{n} and n > 0.

Step1. Realization of E with exponent p. (Type I.a)

Claim: If f_{1}(X) := X^{1+q} then E ≃ G_{∞,1}(f_{1}) and \frac{2p}{p−1} < \frac{|G_{∞,1}(f_{1})|}{g(f_{1})} = \frac{2p}{(p−1)}q

Proof. One has Δ(f_{1})(X, Y) = Y^{q}X + Y X^{q} = (Y^{q^{2}} + Y)X^{q} + P (X, Y) − P (X, Y)^{p}, where P (X, Y) = ∑_{0≤i≤n−1}(Y^{q}X)^{p^{i}}. It follows that Ad_{f_{1}}(Y) = Y^{q^{2}} +Y and for y ∈ Z(Ad_{f_{1}}), P_{f_{1}}(X, y) = P (X, y) (Prop. 5.5). We remark that the polynomial P (X, Y) is an additive polynomial. Therefore if y ∈ Z(Ad_{f_{1}}) and if σ_{y} ∈ G_{∞,1}(f_{1}) is such that σ_{y}(X) = X + y, then σ_{y}(W) = W + P (X, y) + c, for c ∈ 𝔽_{p} (Prop. 5.5), then σ_{y}^{p}(W) = W + P (X, y) + P (X + y, y) + ... + P (X + (p − 1)y, y) = W + \frac{p(p−1)}{2}P (y, y). So if p > 2 then σ_{y}^{p} = Id and consequently the exponent of G_{∞,1}(f_{1}) is p. If y, z ∈ Z(Ad_{f_{1}}) then (Prop. 5.5) ϵ_{f_{1}}(y, z) = P (X, y) + P (X + y, z) − P (X, z) − P (X + z, y) = P (y, z) − P (z, y) = ∑_{0≤i≤n−1}(yz^{q})^{p^{i}} − ∑_{0≤i≤n−1}(y^{q}z)^{p^{i}}. Assume y ≠ 0 is such that σ_{y} is in the center of G_{∞,1}(f_{1}) (this condition doesn’t depend on c ∈ 𝔽_{p} with σ_{y}(W) = W + P (X, y) + c). Then ϵ_{f_{1}}(y, z) as a polynomial in z should have p^{2n} roots which is a contradiction. So the center is < ρ > where ρ(X) = X and ρ(W) = W + 1.

Step 2. Realization of E with exponent p^{2}. (Type I.b)

We will use the Witt vectors of length 2 and modifications of the cover built for type I.a. Let c(X, Y) = \frac{(X+Y)^{p}−X^{p}−Y^{p}}{p} = ∑_{1≤i≤p−1} \frac{(−1)^{i−1}}{i} X^{i}Y^{p−i} and f_{0}(X) := c(X^{p}, −X) = ∑_{1≤i≤p−1} \frac{1}{i}X^{p+(p−1)i}. A straightforward calculation in W_{2}(𝔽_{p}) shows that:

Δ(f_{0})(X, Y) = c((F − Id)X, (F − Id)Y) + (F − Id)c(X, Y). As c((F − Id)X, (F − Id)Y) = (Y^{p} − Y)(X^{p} − X)^{p−1} + lower X-degree terms ∈ (Y^{p} − Y)𝔽_{p}[X, Y] it follows that Ad_{f_{0}}(Y) = Y^{p}−Y and for y ∈ Z(Ad_{f_{0}}(Y)) (i.e. y ∈ 𝔽_{p}) P_{f_{0}}(X, y) = −c(X, y) = ∑_{1≤i≤p−1} \frac{(−1)^{i}}{i} y^{p−i}X^{i} (cf. Prop.5.5). One can prove that G_{∞,1}(f_{0}) is p^{2}-cyclic; in order to get an extraspecial group of exponent p^{2} we use modifications of f_{0}. Let q := p^{n}, f_{1}(X) = X^{1+q}, θ a q^{2}−1-th root of −1 (so that θ^{q^{2}} +θ =

0) and A(X) := θX^{q} − θ^{q}X. Then S(X) := A(X) + A(X)^{p} + ... + A(X)^{q/p} is an additive polynomial and S(X)^{p} − S(X) = A(X)^{q} − A(X) = θ^{q}(X + X^{q^{2}}).

Claim. If f_{2}(X) := red(f_{0}(S(X))) + f_{1}(X) then E ≃ G_{∞,1}(f_{2}) and \frac{|G_{∞,1}(f_{2})|}{g(f_{2})} = \frac{2p}{(p−1)^{2}} < \frac{p}{p−1}.

Proof. We have

Δ(f_{0})(S(X), S(Y)) = c(S(X)^{p}−S(X), S(Y)^{p}−S(Y))+(Id−F)(−c(S(X), S(Y)))

= c(A(X)^{q} − A(X), A(Y)^{q} − A(Y)) + (Id − F)(−c(S(X), S(Y))) (5)

and Δ(f_{1})(X, Y) = Y^{q}X+Y X^{q}. We now show that Ad_{red(f_{0}∘S)}(Y) = θ^{−q}Ad_{f_{0}}(S(Y)) = Y +Y^{q^{2}} for y+y^{q^{2}} = 0. To this end notice that f_{0}(S(X)) = c(S(X)^{p}, −S(X)) = ∑_{1≤i≤p−1} \frac{1}{i}S(X)^{p+(p−1)i} has conductor 1 + (p − 1)q^{2} hence, by Prop. 6.6 III. ii), deg Ad_{red(f_{0}∘S)} ≤ q^{2}. As, by Prop. 6.2, Ad_{f_{0}}(S(Y)) = S(Y)^{p} − S(Y) = A(Y)^{q} − A(Y) = θ^{q}(Y + Y^{q^{2}})|Ad_{red(f_{0}∘S)}(Y), we get the equality.

Note that red(f_{0} ∘ S) and f_{1} have the same additive polynomial so, due to the property of second type modifications (Prop. 6.5), we obtain Y + Y^{q^{2}}|Ad_{f_{2}}(Y). As f_{2} and red(f_{0} ∘ S) have the same conductor, the same argument as above gives deg Ad_{f_{2}}(Y) ≤ q^{2}. Finally we conclude Ad_{f_{2}}(Y) = Y + Y^{q^{2}} showing the claim.

Next we show that G_{∞,1}(f_{2}) has exponent p^{2}. Let y be such that Ad_{f_{2}}(y) = Ad_{red(f_{0}∘S)}(y) = Ad_{f_{1}}(y) = 0. We need to calculate ∑_{0≤i≤p−1} P_{f_{2}}(X + iy, y) (see Cor. 3.5). One has P_{f_{2}}(X, y) = P_{0}(X, y) + P_{1}(X, y), where P_{1}(X, Y) := P_{f_{1}}(X, Y) = y^{q}X+(y^{q}X)^{p}+...+(y^{q}X)^{q/p} (step 1) and P_{0}(X, Y) := P_{red(f_{0}∘S)}(X, Y) can be deduce from (5) in the following way: for y ∈ Z(Ad_{f_{2}}) we have S(y)^{p} − S(y) = θ^{q}(y+y^{q^{2}}) = 0, so (5) gives Δ(f_{0})(S(X), S(y)) = (Id−F)(−c(S(X), S(y))). As we can write red(f_{0} ∘ S)(X) = f_{0}(X) + (Id − F)h(X) it follows that

P_{0}(X, y) = −c(S(X), S(y)) + Δ(h)(X, y)  (6)

We easily see that ∑_{0≤i≤p−1} Δ(h)(X + iy, y) = 0, so we need to calculate ∑_{0≤i≤p−1} c(S(X+iy), S(y)). Note that c(S(X), S(y)) = ∑_{1≤i≤p−1} \frac{(−1)^{i−1}}{i} S(y)^{i}S(X)^{p−i} and that S is an additive polynomial. If we set Z := S(y)S(X)^{p} − S(y)^{p}S(X) then ∑_{0≤i≤p−1} P_{0}(X + iy, y) = Tr_{k(S(X))/k(Z)}(−c(S(X), S(y))) = S(y)^{p}, where k = 𝔽_{p}^{alg}.

For the contribution of P_{1} we remark that f_{1} induces an extraspecial group of exponent p so ∑_{1≤i≤p−1} P_{1}(X + iy, y) = 0.

Finally we have shown that ∑_{0≤i≤p−1} P_{f_{2}}(X + iy, y) = S(y)^{p} and so for y ∈ Z(Ad_{f_{2}}(Y)) − Z(S(Y)) the element σ_{y} ∈ G_{∞,1}(f_{2}) has order p^{2}.

Now we show that the center is < ρ >. If y, z ∈ Z(Ad_{f_{2}}(Y)) then ϵ_{j}(y, z) = P_{j}(X, z) + P_{j}(X + z, y) − P_{j}(X, y) − P_{j}(X + y, z) for j = 1, 2.

We also saw that Ad_{f_{0}}(Y) = Y^{p} − Y and G_{∞,1}(f_{0}) is abelian (in fact cyclic), so ϵ_{red(f_{0}∘S)}(y, z) = ϵ_{f_{0}}(S(y), S(z)) = 0 (see Cor. 3.5, Prop. 6.5). Finally ϵ_{f_{2}}(y, z) = ϵ_{1}(y, z) = (z^{q}y +z^{qp}y^{p} +...+z^{q^{2}/p}y^{q/p})−(zy^{q} +z^{p}y^{pq} +...+z^{q/p}y^{q^{2}/p}). For z ≠ 0 this is a polynomial in y of degree q^{2}/p, so it has at most q^{2}/p roots and hence z ∈ Z(G_{∞,1}(f_{2})) iff z = 0.

Step 3. Realization of saturated subgroups of an extraspecial group E of exponent p^{2}. (type I.b)

Note that E/Z(E) is the group of automorphisms of k[X] whose elements are σ_{y}(X) = X +y where y ∈ Ad_{f_{2}}(Y). Then a subgroup V ⊂ E/Z(E) corresponds to a monic additive polynomial S_{V} which divides Ad_{f_{2}}(Y) (section 2). Note that necessarily S_{V} has distinct roots so S_{V} = s_{0}X + s_{1}X^{p} + .... + X^{p^{r}} and s_{0} ≠ 0. As π^{−1}(E/Z(E)) = E we can assume that 0 < r < 2n. We consider ℓ > 1 such that (ℓ(ℓ + 1), p) = 1 and f(X) := red(S_{V} (X)^{ℓ+1}) + f_{2}(X) = red(S_{V} (X)^{ℓ+1}) + red(f_{0}(S(X))) + f_{1}(X).

Our aim is first to find ℓ such that Ad_{f}(Y) = S_{V} (Y).

We remark that the conductors for these three contributions are respectively 1 + ℓp^{r}, 1 + (p − 1)q^{2} and 1 + q. We adopt the notations of Lemma 5.1 and, like in the proof of Cor. 5.3, we consider j_{0} = 1 + (ℓ − 1)p^{r} and we calculate the coefficient of X^{j_{0}p^{n(j_{0})}} in R(X, Y).

– We show that the contribution from Δ(S_{V}^{ℓ+1})(X, Y) is [\frac{(ℓ+1)!}{1!...1!(ℓ−1)!}S_{V} (Y)s_{0}]^{p^{n(j_{0})}}. One has pj_{0} > (ℓ+1)p^{r}, so the only contribution comes from the monomial X^{j_{0}} and we claim that its coefficient is \frac{(ℓ+1)!}{1!...1!(ℓ−1)!}S_{V} (Y)s_{0}. Namely one has Δ(S_{V}^{ℓ+1})(X, Y) = (S_{V} (Y) + s_{0}X + ... + X^{p^{r}})^{ℓ+1} − S_{V} (Y)^{ℓ+1} − S_{V} (X)^{ℓ+1}, so one needs to solve the following system of equations i+i_{0}+...+i_{r} = ℓ+1 and i_{0}+i_{1}p+i_{2}p^{2}+...+i_{r}p^{r} = j_{0} where i ∈ {1, 2, .., ℓ}. We get p^{r} − 1 = (i − 1)p^{r} + i_{0}(p^{r} − 1) + i_{1}(p^{r} − p) + ... + i_{r−1}(p^{r} − p^{r−1}). It follows that p|i_{0} − 1 and i_{0} ≤ 1 so i_{0} = 1 and i = 1, i_{1} = i_{2} = ... = i_{r−1} = 0 and so i_{r} = ℓ − 1. This proves the claim.

– For the contribution from Δ(red(f_{0} ∘ S))(X, Y) we distinguish two cases. Case p > 3. If ℓ is such that ((ℓ − 1)ℓ(ℓ + 1), p) = 1 (for example ℓ = 2) then there is no contribution of Δ(red(f_{0} ∘ S))(X, Y).

Proof. We have (step 2) Δ(f_{0} ∘ S)(X, Y) = c(A(X)^{q} − A(X), A(Y)^{q} − A(Y)) + (F − Id)c(S(X), S(Y)) and c(A(X)^{q} − A(X), A(Y)^{q} − A(Y)) = c(θ^{q}(X^{q^{2}} + X), θ^{q}(Y^{q^{2}} + Y)) = θ^{pq} ∑_{1≤i≤p−1} \frac{(−1)^{i−1}}{i} (Y^{q^{2}} + Y)^{p−i}(X^{q^{2}} + X)^{i}. We remark that the equation i_{0} +i_{1}q^{2} = p^{α}j_{0} with i_{0} +i_{1} = i, 1 ≤ i ≤ p−1 and α ∈ ℕ is equivalent to α = 0, i_{0} = 1 and ℓ − 1 = i_{1}p^{2n−r}. Hence if ((ℓ − 1)ℓ(ℓ + 1), p) = 1 there is no contribution.

Case p = 3. Take ℓ = 4. Then the equation above gives i_{1}3^{2n−r} = 3 which has a solution iff r = 2n − 1 and then i_{1} = 1 and i = i_{0} + i_{1} = 2. Let us assume that r = 2n − 1. Then the contribution to the coefficient of X^{j_{0}p^{n(j_{0})}} is [2θ^{3q}(Y^{q^{2}} + Y)]^{p^{n(j_{0})}}.

– Clearly there is no contribution from Δ(f_{1})(X, Y).

Let us now show that Ad_{f}(Y) = S_{V} (Y).

If p > 3 and ((ℓ−1)ℓ(ℓ+1), p) = 1, it follows that Ad_{f}(Y) divides S_{V} (Y) which itself divides Ad_{f_{2}}(Y) = Y + Y^{q^{2}} and so Ad_{f}(Y) = S_{V} (Y).

If p = 3 take ℓ = 4 then the same proof works unless r = 2n − 1 and then Ad_{f}(Y) divides T (Y)^{p^{n(j_{0})}} where T (Y) := 2S_{V} (Y)s_{0} + 2θ^{pq}(Y^{q^{2}} + Y). We can write Y^{q^{2}} +Y = (Y −α)S_{V} (Y). Then s_{0}α = −1 and T (Y) = 2θ^{pq}S_{V} (Y)(Y −β), with β = α − θ^{−pq}s_{0}. We remark that s_{0}^{q^{2}} + s_{0} = 0 and θ^{q^{2}} + θ = 0, so β^{q^{2}} + β = −2θ^{−pq}s_{0} ≠ 0. In particular β is not a root of Y^{q^{2}} + Y (and so of S_{V} (Y)). As Ad_{f}(Y) is an additive polynomial we still get that it divides S_{V} (Y).

We remark that, by Cor. 5.3, Ad_{X^{ℓ+1}}(Y) = Y and so Ad_{S_{V}^{ℓ+1}(Y)} = S_{V} (Y) by Prop. 6.2. Then, by Prop. 6.5, for y, z ∈ Z(S_{V} (Y)) one has ϵ_{f}(y, z) = ϵ_{S_{V}^{ℓ+1}}(y, z) + ϵ_{f_{2}}(y, z). As ϵ_{S_{V}^{ℓ+1}}(y, z) = 0 (Prop. 6.2), the commutation rule is that of E.

• p = 2. Let E be the extraspecial group of type II.b. and order pq^{2}, i.e. it is the central product of Q_{8} and n − 1 copies of D_{8}.

Step1. Realization of E. Claim: Let f_{1}(X) := X^{1+q} then E ≃ G_{∞,1}(f_{1}) and \frac{|G_{∞,1}(f_{1})|}{g(f_{1})} = \frac{2p}{(p−1)}q > \frac{2p}{p−1}.

The proof is the same as for type I.a. We use the same calculation up to the point where we used specifically that p = 2. We have seen that σ_{y}^{p}(W) = W + P (X, y) + P (X + y, y) + ... + P (X + (p − 1)y, y) = W + \frac{p(p−1)}{2}P (y, y). So this time as p = 2, σ_{y}^{2}(W) = W + P (y, y) where P (Y, Y) = ∑_{0≤i≤n−1} Y^{(q+1)p^{i}} and Q(Y) := P (Y, Y)Y^{−q} = ∑_{0≤i≤n−1} Y^{(q+1)p^{i}−q} is a separable polynomial of degree 2^{2n−1} + 2^{n−1} − 2^{n}. Define d(n) to be the number of elements of order ≤ 2, they correspond to those y ∈ Z(Ad_{f_{1}}) such that P (y, y) = 0 (this condition doesn’t depend on c). Moreover P (Y, Y)^{2} −P (Y, Y) = Y^{2^{n}}(Y^{2^{2n}} +Y), hence the roots of Q(Y) are simple and among those of Ad_{f_{1}}. It follows that d(n) = 2(2^{2n−1}+2^{n−1}−2^{n}) = 2^{2n}−2^{n} = d_{b}(n) and so G_{∞,1}(f_{1}) is an extraspecial group of type II.b. (section 4.1).

Step 2. Realization of saturated subgroups in E.

If S_{V} (Y) is the additive polynomial corresponding to a saturated subgroup we take f(X) = (redS_{V} (X))^{7} + X^{1+2^{n}}. The conductor is 1 + 6 ∗ 2^{2(n−1)}. We look at the contribution of Δ(S_{V}^{7} )(X, Y) = (S_{V} (Y)+s_{0}X+...+X^{2^{r}})^{7}−S_{V} (Y)^{7}−S_{V} (X)^{7}. The contribution S_{V} (Y)^{2}s_{0}X^{1+2^{r+2}} is the only one in degrees (1 + 2^{r+2})2^{ℕ} and we conclude as in the p > 2 case.

Remark 7.2. Let p > 2 and G be a p-group belonging to the class C_{1}. The proof of the theorem above gives a realization of this group as a group G_{∞,1}(f) through an embedding in an extraspecial group of type I.b hence of exponent p^{2}. In the case that the group G has exponent p one gets simpler realizations in considering G as a saturated subgroup of an extraspecial group of the same exponent p, i.e. of type I.a (see remark 4.5). Following the same line as in step 3 for p > 2, one can show that f(X) = red(S_{V} (X)^{ℓ+1}) + f_{1}(X) gives a realization for G.

7.2 An application: realization of extraspecial groups of type II.a.

We have not yet given an explicit realization of extraspecial groups of type II.a (see Def. 4.3). Such a group with cardinal 2^{2(n−1)+1} is the central product D_{8}∗...∗D_{8} (n−1 times). Let us explain how we get such a realization using the method above. This group is a saturated subgroup of the extraspecial group of type II.b, Q_{8}∗D_{8}∗...∗D_{8} of cardinal 2^{2n+1}. The construction above gives the existence of f(X) = S_{V} (X)^{7}+X^{1+2^{n}} where S_{V} is an additive polynomial of degree 2^{n−1} such that the automorphism group G_{∞,1}(f) is the saturated subgroup D_{8} ∗...∗D_{8} (n−1 times). Note that the conductor is 1 + 6 ∗ 2^{2(n−1)} and so \frac{|G_{∞,1}(f)|}{g(f)} = \frac{2}{3} gives the sharp bound for such a conductor (proposition 6.6). More concretely we now give an explicit realization of D_{8} (n = 2) with conductor 25 which is the minimal one (see B.ii) below and compare with the realization given in section 5.3.)

We view D_{8} as a saturated subgroup π^{−1}(V) of the extraspecial group E := Q_{8} ∗ D_{8} where π : E → E/Z(E) = W for which we know that f_{1}(X) = X^{1+2^{2}} gives a realization. The corresponding additive polynomial is Ad_{f_{1}}(Y) = Y^{2^{4}} + Y and W = Z(Ad_{f_{1}}) ⊂ 𝔽_{2}^{alg}. Then V is a subgroup of order 4 and V = Z(S_{V}), where S_{V} is an additive polynomial dividing Ad_{f_{1}}. Therefore we can write S_{V} (Y) = Y^{4}+aY^{2}+bY where b ≠ 0.

The remainder of the division of Y^{16} + Y by Y^{4} + aY^{2} + bY is (1 + b^{5} + ba^{6})Y + (b^{2}a^{4} + ab^{4} + a^{7})Y^{2}. Consequently we get the two equations 1 + b^{5} + ba^{6} = 0 and b^{2}a^{4} + ab^{4} + a^{7} = 0. For each couple (a, b) satisfying these equations we consider f_{a,b} := (X^{4} + aX^{2} + bX)^{7} + X^{5} and then we get Ad_{f_{a,b}}(Y) = Y^{4} + aY^{2} + bY. If y ∈ W (= E/Z(E)) then P_{f_{1}}(X, y) + P_{f_{1}}(X + y, y) = y^{4}(y^{6} + y). We remark that Y^{6} + Y divides Ad_{f_{1}}(Y) = Y^{16} + Y and the quotient is Y^{10} + Y^{5} + 1. This gives a partition of W into two sets. W_{2}: the roots of Y^{6}+Y corresponding to the 12 elements of G_{∞,1}(f_{1}) of order ≤ 2 and W_{4}: the roots of Y^{10} + Y^{5} + 1 corresponding to the 20 elements of G_{∞,1}(f_{1}) of order 4. Now V := Z(Ad_{f_{a,b}}) is a subgroup of W and for y ∈ V one has P_{f_{1}}(X, y) = P_{f_{a,b}}(X, y) and so P_{f_{a,b}}(X, y)+P_{f_{a,b}}(X +y, y) = y^{4}(y^{6} +y). Concerning the commutation rule for y, z ∈ V ⊂ W we have ϵ_{f_{a,b}}(y, z) = ϵ_{f_{1}}(y, z) = z^{2}y^{8} + zy^{4} + z^{8}y^{2} + z^{4}y = yz(y + z)(y^{2} + zy + z^{2})(zy^{4} + z^{4}y + 1).

A. a = 0 and 1 + b^{5} = 0. Note that in this case Ad_{f_{a,b}}(Y) = Y^{4} + bY and only the roots y = 0 and y = b^{2} are in W_{2}. Moreover for y, z ∈ V one has yz(y + z)(y^{2} + zy + z^{2}) = yz(y^{3} + z^{3}) = 0. It follows that the group G_{∞,1}(f_{a,b}) is abelian, isomorphic to ℤ/2ℤ × ℤ/4ℤ.

B. Let A := b^{5} +a^{6}b+1 and B := b^{4} +a^{3}b^{2} +a^{6}. Then the resultant of A and B in b is (b^{5} + 1)(b^{10} + b^{5} + 1)^{2}. The case b^{5} + 1 = 0 is case 1. above. Now we can assume that b^{10} + b^{5} + 1 = 0, i.e. b is a primitive 15-th root of 1. The equations A = B = 0 give three sets of covers.

B.i) ab = 1, i.e. a = b^{14}. In this case Y^{4} + aY^{2} + bY divides Y^{6} + Y. Then the group has exponent 2 and it is isomorphic to (ℤ/2ℤ)^{3}.

B.ii) ab = b^{5} i.e. a = b^{4}. In this case Y^{4} + aY^{2} + bY has only one root (b^{7}) in common with Y^{10}+Y^{5}+1. It follows that the group G_{∞,1}(f_{a,b}) has 2 elements of order 4 and so it is D_{8}. We can write red(f_{a,b}) = (b^{14}+b^{5})X +(b+b^{8})X^{3}+(1+b^{14}+b^{13})X^{5}+ (b^{7}+1)X^{7}+(b^{13}+b^{10})X^{9}+(b^{4}+b+b^{6})X^{11}+(b^{2}+1)X^{13}+b^{2}X^{17}+b^{3}X^{19}+b^{9}X^{21}+bX^{25}, which is defined over 𝔽_{16}.

B.iii) ab = b^{10}, i.e. a = b^{9}. In this case (Y^{4} + aY^{2} + bY)/Y divides Y^{10} + Y^{5} + 1 and it follows that the group G_{∞,1}(f_{a,b}) has 6 elements of order 4, so this is Q_{8}.

Remark 7.3. We could as well obtain families. For this it suffices to deal with f_{1} giving a family, for example f_{1} = tX^{3} + X^{5} (see Prop. 8.1 ii)). The corresponding discussion is thought more delicate as the above.

8 Maximal curves

8.1 Universal family

We have seen that if g(C_{f}) ≥ 2 then \frac{|G_{∞,1}(f)|}{g(f)^{2}} ≤ \frac{4p}{(p−1)^{2}} and we have equality for f = X^{1+p^{n}}. In fact this equality characterizes the f with red(f) ≠ 0 and of the form S_{1}(X)S_{2}(X) where for i = 1, 2 the S_{i} are additive polynomials. Namely for a given genus there is a universal family given in the following:

Proposition 8.1. Let n > 1, A_{n} := 𝔽_{p}^{alg}[t_{i}, 0 ≤ i ≤ n] and f := t_{0}X^{1+p^{0}} + t_{1}X^{1+p} + ... + t_{n−1}X^{1+p^{n−1}} + t_{n}X^{1+p^{n}} ∈ A_{n}[X]. Let k be an algebraically closed field and φ : A_{n} → k a specialization homomorphism such that φ(t_{n}) ≠ 0. Then Ad_{φ(f)}(Y) = φ(∑_{0≤i≤n}(t_{i}^{p^{n−i}}Y^{p^{n−i}} + t_{i}^{p^{n}}Y^{p^{n+i}})) = φ(t_{n})Y + ... + (φ(t_{n}))^{p^{n}}Y^{p^{2n}} and |G_{∞,1}(φ(f))| = p^{2n+1}. Further

i) If p > 2 then φ(f) is reduced and G_{∞,1}(φ(f)) is an extraspecial group of type I.a.

ii) If p = 2 and t_{0} = 0 then φ(f) is reduced and G_{∞,1}(φ(f)) is an extraspecial group of type II.b.

Let π : X = Spec\frac{A_{n}[t_{n}^{−1}][X,W]}{(W^{p}−W −f)} → SpecA_{n}[t_{n}^{−1}] be the structure morphism. This is a flat family of curves and the fibers are highly singular. Let us consider the normalization π̃ : X̃ → SpecA_{n}[t_{n}^{−1}]. As the fibers have the same conductor, and flatness is an open condition, after a suitable localization we get a smooth family over some non empty open SpecA_{n}[Q^{−1}] ⊂ SpecA_{n}[t_{n}^{−1}]. Each geometric fiber of this family has genus g_{n} := \frac{p^{n}(p−1)}{2}. The geometric generic fiber is the curve C_{Id(f)}. The group G_{∞,1}(Id(f)) acts faithfully on each fiber C̃_{φ(f)} and is identified with the group G_{∞,1}(φ(f)).

Proof. Write f_{i} = t_{i}X^{1+p^{i}} with t_{0} = 0 for p = 2. Then Δ(f)(X, Y) = ∑_{0≤i≤n} Δ(f_{i}), where Δ(f_{i}) = A_{i} + B_{i}, A_{i} = t_{i}Y X^{p^{i}}, B_{i} = t_{i}Y^{p^{i}}X. We can write Δ(f_{i}) = (t_{i}^{p^{n−i}}Y^{p^{n−i}} + t_{i}^{p^{n}}Y^{p^{n+i}})X^{p^{n}} + P_{i} − P_{i}^{p} where P_{i} = A_{i} + ... + A_{i}^{p^{n−i−1}} + B_{i} + ... + B_{i}^{p^{n−1}} for i ≤ n − 1 and B_{i} + ... + B_{i}^{p^{n−1}} for i = n. One obtains Ad_{φ(f)}(Y) = φ(∑_{0≤i≤n}((φ(t_{i}))^{p^{n−i}}Y^{p^{n−i}} + (φ(t_{i}))^{p^{n}}Y^{p^{n+i}})) = φ(t_{n})Y + ... + (φ(t_{n}))^{p^{n}}Y^{p^{2n}} is a separable additive polynomial with constant degree p^{2n}. A geometric argument on automorphisms groups of relative curves gives the results as we know it for the specialisation morphism φ_{0} given by φ_{0}(t_{i}) = 0 for i < n and φ_{0}(t_{n}) = 1 (see step 1 for p > 2 case in 7.1). One can as well mimic the proof we gave in case of φ_{0}; for this we remark that P (X, Y) = ∑_{0≤i≤n} P_{i}(X, Y) is again an additive polynomial so the calculations we have done for φ_{0} can be done in general. It is easy from the above formulas for the P_{i} to check that the degree conditions we need are determined uniquely by P_{n} and so they are still satisfied . The rest of the proposition is standard.

Remark 8.2. Such a family was previously considered in the literature in connection with coding theory. Namely in [vdG-vdV] the specialization morphisms φ taking values in 𝔽_{p^{2n}} are studied. The additive polynomial Ad_{φ(f)} is then their polynomial E_{n}(Y) = S(Y)^{p^{n}} + ∑_{0≤i≤n}(t_{i}Y)^{p^{n−i}} where S(X) = ∑_{0≤i≤n} t_{i}Y^{p^{i}}. The zeroes are interpreted as the 𝔽_{p}-vector space which is the kernel of the 𝔽_{p}-bilinear form Tr_{𝔽_{p^{2n}}/𝔽_{p}}(xS(y) + yS(x)) (which is symmetric if p > 2 and alternating if p = 2). In case p = 2 they also prove a factorization of E_{n}(Y) = Y E_{n}^{−}(Y)E_{n}^{+}(Y) which corresponds to a partition of the roots depending on the order of the corresponding automorphism of C_{f}. Such a decomposition works in general. They give the structure of the groups G_{∞,1}(φ(f)).

An approach more closely related to ours can be found in [El], notably paragraph 5. There the polynomials Ad_{φ(f)}(Y) = φ(∑_{0≤i≤n}(t_{i}^{p^{n−i}}Y^{p^{n−i}} + t_{i}^{p^{n}}Y^{p^{n+i}})) are studied on their own. They form the class of “the palindromic polynomials”.

8.2 Application to the moduli space of curves

We keep the notations of Prop. 8.1. For a fixed n > 1 let A_{n} := 𝔽_{p}^{alg}[t_{i}, 0 ≤ i ≤ n − 1] and f := t_{0}X^{1+p^{0}} + t_{1}X^{1+p} + ... + t_{n−1}X^{1+p^{n−1}} + t_{n}X^{1+p^{n}} ∈ A_{n}[X] with t_{n} = 1. (As soon as t_{n} ≠ 0, after a change of variable, this condition is satisfied). For p = 2 we take t_{0} = 0 (in order to have f be reduced) and set g_{n} := \frac{p^{n}(p−1)}{2}. Let θ ∈ 𝔽_{p}^{alg} be a primitive (p − 1)(p^{n} + 1)-th root of 1. Then Θ : (t_{0}, ..., t_{n−1}) → (θ^{p^{n}−p^{0}}t_{0}, θ^{p^{n}−p^{1}}t_{1}, ..., θ^{p^{n}−p^{n−1}}t_{n−1}) induces an 𝔽_{p}^{alg}-automorphism of SpecA_{n} of order p^{n} + 1. Let B_{n} := A_{n}^{<Θ>} be the quotient of the affine space by the cyclic group of automorphisms < Θ >. The structure morphism π : C̃_{f} → SpecA_{n}[Q^{−1}] is a family of curves of genus g_{n} (Prop. 8.1). Moreover by Prop. 3.3 two specialization morphisms φ_{i} : A_{n} → k, for i = 1, 2 will give isomorphic k-curves iff ∃c ∈ 𝔽_{p}^{×} and (a, b) ∈ (k^{×}, k) such that ∑_{0≤i≤n−1} φ_{2}(t_{i})(aX + b)^{1+p^{i}} + (aX + b)^{1+p^{n}} = c(∑_{0≤i≤n−1} φ_{1}(t_{i})X^{1+p^{i}} + X^{1+p^{n}}) mod (Id − F)k[X]. It follows that a^{1+p^{n}} = c ∈ 𝔽_{p}^{×} and for i ≥ 0 one has φ_{2}(t_{i}) = ca^{−(1+p^{i})}φ_{1}(t_{i}) = a^{p^{n}−p^{i}}φ_{1}(t_{i}). (Note that for p = 2 we have assumed t_{0} = 0 and so f is reduced). Reciprocally these conditions (take b = 0) define two specialization morphisms which give isomorphic k-curves which are in the same orbit under the action of the group Θ. By the definition of the coarse moduli space M_{g_{n}} we deduce from the existence of the family C̃_{f} → SpecA_{n}[Q^{−1}] a map from SpecA_{n}[Q^{−1}] to M_{g_{n}}. This map factors through SpecB_{n}[N(Q)^{−1}] (where N(Q) := ∏_{0≤i≤p^{n}} Θ^{i}(Q)) in an injective morphism. The image is an algebraic subset of M_{g_{n}} of the same dimension as that of SpecB_{n}. A measure of the size of families of curves which are étale covers of the affine line, and with given extraspecial group of type I.a and order p^{2n+1} as an automorphism group, is given by the dimension of this image, which is O(log(g_{n})).

8.3 Characterization of maximal p-cyclic covers of the affine line

It is remarkable that these families for varying n can be characterized by the following Hurwitz-type bound.

Proposition 8.3. Let k be an algebraically closed field, p = char(k) > 0 and f(X) ∈ Xk[X] a polynomial of degree m := deg f prime to p. We assume f is reduced. If \frac{|G_{∞,1}(f)|}{g(f)} > \frac{p}{p−1} (2/3 for p = 2 ) then f(X) = XS(X), where S(X)−S(0) is an additive

polynomial. Moreover if deg S = p^{n} then \frac{|G_{∞,1}(f)|}{g(f)} = 2p^{n} \frac{p}{p−1} and \frac{|G_{∞,1}(f)|}{g(f)^{2}} = \frac{4p}{(p−1)^{2}} (see Prop. 8.1).

Proof. The proof works by elimination of bad monomials. We saw in Prop. 6.6 that only for m = 1 + p^{s} with s > 0, \frac{|G_{∞,1}(f)|}{g(f)} can be > \frac{p}{p−1} ( \frac{2}{3} if p = 2). Now we show that any other monomial in f(X) has exponent 1 or 1 + p^{t} with t < s.

Let us assume that this is not the case and denote by X^{a} the monomial of highest degree which is not of the above form. We first assume that p doesn’t divide a − 1 (such a case doesn’t occur if p = 2) and consider the integer k such that p^{k−1} < a − 1 < p^{k}. Then k ≤ s and p^{s−1} < (a − 1)p^{s−k} < p^{s}. Further the monomial in X^{a} has a contribution in R(X, Y) (defined in Lemma 5.1) which is equal to a^{p^{s−k}}Y^{p^{s−k}}X^{(a−1)p^{s−k}} +other terms. Moreover as for α > 0 we have p^{α}(a−1) > (a−1) and it follows that a^{p^{s−k}}Y^{p^{s−k}} is exactly the coefficient of X^{(a−1)p^{s−k}} in R. This would imply that |G_{∞,1}(f)| ≤ p and \frac{|G_{∞,1}(f)|}{g(f)} ≤ \frac{2p}{(p−1)p^{s}} ≤ \frac{p}{(p−1)} - a contradiction.

Let us now assume that a − 1 = ℓp^{t}, where ℓ > 1, (ℓ, p) = 1 and t > 0. Let j_{0} = 1 + (ℓ − 1)p^{t} and say p^{k−1} < j_{0} < p^{k}. Then p^{s−1} < j_{0}p^{s−k} < p^{s} and pj_{0} < a iff ℓ < \frac{p}{p−1} − \frac{1}{p^{t}} but this is not the case. Hence the monomial X^{a} contributes to R(X, Y) in the monomial X^{j_{0}p^{s−k}} with only one term - equal to (ℓY^{p^{t}})^{p^{s−k}}X^{j_{0}p^{s−k}}. Any other contribution to R in the monomial X^{j_{0}p^{s−k}} can only occur from a monomial X^{b} with j_{0} < b < a. Now such a contribution will be (\binom{b}{j_{0}}Y^{b−j_{0}})^{p^{s−k}} whose degree is ≤ (b − j_{0})p^{s−k} < (a − j_{0})p^{s−k} = p^{t+s−k}. Finally we get that deg Ad_{f}(Y) ≤ p^{t+s−k}. From this we get \frac{|G_{∞,1}(f)|}{g(f)} ≤ \frac{2p^{t+s−k+1}}{p^{s}(p−1)} and so 1 < 2p^{t−k}, i.e. t ≥ k for p > 2 (\frac{1}{3} < 2^{t−k+1} and t ≥ k − 1 for p = 2). On the other hand (ℓ − 1)p^{t} < p^{k}, so (ℓ − 1) < \frac{1}{p^{t−k}} - a contradiction.

8.4 Maximal p-group actions

In [St1], Stichtenoth proved that if C is a non singular projective curve over k with genus g_{C} ≥ 2 then |Aut_{k}(C)| ≤ 16g_{C}^{4} unless C is birational to a curve W^{p^{n}} + W = X^{1+p^{n}} (p = char k). In fact he has gotten this general bound as a corollary to his study of p-cyclic (étale) covers of the affine line with conductor m. Namely it follows from the bound for p-Sylow subgroups |G_{∞,1}(f)| ≤ p(m − 1)^{2} for f ∈ k[X] of degree m prime to p ([St2, Satz 4] and [Cor. 5.3 i)]). In the same spirit Prop. 8.3 has a corollary on automorphisms groups of curves. Namely we can describe the curves C with a “big” p-group of automorphisms G. First we need to define what we mean by big.

For this we use Nakajima’s extension of Stichtenoth’s work in order to circumvent group actions on curves contradicting Hurwitz-type bounds. Namely in [Na, Theorem 1] he concentrates on p-group actions and gives bounds on the size of the group in terms of the Hasse-Witt invariant of the curve. The following proposition is a translation of his results. Let us first make a definition.

Definition 8.4. Let C be a non singular projective curve over k and G a p-subgroup of Aut_{k}C. We say that (C, G) satisfies condition (N) if g_{C} > 0 and \frac{|G|}{g_{C}} > \frac{2p}{p−1}.

Proposition 8.5. Assume (C, G) with g_{C} ≥ 2 satisfies condition (N). Then there is a point, say ∞ ∈ C, such that G is the wild inertia subgroup of G at ∞. Moreover C/G is isomorphic to ℙ_{k}^{1} and the ramification locus (resp. branch locus) of the cover π : C → C/G is the point ∞ (resp. π(∞)). We define the ramification groups in lower notation G_{i} (i ≥ 0) by

G_{0} := {σ ∈ G | σ(∞) = ∞} = G

and for i ≥ 1

G_{i} := {σ ∈ G_{0} | ord_{∞}(σ(π_{∞}) − π_{∞}) ≥ i + 1},

where π_{∞} is a uniformizing parameter at ∞ ∈ C and ord is the order function at ∞. Let i_{0} be the integer such that G_{2} = G_{3} = .... = G_{i_{0}} ⫌ G_{i_{0}+1}.

i) Then G_{2} ≠ G_{1} and the quotient curve C/G_{2} is isomorphic to ℙ_{k}^{1}.

ii) Let H be a subgroup which is normal in G and such that g_{C/H} > 0. Then G/H is a p-subgroup of Aut_{k}C/H and \frac{|G|}{g_{C}} ≤ \frac{|G/H|}{g_{C/H}}. In particular (C/H, G/H) satisfies

condition (N). Moreover if M ≤ \frac{|G|}{g_{C}^{2}} for some M one gets |H| ≤ \frac{1}{M} \frac{|G/H|}{g_{C/H}^{2}}.

iii) Let H be a subgroup which is normal in G and G_{2} ⫌ H ⊃ G_{i_{0}+1}. Then g_{C/H} = \frac{(|G_{2}/H|−1)(i_{0}−1)}{2} > 0 and (C/H, G/H) satisfies condition (N).

Proof. i) When we compare the condition \frac{|G|}{g_{C}} > \frac{2p}{p−1} to [Na, theorem 1], we see that C has Hasse-Witt invariant zero; moreover it follows from the proof that the branch locus is reduced to one point and the ramification is total. Now we apply the Hurwitz formula to the cover C → C/G. We have

2(g_{C} − 1) = |G|2(g_{C/G} − 1) + ∑_{0≤i}(|G_{i}| − 1).  (7)

As G = G_{0} = G_{1} we get

g_{C} ≥ |G|g_{C/G}.  (8)

This together with the assumption \frac{|G|}{g_{C}} > \frac{2p}{p−1} implies g_{C/G} = 0. Now if G_{2} = G_{1} then (7) yields 2(g_{C} − 1) ≥ |G|2(0 − 1) + 3(|G| − 1) which is a contradiction.

ii) Let H be a normal subgroup of G then G/H is a p-subgroup of Aut_{k}C/H so, as in (8), we have

g_{C} ≥ |H|g_{C/H}.  (9)

If g_{C/H} ≠ 0 we get \frac{|G|}{g_{C}} ≤ \frac{|G/H|}{g_{C/H}}. Now if M ≤ \frac{|G|}{g_{C}^{2}}, one has

|H| ≤ \frac{1}{|H|} \frac{g_{C}^{2}}{g_{C/H}^{2}} ≤ \frac{1}{M} \frac{|G/H|}{g_{C/H}^{2}}.

iii) Under the condition G_{2} ⫌ H ⊃ G_{i_{0}+1} we can calculate g_{C/H}. For this we remark that the cover C/H → C/G_{2} is Galois with group G_{2}/H which is pelementary abelian. The ramification groups are G_{2}/H = (G_{2}/H)_{0} = (G_{2}/H)_{1} = ... = (G_{2}/H)_{i_{0}} ⊃ (G_{2}/H)_{i_{0}+1} = {0} (exercise). It follows from (7) that 2(g_{C/H} − 1) = |G_{2}/H|(0 − 2) + (i_{0} + 1)(|G_{2}/H| − 1). Hence g_{C/H} = \frac{(|G_{2}/H|−1)(i_{0}−1)}{2} > 0 and so \frac{2p}{p−1} < \frac{|G|}{g_{C}} ≤ \frac{|G/H|}{g_{C/H}}.

For the action of a p-group G on a curve C with genus g_{C} ≥ 2, we calculate \frac{|G|}{g_{C}}. If it is > \frac{2p}{p−1} we will see that the values of \frac{|G|}{g_{C}^{2}} sieve the actions at least when they are big enough. We will concentrate on this case of “maximal curves” in a sequel paper ([Le-Ma3]) we go further in the classification.

Theorem 8.6. Let C be a non singular projective curve over k of genus g_{C} ≥ 2 and G a p-subgroup of Aut_{k}C. We assume that (C, G) satisfies condition (N) and keep the notations of Prop. 8.5.

i) There is a normal subgroup H of G with G_{2} ⫌ H ⊃ G_{i_{0}+1} and |G_{2}/H| = p. For any such H g_{C/H} = \frac{(p−1)(i_{0}−1)}{2} > 0 and there is a polynomial f(X) = XS(X) ∈ k[X], where S(X) − S(0) is an additive polynomial of degree i_{0} − 1 = p^{n_{0}} ∈ p^{ℕ^{∗}}, such that C/H is birational to the curve C_{f}. Moreover g_{C/G_{i_{0}+1}} = \frac{1}{2}(|G_{2}/G_{i_{0}+1}|−1)(i_{0} −1) and |G/G_{2}| = \frac{1}{p}|G/H| ≤ (i_{0} − 1)^{2}.

ii) Let M_{1} := \frac{4p}{(p−1)^{2}} . Then M_{1} ≤ \frac{|G|}{g_{C}^{2}} iff C is birational to a curve C_{f} : W^{p} −W =

f(X) := XS(X) where S(X)−S(0) = ∑_{0≤i≤μ} t_{i}X^{p^{i}} is an additive polynomial ∈ k[X] with μ ≥ 1 and t_{μ} ≠ 0. In this case G = G_{∞,1}(f) is an extraspecial group of type I.a (resp. II. b) if p > 2 (resp. p = 2), M_{1} = \frac{|G|}{g_{C}^{2}}, and G_{2} ≃ ℤ/pℤ.

iii) Let M_{2} := \frac{4}{(p−1)^{2}} . Then M_{2} ≤ \frac{|G|}{g_{C}^{2}} < M_{1} iff C is as in ii). In this case G has

index p in G_{∞,1}(f) and M_{2} = \frac{|G|}{g_{C}^{2}}. Note that G ⊃ [G_{∞,1}(f), G_{∞,1}(f)] = Z(G_{∞,1}(f)).

Proof. i) The existence of such a group H is a classical fact for p-groups [Su1, Th. 1.12 Chap. 2]. The assertion follows from the propositions 8.5 iii), 8.3 and 6.6 I.

One common ingredient for ii) and iii) is that a lower bound M ≤ \frac{|G|}{g_{C}^{2}} will produce, following i) and Prop. 8.5 ii), an upper bound for G_{i_{0}+1}; namely

|G_{i_{0}+1}| ≤ \frac{1}{M} \frac{|G/G_{i_{0}+1}|}{g_{C/G_{i_{0}+1}}^{2}} ≤ \frac{1}{M} \frac{4|G_{2}/G_{i_{0}+1}|}{(|G_{2}/G_{i_{0}+1}| − 1)^{2}} .

ii) We have 1 ≤ |G_{i_{0}+1}| ≤ \frac{1}{M_{1}} \frac{4|G_{2}/G_{i_{0}+1}|}{(|G_{2}/G_{i_{0}+1}|−1)^{2}} . Observe that \frac{4p^{n}}{(p^{n}−1)^{2}} as a function in n is strictly decreasing and hence it follows that |G_{i_{0}+1}| = 1 and |G_{2}/G_{i_{0}+1}| = p. Finally this gives |G_{2}| = p and the result follows from Prop. 8.3.

iii) If |G_{2}/G_{i_{0}+1}| = p^{2+n} with n ∈ ℕ then \frac{1}{M_{2}} \frac{4p^{2+n}}{(p^{2+n}−1)^{2}} ≤ \frac{1}{M_{2}} \frac{4p^{2}}{(p^{2}−1)^{2}} < 1; so 1 ≤ |G_{i_{0}+1}| < 1 - a contradiction. It follows that |G_{2}/G_{i_{0}+1}| = p and |G_{i_{0}+1}| = 1 or p. Assuming that |G_{i_{0}+1}| = p we obtain that there is i_{1} > 0 such that G = G_{0} = G_{1} ⊇ G_{2} = ⋯ = G_{i_{0}} ⫌ G_{i_{0}+1} = ⋯ = G_{i_{0}+i_{1}} ⫌ G_{i_{0}+i_{1}+1} = 1. Further one has |G_{2}| = p^{2} and 2(g_{C} − 1) = |G_{2}|2(0 − 1) + ∑_{0≤i}(|G_{i}| − 1) = −2p^{2} + (i_{0} + 1)(p^{2} − 1) + i_{1}(p − 1). In particular, as i_{1} > 0, we get g_{C} > \frac{(i_{0}−1)(p^{2}−1)}{2} and so \frac{|G|}{g_{C}^{2}} < \frac{4p^{2}}{(p^{2}−1)^{2}} < M_{2} - a contradiction! We therefore have proven that |G_{i_{0}+1}| = 1 and G_{2} ≃ ℤ/pℤ and conclude that G is a subgroup of index p of an extraspecial group by 8.1.

Thanks. We would like to thank Aristides Kontogeorgis for his correspondence along the elaboration of this paper. Its final version was written while the second author was a guest of the Max-Planck-Institut in Bonn.

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Claus LEHR Università di Padova Dipartimento di Matematica Pura ed Applicata, Via Belzoni 7, 35131 Padova, Italy

e-mail : lehr@math.unipd.it

Bât. M2, Michel MATIGNON Laboratoire de Théorie des Nombres et d’Algorithmique Arithmétique, UMR 5465 CNRS Université de Bordeaux I, 351 cours de la Libération, 33405 Talence Cedex, France

e-mail : matignon@math.u-bordeaux.fr