Isotypic Decomposition of the Cohomology and Factorization of the Zeta Functions of Dwork Hypersurfaces

HAL Id: hal-00440358

https://hal.archives-ouvertes.fr/hal-00440358

Preprint submitted on 10 Dec 2009

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Isotypic Decomposition of the Cohomology and Factorization of the Zeta Functions of Dwork

Hypersurfaces

Philippe Goutet

To cite this version:

Philippe Goutet. Isotypic Decomposition of the Cohomology and Factorization of the Zeta Functions

of Dwork Hypersurfaces. 2009. �hal-00440358�

Isotypic Decomposition of the Cohomology and Factorization of the Zeta Functions of Dwork Hypersurfaces

Philippe Goutet December 10, 2009

Abstract

The aim of this article is to illustrate, on the Dwork hypersurfacesxⁿ1+· · ·+xⁿn−nψx¹. . . xn= 0 (withn an integer ≥3 andψ ∈F_q^∗ a parameter satisfyingψⁿ 6= 1), how the study of the representation of a finite group of automorphisms of a hypersurface in its etale cohomology allows to factor its zeta function.

1 Introduction

Letnbe an integer≥3 andF_q a finite field of characteristicp6= 2 not dividing n; to simplify the results, we will assume that q ≡1 modn. We consider the projective hypersurface Xψ ⊂ P_Fqⁿ⁻¹ given by

xⁿ₁ +· · ·+xⁿ_n−nψx1. . . xn= 0,

whereψ is a non zero parameter belonging toFq. The zeta function ofXψ is defined as ZXψ/Fq(t) = exp

^+∞

X

r=1

#Xψ(Fq^r)t^r r

.

We assume thatψⁿ 6= 1, so thatXψ=Xψ⊗^FqF_qis nonsingular. AsXψis a non-singular hypersurface ofPⁿ⁻¹, we know that the dimension of the etaleℓ-adic cohomology spacesH_etⁱ (Xψ,Q_ℓ) is zero for i >2n−4 ori <0 and that, for 0≤i≤2n−4,

dimH_etⁱ(Xψ,Q_ℓ) =

(δi ifi6=n−2, δi+^{(n−1)n+(−1)}_n ⁿ⁽ⁿ⁻¹⁾ ifi=n−2,

where δi = 0 ifiis odd and δi = 1 ifi is even (see §2.2). As we will recall in Remark 2.3 page 3, the zeta function ofXψ is related to how the Frobenius acts onH_etⁿ⁻²(Xψ,Q_ℓ).

We set

A={(ζ1, . . . , ζn)∈µ_n(Fq)ⁿ|ζ1. . . ζn= 1}/{(ζ, . . . , ζ)}; Aˆ={(a1, . . . , an)∈(Z/nZ)ⁿ |a1+· · ·+an= 0}/{(a, . . . , a)},

and denote by [ζ1, . . . , ζn] the class of (ζ1, . . . , ζn) in A and [a1, . . . , an] that of (a1, . . . , an) in ˆA.

We will identify the group ˆA with the group of characters ofA taking values inF_q^∗. The groupA

2000 Mathematical Subject Classification:Primary 14G10; Secondary 11G25, 14G15, 20C05.

Keywords: Zeta function factorisation, Dwork hypersurfaces, isotypic decomposition.

acts onXψ by coordinatewise multiplication; the symmetric groupS_n acts on the right on Xψ by permutation of the coordinates

[x1:. . .:xn]^σ= [xσ(1):. . .:xσ(n)], and on the left onAand ˆA by

σ[ζ1, . . . , ζn] = [ζσ⁻¹(1), . . . , ζσ⁻¹(n)];

σ[a1, . . . , an] = [aσ⁻¹(1), . . . , aσ⁻¹(n)].

The semidirect productG=A⋊S_n acts on the right onXψ, and hence on the left onH_etⁿ⁻²(Xψ, Qℓ) as the functorg7→g^∗is contravariant.

The aim of this article is to describe the structure ofH_etⁿ⁻²(Xψ,Q_ℓ) as aQ_ℓ[G]-module in order to deduce a factorization of the zeta function ofXψ. More precisely, we will show that the primitive part ofH_etⁿ⁻²(Xψ,Q_ℓ) (as defined in§2.2) admits an isotypic decomposition

M

a,ω

Wa,ω⊗DaVa,ω,

where a describes (S_n ×(Z/nZ)^×)\A,ˆ ω belongs to a certain set of roots of unity (see Corol- lary 5.12 page 19),Wa,ωis a simpleQ[G]-module which is independent ofℓ,Da is the division ring EndQ[G](Wa,ω)^opp, andVa,ωis a free module overDa⊗^QQ_ℓwhose rank is independent ofℓ. Because the Frobenius stabilizes these isotypic spaces, its characteristic polynomial splits in as many factors (the idea to use this method is inspired by an argument given in [Hulek et al., 2006,§6.2]).

The first step is to decompose the Q_ℓ[G]-module H_etⁿ⁻²(Xψ,Q_ℓ); we follow the same method Br¨unjes used for the case ψ = 0 (Fermat hypersurface), but, thanks to a more powerful trace formula, we avoid the tedious induction of [Br¨unjes, 2004, Proposition 11.5]. Our methods can be generalized to other families of hypersurfaces, allow us to obtain factorizations slightly finer than those of Kloosterman [2007] (who uses thep-adic Monsky-Washnitzer cohomology), and also allow us to express each factor as the norm of a polynomial with coefficients in a certain finite extension ofQ, hence explaining a numerical observation of Candelas, de la Ossa and Rodriguez-Villegas in the case n= 5 where this extension is Q(√

5) (see [Candelas et al., 2003, Table 12.1 page 133]¹).

Let us also mention that, in a recent article, Katz [2009] studies the action ofA(but not ofA⋊S_n) on the cohomology ofXψand establishes a motivic link betweenXψ and objects of hypergeometric type.

The article is organized as follows. After preliminaries (§2), we describe the structure ofH_etⁿ⁻²(Xψ, Q_ℓ) as aQ_ℓ[A]-module (§3) and then as aQ_ℓ[G]-module (§4). We then deduce the structure of the Q_ℓ[G]-moduleH_etⁿ⁻²(Xψ,Q_ℓ) (§5) and explain the link between this structure and the existence of a factorisation of the zeta function ofXψ (§6). An index of all notations introduced in the article is given in§A and a table of the main formulas appears in§B.

2 Preliminaries

We begin by recalling a Lefschetz-type trace formula by Deligne and Lusztig which allows to express the alternating sum of the traces of an automorphism on theℓ-adic cohomology spaces as the Euler–

Poincar´e characteristic of the fixed-point scheme of this automorphism. We then recall the value of this Euler–Poincar´e characteristic in the cases we will encounter in what follows (smooth projective hypersurfaces). Finally, we link the trace of an element ofGto the Euler–Poincar´e characteristic of a subscheme of fixed points.

1They make this observation only in the case ψ = 0, but their numerical data in §13.3 suggests the same phenomenon happens whenψ6= 0 andq≡1 mod 5.

2.1 Lefschetz trace formula

Let us recall that the Euler–Poincar´e characteristic of a proper scheme overFp is given by χ(X) =

2 dimX

X

i=0

(−1)ⁱdimH_etⁱ (X,Q_ℓ), whereℓ is a prime number6=p. It is an integer independent ofℓ.

Theorem 2.1. Let X be a proper scheme over F_p. If f is an automorphism ofX of finite order prime top, and if X^f denotes the fixed-point subscheme of f of the schemeX, then

2 dimX

X

i=0

(−1)ⁱtr(f^∗|H_etⁱ (X,Q_ℓ)) =χ(X^f).

Proof. See [Deligne and Lusztig, 1976, Theorem 3.2, page 119].

2.2 Euler–Poincar´ e characteristic of a non-singular hypersurface

In this§2.2, exceptionally, we do not assume thatn≥3.

Theorem 2.2(Hirzebruch formula). Letnbe an integer≥1andf ∈F_p[x1, . . . , xn]a homogeneous polynomial of degree d such that f, _∂x^∂f

1, . . . , _∂x^∂f_n have no common zero in F_pⁿ except (0, . . . ,0).

Then the hypersurface X ⊂P_Fpⁿ⁻¹ defined by f = 0 is non-singular (and irreducible if n≥3) and its Euler–Poincar´e characteristic is

χ(X) = (n−1) +(1−d)ⁿ+ (d−1)

d .

Proof. Ifn≥3, we use Corollary 7.5.(iii) of [SGA5, expos´e VII]: indeed, the subschemeX ofP_Fⁿ⁻¹ is smooth, connected and of dimensionn−2; its Euler–Poincar´e characteristic is hence p

χ(X) =d

n−2

X

i=0

(−1)ⁿ⁻ⁱ n

i

dⁿ⁻²⁻ⁱ= 1 d

n−2

X

i=0

(−1)ⁿ⁻ⁱ n

i

dⁿ⁻ⁱ

=(1−d)ⁿ+nd−1

d ,

which is the announced formula. If n= 2, the hypersurfaceX of P_Fp¹ consists of d distinct points and soχ(X) =d, which shows the result as (2−1) +¹_d[(1−d)²+ (d−1)] =d. Finally, ifn= 1, X=∅and soχ(X) = 0, which also shows the result in this case.

Remark 2.3. Whenn≥3, Theorem 2.2 can be refined as follows. We keep the same notations and denote byj the canonical injectionX →P_Fⁿ⁻¹

p . By the Weak Lefschetz Theorem, (see for example [Freitag and Kiehl, 1988, Corollary 9.4, page 106]), fori < n−2 (respectivelyi=n−2), the linear mapj^∗:H_etⁱ(P_Fⁿ⁻¹_p ,Q_ℓ)→H_etⁱ (X,Q_ℓ) is bijective (respectively injective). If we set δi = 0 if i odd andδi= 1 ifi is even, we thus have dimH_etⁱ(X,Qℓ) =δi fori < n−2, and this result stays valid forn−2< i≤2(n−2) by Poincar´e duality. Fori=n−2, the image of the mapj^∗:H_etⁿ⁻²(P_Fpⁿ⁻¹, Q_ℓ)→ H_etⁿ⁻²(X,Q_ℓ) has dimension δi. We will denote it by H_etⁿ⁻²(X,Q_ℓ)^inprim and set H_etⁿ⁻²(X, Q_ℓ)^prim = H_etⁿ⁻²(X,Q_ℓ)/H_etⁿ⁻²(X,Q_ℓ)^inprim. Because the Frobenius acts as the multiplication by q^(n−2)/2 onH_etⁿ⁻²(Xψ,Q_ℓ)^inprimand by multiplication by qⁱ on eachH_et²ⁱ(Xψ,Q_ℓ), we have

ZX_ψ/Fq(t) =det(1−tFrob^∗|H_etⁿ⁻²(Xψ,Q_ℓ)^prim)^(−1)n−1 (1−t)(1−qt). . .(1−qⁿ⁻²t) .

2.3 Character of G acting on H

(X

, Q

)

The isomorphism class of aQ_ℓ[G]-module is completely determined by its character. In this§2.3, we will express in terms of Euler–Poincar´e characteristics the values of the character of theQ_ℓ[G]- moduleH_etⁿ⁻²(Xψ,Q_ℓ)^primfor the elementsg∈Gwhich are of order prime to p.

Lemma 2.4. Each g ∈G acts as the identity on H_etⁿ⁻²(Xψ,Qℓ)^inprim and on H_etⁱ(Xψ,Qℓ) when i6=n−2.

Proof. As g is the restriction of an automorphism of P_Fⁿ⁻¹

q , it results from Remark 2.3 and the following lemma.

Lemma 2.5. If his an automorphism of P_Fqⁿ⁻¹, thenh^∗ acts as the identity on H_etⁱ(P_Fqⁿ⁻¹,Q_ℓ) for alli.

Proof. The groupP GL_n(Fq) acts on the right onH_etⁱ (P_Fⁿ⁻¹_q ,Qℓ) byu7→u^∗; asH_etⁱ(P_Fqⁿ⁻¹,Qℓ) is of dimension 0 or 1, this action is by homothety, and thus factors by an abelian quotient ofP GL_n(F_q).

Since F_q is algebraically closed, P GL_n(F_q) is equal to its commutator subgroup and thus has no nonzero abelian quotient. Hence, for allu∈P GL_n(F_q),u^∗= Id.

Theorem 2.6. If g∈Gis of order prime top, then tr(g^∗|H_etⁿ⁻²(Xψ,Q_ℓ)^prim) = (−1)ⁿ⁻¹

(n−1)−χ(X^g_ψ)

. (2.1)

Proof. Using the trace formula of Theorem 2.1, we can write

2 dimX

X

i=0

(−1)ⁱtr(g^∗|H_etⁱ(Xψ,Q_ℓ)) =χ(X^g_ψ).

By Lemma 2.4, we have (with, as previously,δi= 0 ifiis odd andδi= 1 ifiis even) tr(g^∗|H_etⁱ(Xψ,Q_ℓ)) =

(δi ifi6=n−2, δi+ tr(g^∗|H_etⁱ(Xψ,Q_ℓ)^prim) ifi=n−2, and thus

χ(X^g_ψ) = (n−1) + (−1)ⁿ⁻²tr(g^∗|H_etⁿ⁻²(Xψ,Q_ℓ)^prim), which is exactly the announced formula.

3 Action of A on H

(X

, Q

)

The irreducible representations overQ_ℓof the finite abelian groupAare its characters (of degree 1).

Finding the structure of theQ_ℓ[A]-moduleH_etⁿ⁻²(Xψ,Q_ℓ) hence amounts to figuring out the mul- tiplicity of each character ofA in the representation g7→g^∗ of A in H_etⁿ⁻²(Xψ,Q_ℓ); it is the aim of this §3. The choice, in §3.1, of an isomorphism between µ_n(F_q) and µ_n(Q_ℓ) allows to identify Aˆ to the group of characters of A taking values in Q_ℓ. After determining the character of the Q_ℓ[A]-module H_etⁿ⁻²(Xψ,Q_ℓ)^prim in §3.2, we will prove in §3.3 that the multiplicity of a ∈ Aˆ is ma= #(Z/nZ\ {a1, . . . , an}).

3.1 Characters of A with values in Q

As we only consider the caseq≡1 mod n, the groupµ_n(Fq) consisting of then^th roots of unity of F_q is isomorphic to the group ofn^th roots of unity ofQ_ℓ. We calltan isomorphism of µ_n(Fq) onto µ_n(Q_ℓ) and use it to identify the group ˆA with the group of characters of A taking values in Q_ℓ thanks to the isomorphism [a1, . . . , an]7→([ζ1, . . . , ζn]7→t(ζ1)^a1· · ·t(ζn)^an).

3.2 Character values of the Q

[A]-module H

(X

, Q

)

As pis prime to n by assumption, the elements ofA have an order prime to p; we may thus use Formula (2.1) to obtain the values taken by the characters of theQ_ℓ[A]-moduleH_etⁿ⁻²(Xψ,Q_ℓ)^prim. Theorem 3.1. Consider(ζ1, . . . , ζn)∈µ_n(F_q)ⁿ such thatζ1. . . ζn= 1and let g be the correspond- ing element[ζ1, . . . , ζn]of A. For allζ∈µ_n(F_q), denote by k(ζ) the number ofi∈[[1;n]]such that ζi=ζ. We have

tr(g^∗|H_etⁿ⁻²(Xψ,Q_ℓ)^prim) = (−1)ⁿ n

X

ζ∈µ_n(Fq)

(1−n)^k(ζ). (3.1)

Proof. A point ofXψ with homogeneous coordinates [x1:. . .:xn] is a fixed point ofgif and only if (x1, . . . , xn) is proportional to (ζ1x1, . . . , ζnxn). The proportionality coefficient is necessarily a root of unityζ∈µ_n(F_q), and we must havexi = 0 ifζi 6=ζ. Hence, the subscheme of fixed points of g ofXψ is the disjoint union overζ∈µ_n(F_q) of the subvarieties

Yζ ={x∈Xψ|xi= 0 ifζi 6=ζ}.

Ifk(ζ) =n, we haveYζ =Xψ. If 2≤k(ζ)≤n−1,Yζ is isomorphic to the hypersurface ofP^k(ζ)−1 defined byy₁ⁿ+yⁿ₂ +· · ·+y_k(ζ)ⁿ = 0. Finally, ifk(ζ) = 0 or 1,Yζ is empty. In each of these cases, we can apply Theorem 2.2 and obtain

χ(Yζ) =k(ζ)−1 +(1−n)^k(ζ)+n−1

n =k(ζ)−1

n+(1−n)^k(ζ)

n .

Consequently, sinceP

ζ∈µ_n(Fq)k(ζ) =nandP

ζ∈µ_n(Fq)1 n = 1, χ(X^g_ψ) = X

ζ∈µ_n(Fq)

χ(Yζ) =n−1 + X

ζ∈µ_n(Fq)

(1−n)^k(ζ)

n .

Using trace formula (2.1) page 4, we deduce the announced result.

Remark 3.2. A recent preprint proves, in a more general setting, formulas of the type given in Theorem 3.1 and Theorem 4.12 page 10; see [Chˆenevert, 2009, Corollary 2.5].

3.3 Decomposition in irreducible representations

The following theorem gives a simple expression for the multiplicityma of a charactera∈Aˆin the Q_ℓ[A]-moduleH_etⁿ⁻²(Xψ,Q_ℓ)^prim.

Theorem 3.3. The multiplicity of the irreducible character a = [a1, . . . , an] of A in the Q_ℓ[A]- moduleH_etⁿ⁻²(Xψ,Q_ℓ)^primis

ma = #(Z/nZ\ {a1, . . . , an}) =n−(number of distinctai).

Proof. Consider (ζ1, . . . , ζn) ∈ µ_n(Fq)ⁿ such that ζ1. . . ζn = 1 and let g be the corresponding element [ζ1, . . . , ζn] ofA. From the definition, we have

tr(g^∗|H_etⁿ⁻²(Xψ,Q_ℓ)^prim) =X

a∈Aˆ

maζ₁^a1. . . ζ_n^an

= 1 n

X

(a₁,...,an)∈(Z/nZ)ⁿ a₁+···+an=0

maζ₁^a1. . . ζ_n^an.

We will show that if we replacema by the number of elements ofZ/nZ\ {a1, . . . , an} in the right hand side, we recover Formula (3.1) above, which will show the announced result. We write

1 n

X

(a₁,...,an)∈(Z/nZ)ⁿ a₁+···+an=0

#(Z/nZ\ {a1, . . . , an})ζ₁^a1. . . ζ_n^an

= 1 n

X

(a₁,...,an)∈(Z/nZ)ⁿ a₁+···+an=0

X

k∈Z/nZ

∀i,a_i6=k

1

ζ₁^a1. . . ζ_n^an

= 1 n

X

k∈Z/nZ

X

(a₁,...,an)∈(Z/nZ)ⁿ a₁+···+an=0

∀i,ai6=k

ζ₁^a1. . . ζ_n^an

= 1 n

X

k∈Z/nZ

X

(a₁,...,an)∈(Z/nZ)ⁿ a₁+···+an=0

∀i,ai6=0

ζ₁^a1. . . ζ_n^an

= X

(a₁,...,an)∈(Z/nZ)ⁿ a₁+···+an=0

∀i,ai6=0

ζ₁^a1. . . ζ_n^an

= X

a₁,...,an∈(Z/nZ)\{0}

a₁+···+an=0

ζ₁^a1. . . ζ_n^an.

We now conclude by using the following lemma.

Lemma 3.4. Let r be an integer≥1 and ζ1, . . . , ζr elements of µ_n(Fq). If k(ζ) =k(ζ₁,...,ζr)(ζ) denotes the number ofi∈[[1;r]]such thatζi=ζ, then

X

a₁,...,ar∈(Z/nZ)\{0}

a₁+···+ar=0

ζ₁^a1. . . ζ_r^ar =(−1)^r n

X

ζ∈µ_n(Fq)

(1−n)^k(ζ).

Proof. We proceed by induction onr. Forr= 1, the equality is the relation 0 =−1

n

(1−n)¹+ (n−1)(1−n)⁰ .

We now assume thatr≥2 and that the result is known forr−1. We write X

a₁,...,ar∈(Z/nZ)\{0}

a₁+···+ar=0

ζ₁^a1. . . ζ_r^ar = X

a₁,...,ar−1∈(Z/nZ)\{0}

a₁+···+ar−16=0

ζ₁^a1. . . ζ_r−1^ar−1ζ_r^{−a1−···−ar−1}

= X

a₁,...,ar−1∈(Z/nZ)\{0}

ζ1

ζr

a₁

. . . ζr−1

ζr

ar−1

− X

a₁,...,ar−1∈(Z/nZ)\{0}

a₁+···+ar−1=0

ζ₁^a1. . . ζ_r−1^ar−1.

Givenζ∈µ_n(F_q), we have

X

a∈(Z/nZ)\{0}

ζ^a =

(−1 ifζ6= 1, n−1 ifζ= 1.

This little remark allows to compute the first sum:

X

a₁,...,ar−1∈(Z/nZ)\{0}

ζ1

ζr

^a1

. . . ζr−1

ζr

^ar−1

= (−1)^r−k(ζr)(n−1)^k(ζr)−1,

wherek(ζ) =k(ζ₁,...,ζr)(ζ). To compute the second sum, we use the induction assumption:

X

a₁,...,ar−1∈(Z/nZ)\{0}

a₁+···+ar−1=0

ζ₁^a1. . . ζ_r−1^ar−1 =(−1)^r−1 n

X

ζ6=ζr

(1−n)^k(ζ)+ (1−n)^k(ζr)−1

.

We conclude by noting that

(−1)^r−k(ζr)(n−1)^k(ζr)−1−(−1)^r−1

n (1−n)^k(ζr)−1

=−(−1)^rn(1−n)^k(ζr)−1

n +(−1)^r

n (1−n)^k(ζr)−1= (−1)^r

n (1−n)^k(ζr).

Remark 3.5. As a consequence of Theorem 3.3, the multiplicity ma of the character a ∈ Aˆ is nonzero unlessabelongs to the orbit of [0,1,2, . . . , n−1] underS_n (which imposesnodd, or else 1 + 2 +· · ·+ (n−1) is not divisible byn).

4 Action of G on H

(X

, Q

)

4.1 A decomposition of the Q

[G]-module H

(X

, Q

)

For everyabelonging to ˆAidentified to the group of characters ofAtaking values inQ_ℓ, we denote byHathe isotypic component relatively toaof theQ_ℓ[A]-moduleH_etⁿ⁻²(Xψ,Q_ℓ)^prim(see [Bourbaki, 1958,§3.4]). It is aQ_ℓ-vector space of dimensionma, wherema is the multiplicity computed in§3.3, and we have

H_etⁿ⁻²(Xψ,Q_ℓ)^prim=M

a∈Aˆ

Ha.

The groupGacts on the left onAby inner automorphisms, and thus acts on the left on ˆA: if g∈Aσ, withσ∈S_n, and ifa= [a1, . . . , an], we have^ga=^σa= [aσ⁻¹(1), . . . , aσ⁻¹(n)].

Considera∈A. Denote byˆ haithe orbit ofaunderS_n. The stabilizerGa ofain Gis equal to A⋊Sa, whereSa ={σ∈S_n | ^σa=a}. We have gHa =H^ga for all g ∈G and the space Ha is stable byGa. The subspaceL

a^′∈haiHa^′ ofH_etⁿ⁻²(Xψ,Q_ℓ)^primis stable byG; it is aQ_ℓ[G]-module canonically isomorphic to Ind^G_GaHa. We thus deduce the following result.

Theorem 4.1. Denote by R⊂Aˆ a set of representatives ofS_n\A. Theˆ Q_ℓ[G]-module H_etⁿ⁻²(Xψ, Q_ℓ)^prim is isomorphic to

M

a∈R

Ind^G_GaHa.

The aim of the rest of this§4 is to determine how the groupSa acts onHa. The strategy is the following: after showing thatSais a semi-direct productS_a^′⋊Σa(§4.2), we compute tr(σ^∗|H_etⁿ⁻²(Xψ, Q_ℓ)^prim) for σa generator ofS_a^′ and compare it to the trace of the identity (§4.4) to deduce that Sa^′ acts asǫ(σ) Id_HaonHa (see§4.5). We then show, using a method similar to§3, that Σa acts as a multiple of the regular representation (§4.6–4.8).

The approach we use to study the action ofS_a^′ is the same that Br¨unjes used in [Br¨unjes, 2004, Proposition 11.5, page 197] for the case ψ = 0, the only difference being that our trace formula allows us to avoid a tedious proof by induction.

4.2 Structure of S

Considera= [a1, . . . , an]∈A, where (aˆ 1, . . . , an) is an element of (Z/nZ)ⁿsuch thata1+· · ·+an= 0.

The set ofj∈Z/nZsuch that (a1+j, . . . , an+j) is a permutation of (a1, . . . , an) is a subgroup of Z/nZ; it can be written asn^′_aZ/nZfor some integern^′_a≥1 dividingn; letda=n/n^′_a be the order of this group. These two integers only depend onaand not on the choice ofa1, . . . ,an.

Remark 4.2. For all b ∈Z/nZ, denote by I(b) the set ofi∈ {1, . . . , n} such thatai =b. The set n^′_aZ/nZis the set of j∈Z/nZsuch that I(b+j) has the same number of elements asI(b) for all b∈Z/nZ.

Lemma 4.3. There is a permutation σ∈S_n such that a) if 1≤i≤n, we have aσ(i)=ai+n^′_a ;

b) σ is the product ofn^′_a disjoint cycles of lengthda.

Proof. Let us note that the condition 4.3.ais equivalent to the fact thatσ(I(b)) =I(b+n^′_a). For all b∈Z/nZsuch that I(b)6=∅, choose a numbering i1(b), . . . ,i#I(b)(b) of the elements ofI(b) and denote byσthe element ofS_n which sendsil(b) toil(b+n^′_a) for allb∈Z/nZand 1≤l≤#I(b).

From the definition, we have aσ(i) =ai+n^′_a and, inspecting the orbits of each of theai under b7→b+n^′_a, we see thatσis a product ofn^′_a disjoint cycles of lengthda.

Denote byS_a^′ the fixator of (a1, . . . , an)∈(Z/nZ)ⁿ in S_n; it is a group which can be identified with Q

b∈Z/nZS_I(b) (it is hence generated by transpositions) and we set γa = [S_n :S_a^′]. Consider σ∈S_n satisfying the conditions of the preceding lemma and let Σa =hσibe the cyclic subgroup of orderda ofS_n generated byσ.

Proposition 4.4. The fixatorSa ofa= [a1, . . . , an]∈Aˆ can be written as the semi-direct product Sa=S_a^′ ⋊Σa.

Proof. If s∈Sa, there exists a uniquej ∈n^′_aZ/nZsuch that ^s(a1, . . . , an) = (a1+j, . . . , an+j).

This element only depends on a, not on the choice ofa1, . . . ,an; we denote it byja(s). The map ja:Sa →n^′_aZ/nZ thus defined is a group homomorphism. This homomorphism is surjective and its kernel is the fixatorS_a^′ of (a1, . . . , an)∈(Z/nZ)ⁿ in S_n.

Moreover, asaσ(i)=ai+n^′_a and thusaσ⁻¹(i)=ai−n^′_a, we haveja(σ) =−n^′_a by construction, henceja induces an isomorphism of Σa=hσionto the imagen^′_aZ/nZofja, which shows that

Sa=S_a^′ ⋊Σa.

Remarks 4.5. a) In particular, the groupS_a^′ is a normal subgroup ofSa and the quotient group Sa/S_a^′ is isomorphic ton^′_aZ/nZand hence of orderda.

b) Let us insist on the fact that n^′_a, da, S_a^′ and ja only depend on a and not on the choice of the representative (a1, . . . , an) ∈ Z/nZ. The group Σa also only depends on a, but its construction is not canonical as it depends on an arbitrary choice of numbering.

c) Let us also note that ifk∈(Z/nZ)^×, thendka=da,n^′_ka=n^′_a,S_ka^′ =S_a^′ andSka=Sa, but jka=kja.

4.3 Character values on a transposition τ

Theorem 4.6. For any transposition τ∈S_n, we have tr(τ^∗|H_etⁿ⁻²(Xψ,Q_ℓ)^prim) = (−1)ⁿ

(1−n)ⁿ⁻¹+ (n−1)

n −δn

, (4.1)

where, as previously,δn= 0if nis odd and δn = 1if nis even.

Proof. We may assume that τ = (1,2). We look for the fixed points of τ, i.e. the set of points [x1:. . .:xn] such that [x1:x2:x3:. . .:xn] = [x2:x1:x3:. . .:xn] andxⁿ₁+· · ·+xⁿ_n−nψx1. . . xn = 0.

For such a point, we havex²₁=x²₂, so that we are in one of the following two cases.

a) We have x1 = x2 and 2xⁿ2 +xⁿ3 +· · ·+xⁿn−nψx²2x3. . . xn = 0. The hypersurface of Pⁿ⁻² defined by this equation is smooth becauseψⁿ 6= 1 and its Euler–Poincar´e characteristic is (n−2) +_n¹[(1−n)ⁿ⁻¹+ (n−1)] (Theorem 2.2).

b) We have x1 =−x2 6= 0, in which casex3 =· · ·=xn = 0 and xⁿ₁ +xⁿ₂ = 0. This can only happen if nis odd and [x1:. . .:xn] = [1 :−1 : 0 :. . .: 0].

The Euler–Poincar´e characteristic of the fixed-point subvariety ofτ ofXψ is thus χ(X_ψ^τ) = (n−2) +(1−n)ⁿ⁻¹+ (n−1)

n + 1−δn

= (n−1) +(1−n)ⁿ⁻¹+ (n−1)

n −δn,

and consequently, asτ is of order 2 andF_q is of characteristic6= 2, Theorem 2.6 applies:

tr(τ^∗|H_etⁿ⁻²(Xψ,Q_ℓ)^prim) = (−1)ⁿ⁻¹

(n−1)−χ(X^τ_ψ)

= (−1)ⁿ

(1−n)ⁿ⁻¹+ (n−1)

n −δn

.

4.4 Sum of the dimensions of the spaces H

for a ∈ A ˆ

Proposition 4.7. Letτ ∈S_n be a transposition. Denote byAˆ^τ the set of elements ofAˆfixed byτ.

We have

X

a∈Aˆ^τ

ma= (−1)ⁿ⁻¹

(1−n)ⁿ⁻¹+ (n−1)

n −δn

,

where, as previously,δn= 0if nis odd and δn = 1if nis even.

Proof. We may assume thatτ= (1,2). Denote byBthe set of elements (b1, . . . , bn)∈(Z/nZ\{0})ⁿ such thatb1=b2andb1+· · ·+bn= 0. The map (b1, . . . , bn)7→[b1, . . . , bn] fromBto ˆA^τ is surjective and each elementa∈Aˆ^τ has exactlyma elements in its preimage. We thus haveP

a∈Aˆ^τma= #B and conclude thanks to the following lemma.

Lemma 4.8. Letrbe an integer≥2. The number ofr-uples(b1, . . . , br)belonging to(Z/nZ\{0})^r such thatb1=b2 andb1+· · ·+br= 0 is

(−1)^r−1

(1−n)^r−1+ (n−1)

n −δn

.

Proof. Denote byurthe number we want to compute. We haveu2=δnandur+ur+1is the number of (r+ 1)-uples (b1, . . . , br, br+1)∈(Z/nZ\ {0})^r×Z/nZsuch thatb1=b2andb1+· · ·+br+1= 0, that is,ur+ur+1= (n−1)^r−1. We deduce the announced result by induction onr.

4.5 Action of S

on H

We start with a general result on automorphisms of finite order with trace equal to the dimension of the space.

Lemma 4.9. Let k be a field of characteristic zero, V a vector space of finite dimension over k anduan automorphism ofV of finite order. If tru= dimV, thenu= IdV.

Proof. LetM be the matrix ofuin a certain basis ofV overk. The subfield k^′ ofk generated by the coefficients ofM embeds itself inC; we can thus restrict ourselves to the casek=C.

Let λ1, . . . ,λm (where m = dimV) be the (complex) eigenvalues of M, each repeated with multiplicity. They are all roots of unity. As we have, according to the assumptions of the lemma,

|λ1+· · ·+λm|=|tru|=m=|λ1|+· · ·+|λm|,

theλi’s are positively proportional, hence equal. As their sum is m, they are all equal to 1. The endomorphismuofV is thus unipotent; as it is of finite order, it is equal to IdV.

Remark 4.10. Let k be a field having characteristic zero, and (Vi)i∈I a finite sequence of vector space of finite dimensions overk. For each i∈I, let ui be an automorphism of Vi of finite order.

IfP

i∈Itrui is equal toP

i∈IdimVi (respectively to −P

i∈IdimVi), then ui = IdVi (respectively ui = −IdVi) for all i ∈ I. This results from Lemma 4.9 applied to the automorphism uof V = L

i∈IVi which is equal toui (respectively to−ui) overVi for alli∈I.

Letτ ∈S_nbe a transposition. AsH_etⁿ⁻²(Xψ,Q_ℓ)^prim=L

a∈AˆHa and asτ^∗sendsHa intoH^τa, we have

tr(τ^∗|H_etⁿ⁻²(Xψ,Q_ℓ)^prim) = X

a∈Aˆ^τ

tr(τ^∗|Ha).

By Theorem 4.6 and Proposition 4.7, we also have

tr(τ^∗|H_etⁿ⁻²(Xψ,Q_ℓ)^prim) =− X

a∈Aˆ^τ

dimHa.

We thus deduce from Remark 4.10 that, for eacha∈Aˆ^τ, τ^∗ acts onHa by−Id_Ha.

Theorem 4.11. Considera∈Aˆandσ∈Sa^′. If we denote byǫ(σ)the signature of σ, we have σ^∗|Ha=ǫ(σ) Id_Ha.

Proof. The subgroupS^′_aofS_n is generated by the transpositionsτ satisfying^τa=a(see§4.2) and we have just seen thatτ^∗|Ha=−Id_Ha =ǫ(τ) Id_Ha.

4.6 Character values on Aσ where σ is a product of n

disjoint cycles of length d

Letn^′ anddbe integers≥1 such thatn^′d=nand letσ∈S_n be a product ofn^′ disjoint cycles of lengthd. Letζ1, . . . ,ζn be elements of µ_n(Fq) such thatζ1. . . ζn= 1 and denote byg the element [ζ1, . . . , ζn]σ of G = A⋊S_n. Let O1, . . . , On^′ be the n^′ orbits of σ in {1, . . . , n} and, for each ζ ∈µ_n(Fq), denote byk(ζ) the number ofj ∈ {1, . . . , n^′} such that Q

i∈Ojζi =ζ. The following theorem generalizes Theorem 3.1 (which is recovered by takingd= 1 and n^′=ni.e.σ= Id).

Theorem 4.12. Under the preceding assumptions, tr(g^∗|H_etⁿ⁻²(Xψ,Q_ℓ)^prim) = (−1)ⁿ

n^′

X

ζ∈µ_n′(Fq)

(1−n)^k(ζ).

Proof. We may assume that σ is the product of ((j−1)d+ 1, . . . , jd) for 1 ≤ j ≤ n^′ and that Oj ={(j−1)d+ 1, . . . , jd}. The fixed points of g inXψ(F_q) are the points [x1:. . .:xn] ofXψ(F_q) such that

[ζσ⁻¹(1)xσ⁻¹(1):. . .:ζσ⁻¹(n)xσ⁻¹(n)] = [x1:. . .:xn] i.e.

[ζ1x1:. . .:ζnxn] = [xσ(1):. . .:xσ(n)].

The subscheme X^g_ψ of these fixed points is thus the disjoint union, over λ ∈ F_q^∗, of the closed subschemesYλ ofXψ defined by

(Yλ)

(xⁿ₁+· · ·+xⁿ_n−nψx1. . . xn= 0, xσ(i)=λζixi for 1≤i≤n.

Let j ∈ {1, . . . , n^′}. If Q

i∈Ojζi 6= λ^−d, the second relation shows that xi = 0 for alli ∈ Oj. If Q

i∈Ojζi =λ^−d, we haveλ∈µ_nd(Fq) and the second relation shows that X

i∈Oj

xⁿ_i =xⁿ_jd ^d

X

i=1

(λⁿ)ⁱ

=

(dxⁿ_jd ifλ∈µ_n(Fq), 0 ifλ /∈µ_n(F_q).

Considerλ∈F_q^∗ and letζ=λ^−d (asn=n^′d, we haveζ^n′ = 1 ⇐⇒ λⁿ = 1). Denote byJ the set ofj ∈ {1, . . . , n^′} such thatQ

i∈Ojζi =ζ and let yj =xjd for each j ∈J. Ifζ /∈µ_n(F_q), J is empty and henceYλ is empty. Assume now that ζ∈µ_n(F_q). The number of elements ofJ is k(ζ).

We consider two cases.

a) First case:ζ∈µ_n^′(F_q). According to what we have just done, the schemeYλis isomorphic to the hypersurface ofP_F^k(ζ)−1

q defined by d

X

j∈J

yⁿ_j

= 0 ifJ 6={1, . . . , n^′},

d(y₁ⁿ+· · ·+y_n^n′)−nψ^′y₁^d. . . y_n^d′ = 0 ifJ ={1, . . . , n^′},

whereψ^′ is the product ofψby an element ofµ_n(F_q). This hypersurface is smooth (because, in the second case, we have (ψ^′)ⁿ =ψⁿ 6= 1 and thus (ψ^′)^n′ 6= 1), hence, by Theorem 2.2 page 3, we have

χ(Yλ) =k(ζ)−1 + (1−n)^k(ζ)+ (n−1)

n =k(ζ) +(1−n)^k(ζ)−1

n .

b) Second case: ζ ∈ µ_n(Fq)\µ_n′(Fq). This time, the schemeYλ is isomorphic toP_F^k(ζ)−1_q if J 6={1, . . . , n^′}and to the hypersurface ofP_Fq^n′−1defined by (y1. . . yn^′)^d= 0 ifJ={1, . . . , n^′}. In the first case, we haveχ(Yλ) =k(ζ). In the second case, we necessarily haven^′ ≥2 and the Euler–Poincar´e characteristic ofYλ is equal to that of Y_λ^red, which is the union inP_Fq^n′−1 of the hyperplanes defined byyj = 0, hence

χ(Yλ) = X

L⊂{1,...,n^′} L6=∅

(−1)^#L−1(n^′−#L) =

n^′

X

l=1

(−1)^l−1 n^′

l

(n^′−l)

=n^′

n^′−1

X

l=1

(−1)^l−1 n^′−1

l

=n^′(1−(1 + (−1))^n′−1) =n^′=k(ζ).

For eachζ∈µ_n(Fq), there exists exactlydvalues ofλsuch thatλ^−d=ζ. Thus χ(X^gψ) = X

λ∈F_q^∗

χ(Yλ) =d X

ζ∈µ_n(Fq)

k(ζ) +d X

ζ∈µ_n′(Fq)

(1−n)^k(ζ)−1 n

=dn^′+ X

ζ∈µ_n′(Fq)

(1−n)^k(ζ)−1

n^′ =n−1 + X

ζ∈µ_n′(Fq)

(1−n)^k(ζ) n^′ .

The order ofg dividesndand hence is prime toq; thus, by Theorem 2.6, tr(g^∗|H_etⁿ⁻²(Xψ,Q_ℓ)^prim) = (−1)ⁿ⁻¹

(n−1)−χ(X^g_ψ)

= (−1)ⁿ n^′

X

ζ∈µ_n′(Fq)

(1−n)^k(ζ).

4.7 Trace of a product σ of n

disjoint cycles of length d acting on H

when a ∈ A ˆ

We keep the notations of§4.6.

Lemma 4.13. If σ∈S_n is a product of n^′ disjoint cycles of length d, X

a∈Aˆsuch thatσ∈S_a¯′

a(ζ1, . . . , ζn)ma= (−1)^n′ n^′

X

ζ∈µ_n′(Fq)

(1−n)^k(ζ).

Proof. Denote by B the set of (b1, . . . , bn) ∈ ((Z/nZ)\ {0})ⁿ such that b1+· · ·+bn = 0 and

σ(b1, . . . , bn) = (b1, . . . , bn). The image of the map B→A, (bˆ 1, . . . , bn)7→[b1, . . . , bn] is the set of a∈Aˆ such thatσ∈S_a^′; such an elementa has exactlyma elements in its preimage. The sum we must compute can hence be rewritten as

X

(b₁,...,bn)∈B

ζ₁^b1. . . ζ_n^bn.

If (b1, . . . , bn) ∈ B, all the bi, for i belonging to an orbit Oj of σ, are equal to a common cj ∈(Z/nZ)\ {0} and we haved(c1+. . . cn^′) = 0 inZ/nZi.e.c1+· · ·+cn^′ ∈n^′Z/nZ. Our sum can thus be rewritten as

X

c₁,...,c_n′∈(Z/nZ)\{0}

c₁+···+c_n′∈n^′Z/nZ

µ^c₁¹. . . µ^c_nⁿ′^′,

where µj = Q

i∈Ojζi. We conclude by using the following generalization of Lemma 3.4 (which is recovered by takingd= 1 andn^′=ni.e.σ= Id).

Lemma 4.14. Let r be an integer≥1 andµ1, . . . ,µr elements of µ_n(Fq). For each ζ∈µ_n(Fq), we denote byk(ζ)the number of j∈ {1, . . . , r} such that µj=ζ. We have

X

c₁,...,cr∈(Z/nZ)\{0}

c₁+···+cr∈n^′Z/nZ

µ^c₁¹. . . µ^c_r^r =(−1)^r n^′

X

ζ∈µ_n′(Fq)

(1−n)^k(ζ).

Proof. We prove the result by induction onr. Forr= 1, we have X

c₁∈n^′Z/nZ\{0}

µ^c₁¹=

(d−1 = ⁻¹_n′((1−n)¹+ (n^′−1)(1−n)⁰) ifµ1∈µ_n^′(F_q),

−1 = ⁻¹_n′(n^′(1−n)⁰) ifµ1∈/ µ_n′(Fq),

hence the result in that case. Assume now thatr≥2 and that the result is proved forr−1. We