(b) Il faut sans doute ajouter 2−2g−r <0 donc, dans le cas etale,g≥2 (cf

A NON TRIVIAL MORPHISM FROM MODULI SPACES OF G-COVERS TO MODULI SPACES OF ABELIAN VARIETIES

ANNA CADORET

[email protected]

Univ. Bordeaux 1, Laboratoire A2X, I.M.B., 351 Cours de la lib´eration, F-33405 Talence cedex, FRANCE.

Abstract. 2000Mathematic Subject Classification. Primary: 14H10, 14K10; Secondary: 14K30, 14H30, 14H37, 14K02.

1. Introduction Notes:

(a) Dans le cor 2.2 attention, la notion de dualite pour les schema abelien n’a de sens que siA→S est projectif donc il faut verifier si (2) un schema abelien pp est projectif et (1) si un schema abelien polarise isogene a un schema abelien projectif est encore projectif. En fait, il semble que (2) est vrai (cf notes manuscrites) et que (1) fasse partie de la def de pp mais a verifier.

(b) Il faut sans doute ajouter 2−2g−r <0 donc, dans le cas etale,g≥2 (cf. [8]).

(c) Comparer les deux constructions et montrer que ce sont les mˆemes en les explicitant (ex, quand on a une section, en termes de faisceaux inversibles).

(d) Suite a (c), dans la premiere construction la restriction sur la caracteristique premiere a n devrait apparaitre qq part. Sans doute dans la dualite de Cartier Z/n^∨=µn.

(e) Completer les references dans la partie 3.2.

2. Notation for stacks

Given a connected scheme S and an integer g ≥ 1, a curve of genus g is a smooth projective S-scheme f : X → S with connected, 1-dimensional geometric fibers of genus g. In particular, this implies that f_?O_X =O_S [4, Exp. X,§1, Prop. 1.2].

LetGbe a finite group andSa connectedQ-scheme; a G-cover overSis a pair (φ:X →X0→S, α), whereX₀ →S andX →S are curves,φ:X →X₀ is a Galois cover, that is a finite flat surjective and separable morphism such that Aut(X/X0) acts transitively on the fibers andα :GS→Aut(X/X˜ 0) is a group scheme isomorphism.

Given a fieldk of characteristic 0, we will write Γ_k for its absolute Galois group and we will always assume that a compatible system (ζ_n)n≥1 of primitive roots of unity is given in the algebraic closure k ofk (that is,ζ_nmⁿ =ζ_m, n, m≥1). With this convention, two classical invariants can be associated to a G-cover (φ: X →X₀→k, α) overk with groupG: the ramification divisort∈M_gY,[r](k), where M_gY,[r] denotes the coarse moduli scheme for genus g_Y curves with r unordered marked points (or, more precisely, relatively ´etale divisors of degree r) and the inertia canonical invariant C= (Ct)t∈t1. For the general theory of G-covers over arbitrary schemes, we refer to [1, §3]. In particular, given a connected Z[_|G|¹ ]-scheme S, the inertia canonical invariant C_s of the geometric fiber of a G-cover (φ:X →X₀→S, α) does not depend (up to Γ_Q-conjugate) on the geometric points∈S. Hence, we

1Recall that C= (Ct)t∈t is defined as follows. For each t∈ t, choose a place Pt in k(X) abovet and let IPt be the corresponding inertia group, which is cyclic of orderet. Any uniformizing parameteru∈Pt induces a well-defined (independent of the uniformizing parameteru) group monomorphism φPt :IPt ,→k^×, ω→ ^ω(u)_u modPt. The element ωPt :=α(φ⁻¹_Pt(ζet))∈G is called the distinguished generator ofIPt. The set of all ωPt for places Pt abovet is a full conjugacy class Ct inG. In [1, §2.2.1] an equivalent definition of the inertia canonical invariant is given in terms of

“holonomy”.

1

denote it by Cand call it the inertia canonical invariant of (φ:X →X₀ →S, α).

Fix a finite group G, an integer g≥0 and a r-tuple C= (C₁, ..., C_r) of possibly trivial conjugacy classes inG. Assume furthermore that 2−2g−r <0. Then, for any connected Q-schemeS, consider the groupoid H_g,G,C(S) with:

- Objects: G-covers (φ : X → X₀ → S, α) over S with group G, inertia canonical invariant C and g_X0 =g.

-Isomorphisms from (φ₁ : X₁ →X_0,1 → S, α₁) to (φ₂ : X₂ → X_0,2 →S, α₂): pairs (u : X₁→X˜ ₂, u₀ : X_0,1→X˜ _0,2) ofS-isomorphisms such that u₀◦φ₁=φ₂◦u andα₁(u◦ · ◦u⁻¹) =α₂.

Fix an integer g ≥ 1 and a sequence d1, ..., dg ≥ 1 of integers with d1|d2| · · · |dg and write D :=

diag(d₁, ..., d_g). Then, for any connected locally noetherianQ-schemeSconsider the following groupoids.

- A^D(S) with:

- - Objects: g-dimensional polarized abelianS-schemes (A, θ) of typeD.

- -Isomorphisms from (A₁, θ₁) to (A₂, θ₂): isomorphisms of abelian S-schemes u : A₁→A˜ ₂ such that u^∗θ2 =θ1.

When D=I_g, we write A_g(S) instead of A^Ig(S).

- A_D(S) with:

- - Objects: pairs (φ: (A, θ)→(A₀, θ₀), α), where (A, θ)∈ A^D(S), (A₀, θ₀)∈ A_g(S),φ:A→A₀ is an isogeny verifyingφ^∗θ₀ =θ andα :⊕_1≤i≤gZ/d_{i S}→˜ ker(φ) is a S-group scheme isomorphism.

- - Isomorphisms from (φ₁ : (A₁, θ₁) → (A_1,0, θ_1,0), α₁) to (φ₂ : (A₂, θ₂) → (A_2,0, θ_2,0), α₂): pairs (u : (A₁, θ₁) ˜→(A₂, θ₂), u₀ : (A_0,1, θ_0,1) ˜→(A_0,2, θ_0,2)) of isomorphisms in A_D(S)× A_g(S) such that u₀◦φ₁=φ₂◦u andα₁ =α₂◦u|^ker(φ_ker(φ²⁾

1). - A⁰_D(S) with:

- - Objects: pairs ((A₀, θ₀), α), where (A₀, θ₀) ∈ A_g(S), α : Q

1≤i≤gµ_diS ,→ A₀ is a S-group scheme monomorphism.

- - Isomorphisms from ((A_1,0, θ_1,0), α₁) to ((A_2,0, θ_2,0), α₂): isomorphisms u₀ : (A_0,1, θ_0,1) ˜→(A_0,2, θ_0,2) inA_g(S) such thatα₂ =u₀◦α₁.

These define algebraic stacks H_g,G,C, A^D, A_D, A⁰_D over Q-schemes; we will denote their coarse moduli spaces by H_g,G,C, A^D, A_D, A⁰_D respectively.

Now, fix two integers n, g≥ 2 and simply write H_g,n forH_g,Z/n,∅, A_g,n forAdiag(1,...,1,n) and A⁰_g,n forA⁰diag(1,...,1,n). We first compare A_g,n and A⁰_g,n; we will use this result in section 4.

Lemma 2.1. LetA₀ →Cbe ag-dimensional complex abelian variety and θ₀ :A₀−→A˜ ^∨₀ be a principal polarization. Then, for any complex isogeny φ : A −→ A₀ with cyclic kernel ker(φ) = Z/n, the polarization φ^∗θ₀ has type diag(1, ...,1, n).

Proof. One can always assume thatA₀ :=C^g/Λ_A0 andA:=C^g/Λ_Ain such a way that Λ_A⊂Λ_A0 and that the analytic representation ofφis just the identity on C^g; in particular Λ_A0/Λ_A= ker(φ) =Z/n.

Recall that there is a canonical commutative diagram, where the horizontal arrows are isomorphisms [7, Chap I,§2].

NS(A₀) ^cA0//

φ^?

Λ²H¹(A₀(C),Z) = Λ²HomZ(Λ_A0,Z)

|_ΛA×ΛA

NS(A₀) ^{cA //}Λ²H¹(A(C),Z) = Λ²HomZ(Λ_A,Z

So, starting from θ₀ : Λ_A0 ×Λ_A0 −→ Z ∈ Λ²Hom_Z(Λ_A0,Z) with type I_g, one has to construct a symplectic basis λ₁, ..., λ_g, u₁, ..., u_g for Λ_A such that (θ(u_i, u_j))1≤i,j≤g = (θ(λ_i, λ_j))1≤i,j≤g = 0, (θ(u_i, λ_j))_1≤i,j≤g =−(θ(λ_i, u_j))_1≤i,j≤g = diag(1, ...,1, n).

Assume first thatg= 2 and observe that the idealθ(ΛA×ΛA) isZ. Indeed, consider the commutative

diagram of epimorphisms

Λ_A0 ×Λ_A0 ^θ ////

Z ^////Z/θ(Λ_A×Λ_A)

Λ_A0/Λ_A×Λ_A0/Λ_A

θ

3333 hh hh hh hh hh hh hh hh hh

.

Now, θ is an alternating form and Λ_A0/Λ_A = Z/n is cyclic, which implies that θ is the zero-form.

Hence there exists u₁, λ₁ ∈Λ_A such that θ(u₁, λ₁) = 1. Let P :={λ∈Λ_A0 | θ(u₁, λ) = θ(λ₁, λ) = 0}

be the orthogonal of Zu₁⊕Zλ₁ with respect to θ in Λ_A. Then ker(θ(u₁,·)) = Zu₁⊕P and Λ_A0 = (Zλ₁⊕Zu₁)⊕P, Λ_A = (Zλ₁⊕Zu₁)⊕P ∩Λ_A. As θ : P ×P → Z is still a principal polarization, the ideal θ(P ×P ∩Λ_A) is Z. Indeed, denoting by P := HomZ(P,Z) the dual Z-module of P, the canonical morphism

P −→ P λ −→ θ(λ,·)

is aZ-module isomorphism. Soθ(P×P∩Λ_A) ={u(λ)|u∈P , λ∈P∩Λ_A}and, asP/P∩Λ_A=Z/n is cyclic whereasP has rank two we get {u(λ)|u∈P , λ∈P ∩Λ_A}=Z. Hence there existsu₂ ∈P, λ2 ∈P∩ΛAsuch thatθ(u2, λ2) = 1 and (λ1, λ2, u1, u2) is a symplectic basis forθ: ΛA0×ΛA0 −→Zsuch that Λ_A= (Zλ₁⊕Zλ₂⊕Zu₁)⊕(Λ_A∩Zu₂). But recalling that Λ_A0/Λ_A=Z/n, we get (Λ_A∩Zu₂) =nZu₂ and (λ₁, λ₂, u₁, nu₂) is a symplectic basis forθ: Λ_A×Λ_A−→Z with the expected property.

The cases g≥2 are obtained by a straightforward induction ong.

Corollary 2.2. There is a canonical stack isomorphism ρ_g,n:A_g,n→A˜ ⁰_g,n.

Proof. This is a consequence of lemma 2.1 and duality theory for abelian schemes (cf. for instance [3, Chap. I,§1] for a quick survey and references); let∨denote duality of abelian schemes and∨cdenote Cartier duality of group schemes. Define ρ_g,n : A_g,n → A⁰_g,n on objects by sending (φ : (A, θ) → (A₀, θ₀), α) to ((A^∨₀, θ₀^∨), α^∨), where α^∨ is defined as the composition of the canonical morphisms below.

α^∨ :µ_{n S}−→(Z/n˜ _S)^∨c

(α^∨c)⁻¹

˜

−→ ker(φ)^∨c−→˜ ker(φ^∨),→A^∨₀.

And defineρg,n:Ag,n→ A⁰_g,non isomorphisms by sending (u: (A1, θ1) ˜→(A2, θ2), u0 : (A0,1, θ0,1) ˜→(A0,2, θ0,2)) to u^∨₀.

Conversely, defineσg,n:A⁰_g,n→ Ag,non objects by sending ((A0, θ0), α) to (φ^∨ : ((A0/im(α))^∨, φ^∨?θ^∨₀)→ (A^∨₀, θ^∨₀), α^∨), where φ : A₀ → A/im(α) is the canonical quotient isogeny and α^∨ is defined as the composition of the canonical morphisms below.

α^∨ :Z/n_S−→(µ˜ _{n S})^∨c(α

∨c)⁻¹

˜

−→ im(α)^∨c= ker(φ)^∨c−→˜ ker(φ^∨).

Note that lemma 2.1 ensures that φ^∨?θ^∨₀ is of type diag(1, ...,1, n). And define σ_g,n: A⁰_g,n→ A_g,n on isomorphisms by sending u₀ : (A_0,1, θ_0,1) ˜→(A_0,2, θ_0,2) to (φ^?u^∨₀, u^∨₀).

Then, by constructionρ_g,n◦σ_g,n is the identity on A⁰_g,n and σ_g,n◦ρ_g,n is the identity onA_g,n. Note that corollary 2.2 fails for other typeD since, then, lemma 2.1 is no longer true.

In the next section, we give two constructions of a canonical stack morphism c_g,n:H_g,n→ A⁰_g,n

3. The stack morphism c_g,n:H_g,n→ A_g,n

Given a proper S-scheme f : X → S of finite presentation, recall that the relative Picard functor Pic_X/S of f :X →S is the sheaf R¹_{f ppf}f_?G_{m X} that is the sheaf associated for the fppf topology with

the presheaf Sch^op_S →Mod(Z),φ:S⁰ →S7→Pic(X⁰), where the notation is X⁰

f⁰

φ⁰

//

X

f

S⁰ _φ ^//S

.

It classically follows from the isomorphism H²(S_et⁰ ,G_{m S}0) ˜→H²(S_{f ppf}⁰ ,G_{m S}0) that H⁰(S⁰,R¹_{f ppf}f_?G_{m X}) = H⁰(S⁰,R¹_etf_?⁰G_{m X}0) but by the proper base change theorem [5, Chap VI, cor. 2.3] one also has R¹_etf_?⁰G_m,X0) =φ^?R¹_etf_?G_m). So one can compute Pic_X/S using R¹_etf_?G_m).

The following statement sums up the classical representability results for the Picard scheme of a curve [2, Chap. VIII, Th. 4.3 & Chap. IX, Th. 3.1], [6, Chap. 6, Cor. 6.8].

Theorem 3.1. Given a S-curve f : X → S, the relative Picard functor Pic_X/S of f : X → S is representable by a separated S-group scheme Pic_X/S → S. Furthermore, the identity component Pic⁰_X/S → S is a projective abelian S-scheme and is endowed with a canonical principal polarization θ_X/S : Pic⁰_X/S→Pic˜ ⁰_X/S^∨ . In particular Pic⁰_X/S = [

n≥1

n⁻¹Pic⁰_X/S. We will write J_X/S instead of Pic⁰_X/S.

3.1. First construction. Our first construction is based on classical spectral sequences for the fppf topology. Recall first the following fundamental result of cohomological algebra.

Theorem 3.2. Let A,B andC be abelian categories such that Aand Bhave enough injective and let F :A → B and G:B → C be left exact functors.

(1) 1f F takes injective objects to G-acyclic objects then there is a spectral sequence (R^pG)◦(R^qF)(A)⇒R^p+q(G◦F)(A)

for any objectA of A.

(2) (2) If F is exact thenRⁿ(G◦F)(A) is canonically isomorphic to (RⁿG)◦F(A) for any object A of A, n≥0.

Let S be a Q-scheme and p_X0 : X₀ →S be a S-curve of genus g ≥1 One can apply the theorem above to the following two situations.

• First situation:

- A is the category of sheaves ofZ-modules onX_{0f ppf}; - B is the category of sheaves of Z-module on S_{f ppf}; - C is the category ofZ-modules;

and

- F =p_X0?;

- G= HomS(µnS,·).

Since p_X0? has a left adjoint, it preserves injectives and, by adjunction G◦F = Hom_S(µ_nS, p_X0?·) = Hom_X0(p^?_X0µ_nS,·) = Hom_X0(µ_nX0,·).

So the low degree exact sequence becomes for A=G_mX0: (1) 0 →Ext¹_{Sf ppf}(µ_nS, p_X0?G_mX0)→Ext¹_{X0f ppf}(µ_nX0,G_mX0)

→δ Hom_S(µ_nS,R¹_{f ppf}p_X0?G_mX0)→Ext²_{Sf ppf}(µ_nS, p_X0?G_mX0)→Ext²(µ_nX0,G_mX0).

• Second situation:

- A=B is the category of sheaves ofZ-module on S_{f ppf}; - C is the category ofZ-modules;

and

- F = Hom_S(µ_nS,·);

- G= H⁰(S,·).

This is the classical local-global spectral sequence for Exts [5, Chap. III, Th. 1.22] and the low degree exact sequence becomes for A=p_X0?G_mX0:

0 →H¹_{f ppf}(S,Hom_S(µ_nS, p_X0?G_mX0)→Ext¹_{Sf ppf}(µ_nS, p_X0?G_mX0)

→H⁰(S,Ext¹_{Sf ppf}(µ_nS, p_X0?G_mX0)→ · · ·

ReplacingS by X₀ and p_X0?G_mX0 by G_mX0 in the above one also gets 0 →H¹_{f ppf}(X₀,Hom_X0(µ_nX0,G_mX0)→Ext¹_{X0f ppf}(µ_nX0,G_mX0)

→H⁰(X₀,Ext¹_{X0f ppf}(µ_nX0,G_mX0)→ · · ·

The fact thatp_X0?O_X0 =O_S impliesp_X0?G_mX0 =G_mSand hence, by duality, Hom(µ_n,G_m) =Z/n.

By definition R¹_{f ppf}p_X0?G_mX0 =Pic_X0/S. Finally, by [5, Chap III, lemma 4.17] both Ext¹_{X0f ppf}(µ_nX0,G_mX0) and Ext¹_{Sf ppf}(µ_nS,G_mS) are trivial. So, putting all this together one obtains the following commuta- tive diagram ofZ-module with exact rows and where vertical arrows are isomorphisms.

0 ^//Ext¹_{Sf ppf}(µ_nS,G_mS) //Ext¹_{X0f ppf}(µ_nX0,G_mX0) ^δ //Hom_S(µ_n,Pic_X0/S)

0 ^//H¹_{f ppf}(S,Z/n_S)

OO //H¹_{f ppf}(X₀,Z/n_X0)

OO

From this we deduce an exact sequence:

(2) 0→H¹_{f ppf}(S,Z/n_S)→H¹_{f ppf}(X₀,Z/n_X0)→Hom_S(µ_nS,Pic_X0/S)

Now, letGH¹_{f ppf}(X₀,Z/n_X0)⊂H¹_{f ppf}(X₀,Z/n_X0) be the subset corresponding to etale G-covers (X → X₀ →S, α) with groupZ/n then (2) defines an embedding

(3) GH¹_{f ppf}(X0,Z/nX0)/H¹_{f ppf}(S,Z/nS),→HomS(µn,Pic_X0/S)\ {˜0}

and, since any quotient of a G-cover is again a G-cover, the image of (3) actually lies in the subset Mono_S(µ_nS,Pic_X0/S) of monomorphismµ_nS ,→ Pic_X0/S.

From theorem 3.1 Mono_S(µ_nS,Pic_X0/S) = Mono_S(µ_nS,Pic_X0/S) and, asµ_nS is torsion, Mono_S(µ_nS,Pic_X0/S) = Mono_S(µ_nS,J_X0/S).

The above construction is compatible with base changes and commutes with isomorphisms so it yields the announced stack morphism

c_g,n:H_g,n−→ A⁰_g,n

ρ⁻¹g,n

˜

−→ A_g,n

Remark 3.3. Assume thatX0→S admits a sectionσ:S→X0, then (2) becomes a short exact sequence. Indeed, this occurs if and only ifδis surjectivei.e. if and only if the morphism Ext²_{Sf ppf}(µnS, pX0?GmX0)→Ext²_{f ppf}(µnX0,GmX0) in (1) is injective. But there is now a canonical isomorphism: px0?GmX0→σ˜ ^?GmX0, which induces an isomorphism

Ext²(µnX0,GmX0) = R²_{Sf ppf}HomS(µnS, pX0?)(GmX0) ˜−→R²_{Sf ppf}HomS(µnS, σ^?)(GmX0) = Ext²_{f ppf}(µnS, σ^?GmX0) (recall thatσ^?is exact on the fppf site). And the composite:

Ext²_{Sf ppf}(µnS, pX0?GmX0)→Ext²_{f ppf}(µnX0,GmX0)→Ext²_{f ppf}(µnS, σ^?GmX0) is the isomorphism induced bypx0?GmX0→σ˜ ^?GmX0.

3.2. Second construction. Our second construction uses the theory of etale fundamental group.

Let S be a connected Q-scheme² and X₀ →S be aS-curve. Fix a geometric point s: Ω→S and writeX0s→Ω for the corresponding geometric fiber ofX0 →S. Then there is a canonical short exact sequence [8, Prop. 2.7]:

(4) 1→π₁(X_0s)→π₁(X₀)→π₁(S)→1

and etale G-covers (φ : X → X₀ → S, α) over S with group Z/n correspond to continuous group epimorphisms Φ :π₁(X₀)Z/n which restrict to continuous π₁(S)-equivariant group epimorphisms Φ :π₁(X_0s)Z/n, where Z/n is regarded with its trivialπ₁(S)-module structure and π₁(X_0s) with theπ₁(S)-module structure induced from (4). Such an epimorphism factors through

π₁(X_0s) ^{Φ ////}

Z/n

π^ab₁ (X_0s)

Φab

::::

uu uu uu uu u

.

On the other hand, the zero section on J_X0/S → S yields the split short exact sequence of funda- mental groups:

(5) 1 ^//π₁(J_X0s) //π₁(J_X0/S) //π₁(S) //

z

{{

1.

and, similarly, etale G-covers (φ : A → J_X0/S → S, α) over S with group Z/n correspond to con- tinuous group epimorphisms Φ : π₁(J_X0/S) Z/n which restrict to continuous π₁(S, s)-equivariant group epimorphisms Φ : π₁(J_X0s) Z/n. But, actually, because of the splitting of (5), any con- tinuous π1(S, s)-equivariant group epimorphisms Φ : π1(JX0s) Z/nextends to a continuous group epimorphisms Φ : π₁(J_X0/S) Z/n such that φ◦σ = 0 hence corresponds to an etale G-covers (φ:A→J_X0/S →S, α) over S with groupZ/n.

Recall the canonical morphism [6, Chap; VI, Cor. 6.8] X₀ ,→ Pic¹_X0/S; it yields a commutative diagram of fundamental groups with exact rows, whereα_s is an isomorphism:

(6) 1 ^//π₁(X_0s) ^//

π₁(X₀) ^//

π₁(S, s) ^//1

1 ^//π₁^ab(X_0s) ^//

αs

π₁^(ab)(X₀) ^//

α

π₁(S, s) ^//1

π1(Pic¹_X0s) ^{β //}π1(Pic¹_X0/S) //π1(S, s) ^//1,

whereπ^(ab)₁ (X0) denotes the quotient ofπ1(X0) modulo the commutator subgroup ofπ1(X0s).

Lemma 3.4. There exists a canonical π₁(S)-equivariant isomorphism i_s : π₁(J_X0s) ˜−→π₁(Pic¹_X0s);

in particular the morphism β : π₁(Pic¹_X0s) → π₁(Pic¹_X

0/S) is a monomorphism and the morphism α:π^(ab)₁ (X0)→π1(Pic¹_X0/S) is an isomorphism.

Proof. Write P := Pic¹_X0/S and J := J_X0/S. As P is a J-torsor, there is a canonical isomorphism i:J×_SP−→P˜ ×_SP, (j, p)7→(jp, p) commuting with the second projectionsp₂ :J×_SP →P hence,

2We assume here thatS is of characteristic 0 to avoid additional notation but the whole proof works for schemesS of prime toncharacteristic up to classical adjustments (cf. for instance [8]).

on the closed fibers we get the commutative diagram

(7) Js×Ps

p2s

is

//Ps×Ps

p2s

P_s P_s

By Kunneth formula [4, Exp. XIII,§4, Prop. 4.6], the first and second projections induce canonical π₁(S, s)-equivariant isomorphisms

π₁(J_s×P_s) ˜−→π₁(J_s)×π₁(P_s) and

π₁(P_s×P_s) ˜−→π₁(P_s)×π₁(P_s).

So, (7) yields a commutative diagram of π₁(S, s)-equivariant morphisms with exact rows:

(8) 1 ^//π₁(P_s) //π₁(P_s×P_s) //π₁(P_s) //1

1 ^//π₁(J_s) //π₁(J_s×P_s) //

is

OO

π₁(P_s) //1

Thusi_s: π₁(J_s×_SsP_s) ˜→π₁(P_s×_SsP_s) restricts in aπ₁(S, s)-equivariant isomorphismi_s:π₁(J_s) ˜−→π₁(P_s).

Now, letKdenote ker(π₁(P_s)→π₁(P) then from the above we obtain a commutative diagram with exact rows:

(9) 1 ^//K×K ^//π₁(P_s×P_s) ^//π₁(P ×_SP) ^//π₁(S) ^//1

1 ^//1×K ^//

OO

π₁(J_s×P_s)

OO //π₁(J×_SP) ^//

OO

π₁(S) ^//1

And, as the three arrows on the right are isomorphisms, so is the one on the left 1×K−→K˜ ×K. But K ⊂π₁(P_s)'Zb^2g so by the fundamental structure theorem for profinite abelian groupsK 'Q

pZ^r_p^p, with 0≤r_p ≤g and K×K 'Q

pZ^2r_p ^p. Hence, by unicity of the r_p, r_p = 0 for all prime p and K is trivial, which proves the injectivity ofβand, sinceα_sis an isomorphism,αis an isomorphism as well.

To perform the construction of our stack morphism, one only needs the first part of lemma 3.4.

Indeed, one has now theπ₁(S, s)-equivariant group isomorphismi⁻¹_s ◦α_s:π^ab₁ (X_0,s) ˜−→π₁(J_X0,s) which induces a set isomorphism Epi_π1(S,s)(π₁(X_0,s),Z/n) ˜−→Epi_π1(S,s)(π₁(J_X0,s),Z/n) and the zero section in (5) defines a set monomorphism Epi_π1(S,s)(π₁(J_X0s),Z/n),→Epi_π1(S,s)(π₁(J_X0/S),Z/n).

The above construction is again compatible with base change onSand commutes with isomorphisms so yields the announced stack morphism

cg,n:Hg,n→ Ag,n

With the above notation,cg,n(S) can be described by the commutativity of the diagram below.

Hg,n(s) ^cg,n(s)//Ag,n(s)

z

H_g,n(S)^cg,n(S)//

OO

A_g,n(S)

OO ,

wherec_g,n(s) :H_g,n(s) ˜−→A_g,n(s) is the isomorphism induced by thei_s andz:A_g,n(s),→ A_g,n(S) the monomorphism induced by the zero sections.

Remark 3.5. (1) One also recovers as in section 3.2 that two G-covers Φ1, Φ2 :π1(X0) Z/n induce the same G-cover Φ :π1(J_X0/S)Z/n if and only if Φ1|_π1(X0s) = Φ2|_π1(X0s) that is Φ1−Φ2 :π1(X0)→Z/nfactors through π1(X0)/π1(X0s) =π1(S, s) or, in other words, Φ1−Φ2∈H¹_{f ppf}(S,Z/nS)

(2) Similarly, assume that X0 → S admits a section σ : S → X0, then (4) splits; this splitting induces a section σ:Hg,n(s)→ Hg,n(S) and the morphismcg,n(S) :Hg,n(S)→ Ag,n(S) yields an isomorphism ”modulo H¹_{f ppf}(S,Z/nS)”.

4. Applications

4.1. A general construction. One can use the stack morphismsc_g,n:H_g,n→ A_g,nand functoriality of the H_g,G,C to construct more general stack morphisms.

4.1.1. Elementary functoriality properties of the H_g,G,C.

Lemma 4.1. Fix two integers g ≥ 1, r ≥ 3 such that 2−2g −r < 0. Let G_i be a finite group and C_i = (C_i,1, ..., C_i,r) be a r-tuple of possibly trivial conjugacy classes of G_i, i = 1, . . . , n. Set G:=G₁× · · · ×G_n,C_j =C_1,j× · · · ×C_n,j,j= 1, ..., r and C= (C₁, ..., C_r). Then the canonical stack morphism

Hg,G,C→ Hg,G1,C₁×M_g,[r]· · · ×M_g,[r]Hg,Gn,C_n

is an isomorphism.

Given a finite groupG, write C(G) for the set of all conjugacy classes of G. The inertia canonical invariant C of a G-cover f with group G can be regarded as an element of the free abelian monoid R₊(G) :=N^(C(G))of all applicationsn:C(G)→Nwith finite support. More explicitly,C= (C_t)t∈t is identified with P

C∈C(G)n(C)C, wheren(C) is the number oft∈twith C_t =C.

Then any group epimorphism p : ˜G G defines a morphism of monoids µ : R₊( ˜G) → R₊(G), sending ˜C ∈ C( ˜G) to p( ˜C) ∈ C(G). Geometrically, if ˜f : Y → X is a G-cover with group ˜G and inertia canonical invariant ˜C then µ( ˜C) is the inertia canonical invariant of the G-cover Y /G → X with group G.

Similarly, any group monomorphism i : G ,→ G˜ defines a morphism of monoids ν : R₊( ˜G) → R₊(G) as follows. Consider the canonical map C_G : G → C(G), sending g ∈ G to its conjugacy class C_G(g) in G and let s : C(G) ,→ G, C → s(C) a section of it. Then ν sends ˜C ∈ C( ˜G)

to X

g∈G\˜ G/<s( ˜˜ C)>

CG(˜gs( ˜C)

o( ˜C)

|G∩˜g<s( ˜C)>˜g−1|˜g⁻¹) ∈ R+(G), whereo( ˜C) denotes the common order of the elements in conjugacy class ˜Cof ˜G. Geometrically, if ˜f :Y →X is a G-cover with group ˜Gand inertia canonical invariant ˜C thenν( ˜C) := ν( ˜C) is the inertia canonical invariant of the G-cover Y → Y /G with group G.

Finally, given a finite group G and a r-tuple C= (C₁, ..., C_r) of possibly trivial conjugacy classes inG, consider the formulas given by Riemann-Hurwitz:

γ_u(g, G,C) := 1 +|G|

2 (−2 + 2g+r− X

1≤i≤r

1 o(C_i) and

γ_d(g, G,C) := 1−r

2 +g−1

|G| +1 2

X

1≤i≤r

1 o(C_i) With these notation, one gets from [1, §6.2.2],

Lemma 4.2. The map µinduces a stack morphism

H_g,G,˜C˜ → H_g,G,µ(C). Similarly, the map ν induces a stack morphism

H_g,G,˜C˜ → H_γ

d(γu(g,G,˜C˜),G,ν( ˜C)),G,ν( ˜C).

4.1.2. Group theoretic abelianization. Fix two integers g ≥ 1, r ≥ 3 such that 2−2g−r < 0. Let G be a finite group and C = (C₁, . . . , C_r) be a r-tuple of possibly trivial conjugacy classes of G.

Given a subgroupH⊂G consider the mapν_H,G : R₊(G)→R₊(H) induced by the natural inclusion i:H ,→Gand the mapµ_H,Hab: R₊(H)→R₊(H^ab) induced by the canonical projectionp: HH^ab.

Assume that µ_H,Hab ◦ν_H,G ∈N{1} or, equivalently, that for any (X → X₀ → S, α) ∈ H_g,G,C(S) the resulting G-cover X/[H, H]→X/H is etale.

X ^//

H

[H,H]

X/H ^//X₀

X/[H, H]

99

ss ss ss ss s

H^ab

.

This occurs in particular whenCis anr-tuple of conjugacy classes the elements of which are of order prime to|H|.

From lemma 4.2, one gets correspondingly, a stack morphism Hg,G,C→ H_γd(γu(g,G,C),H,νH,G(C)),H^ab,∅

Write simply γ(H, g, G,C) := γ_d(γ_u(g, G,C), H, ν_H,G(C)). then, if H^ab = ⊕_p⊕_n≥1 (Z/pⁿ)^sn, one eventually gets from lemma 4.1 a stack morphism

Hg,G,C→

×

×

H_g,G,C→

×

×

By the universal property of coarse moduli schemes this, in turn, yields aQ-morphism H_g,G,C→

×

×

References

[1] J.Bertinand M.Romagny,Hurwitz spaces, preprint, 2006.

[2] S.Bosch, W.Lutkebohmertand M.Raynaud,N´eron models, E.M.G. vol.21, Springer-Verlag, 1990.

[3] C.-L.Chaiand G.Faltings,Degeneration of Abelian varieties, E.M.G. vol.22, Springer-Verlag, 1990.

[4] A.Grothendieck,Revˆetements ´etales et groupe fondamental - S.G.A.I, L.N.M.224, Springer-Verlag, 1971.

[5] J.Milne,Etale cohomology, Princeton University Press, 1980.

[6] D.Mumfordand J.Fogarty,Geometric invariant theory, 2nd enlarged ed, E.M.G.34, Springer-Verlag, 1982.

[7] D.Mumford,Abelian varieties, E.M.G. vol.34, Oxford University Press, 1970.

[8] J.Stix,A monodromy criterion for extending curves, I.M.R.N.29, p. 1787 - 1802, 2005.